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Daily Lessons & Assessments for AP Calculus AB: Complete Course
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Daily Lessons and Assessments for AP Calculus AB A Complete Course Numerical, Graphical, and Algebraic Analysis of Calculus Concepts AP* is a registered trademark of the College Board, which was not involved in the production of, and does not endorse, this product. Daily Lessons and Assessments for AP* Calculus AB, A Complete Course Mark Sparks 2012 Page 1 Advanced Placement Calculus AB Course Syllabus Teacher: Mr. Mark Sparks Email: msparks@batesvilleschools.org ll Phone: (870) 307 – 9556 An Overview The Advanced Placement program offers courses to high school students that are taught on a college level. After successfully completing the curricular requirements of an advanced placement course in calculus, students are given an opportunity to take an exam administered by the College Board to assess their knowledge of, but more importantly their understanding, of calculus. Though the Advanced Placement program is an open enrollment program, students must have taken the correct sequence of courses leading up to enrollment in calculus in order to be successful. A solid, fast-paced, intense and rigorous foundation must be laid in algebra, geometry and trigonometry in order for students to be successful in AP Calculus. Specifically, students must have a firm understanding of functions and their properties, algebraic manipulations, and graphs. Important concepts of functions such as domain and range, even and odd, periodic, symmetry, zeros, and intercepts must be understood. Additionally, students need to know values of the trigonometric functions of 0, π/6, π/4, π/2 and their multiples. Part of your summer work is to write a paper in which you demonstrate your understanding of these basic algebraic and trigonometric concepts of functions. Instructional Methodology The particular concepts covered in the AP Calculus AB course are listed throughout this syllabus. Particular attention is paid to helping students understand concepts from a variety of different avenues. Students discover numerical, graphical, and analytical methods of understanding concepts and are required to express their understanding through both the spoken word through in class presentations and in papers that will be presented that parallel their presentations. During the last two weeks of the course, each student is assigned a free response problem from one of the AP Exams from 2004 – 2007. They will be required to write a detailed paper in which they discuss the calculus involved in the problem, independent from the problem and then explain how they applied the calculus to solve the problem. They will present these papers in class. Much learning and understanding can be gained from having a classroom that is student-centered in structure. For the first three or four units, learning activities are very teacher directed. Calculus skills are taught and practiced to build a foundation for the application of calculus. During the later two-thirds of the course, in class activities become more focused on the student-centered learning. Many times, students are given free response problems from past AP exams that emphasize the different methods of thinking according to the rule of four. Free response problems from exams from 1998 – 2003 are used extensively in class discussions to show students the many different ways that concepts in calculus can be applied according to the rule of four. Students are given a notebook containing every free response problem from 1998 – 2007 that have been released in a notebook at the beginning of the year. As the time arises for reference to these problems, we discuss these in class. Typically, the teacher introduces the problem and then students discuss how to solve the problems in small groups for a few minutes and then share their ideas to the larger group for discussion. Many times, we only do one or two parts of the free response item at a time to show students that just because you do not have the answer to part a of a free response does not mean you cannot solve subsequent parts of the problem. Role of Technology The graphing calculator plays an integral role in both developing and applying concepts of calculus. Thus, it is required that all students enrolled in AP Calculus AB have access to a graphing calculator both in class and at home. There are calculators that you may check out from the instructor of this course for the year, just as you would check out a textbook. There is a contract that must be signed by both the student and the parent that explains the responsibility of use and the consequences of not returning the calculator in satisfactory condition. Although these calculators are available, free of charge from the school, students are encouraged to have their own graphing calculator. Instruction during class activities and lectures will be using the TI-84 Silver Edition. Any calculator from the TI family from TI 83 or TI84 will be the best. Students are also required to write papers which must be typed using an equation editor of some type so that all papers can be submitted by e mail. Daily Lessons and Assessments for AP* Calculus AB, A Complete Course Mark Sparks 2012 Page 2 Assessment 20% Homework and Timed AP Practice Exercises Homework completion is very important for a number of reasons. First, it helps you to understand the concepts from the lesson that you understood or did not understand. It gives you questions to present in class the next day that initiates discussion of homework. Second, it helps the teacher to know what concepts you understand. The attention to detailed, written explanations sheds light on the level of understanding that you have of mathematical concepts. Additionally, your detailed work and written explanations set you apart from other students, and it is only those students who work the very hardest and who master the content with deep understanding who deserve to make a top grade of an A in an AP course. Homework packets will be due at the end of a unit for a 50 point completion grade. Throughout the unit, you will periodically have homework quizzes during which you will provide solutions to various homework problems. These will be graded on accuracy and will be recorded as 15 point grades. Typically, I will choose 5 problems for which you must provide your solution from your homework assignment. Two or three times each nine weeks, you will be required to complete in-class assessments that consist of AP formatted questions. These will consist of one free response item or 9 multiple choice items and you will be given 20 minutes to complete them. These will be recorded as 18 point grades. 30% Quizzes Quizzes are given periodically and are always announced on the syllabus. If the assignment sheet says that there is supposed to be a quiz, then you will have a quiz whether or not I told you in class that there will be a quiz. Quizzes are typically always a combination of 6 multiple choice problems and 1 free response problem. Quizzes will either be totally calculator permitted or totally non-calculator permitted. There will be a total of 18 points available on each quiz and a grade out of 100 will be recorded. Quizzes are timed to 45 minutes. 50% Exams All exams will be announced and will be in true AP format, with both calculator active and non-calculator active sections. Each calculator active section will have 7 multiple choice and one free response item. Each non calculator active section will have 7 multiple choice items and one free response item. Students will be given the calculator active section at the beginning of the testing administration. After 45 minutes, calculators will be put away and the non-calculator section will be distributed. Students may continue to work on either section of the exam until class is over. Once the bell rings to end class, both sections of the exam will be turned in. 18 100% 16 – 17.9 95% 15 – 15.9 90% 13 – 14.9 85% 11 – 12.9 80% 9 – 10.9 75% 7 – 8.9 70% 5 – 6.9 65% 0 – 4.9* 60% (With serious attempt) You will receive 100 point grades for each test. In other words, a 100 point grade will be recorded for the calculator permitted test and a 100 point grade will be recorded for the non-calculator permitted test. On each section, there is a total of 18 points available. You will receive a percentage grade out of 100 based on the scale in the table to the right. Because exams contain cumulative questions that assess not only material learned for a particular unit but also material from past units, an adjusted scale like this must be used. It is important to understand that you will only be graded according to this scale if all of your homework for the unit has been completed and on time. Any lapse in this expectation and your tests will NOT be graded according to this adjusted scale. Materials Needed 1. 2. 3. 4. 5. 3 Ring Binder (I suggest a 2” binder as we will have LOTS of handouts.) 4 Dividers (General Information, Free-Response Examples, Notes, Homework) Tablet of Graphing Paper that stays in the binder Pencils—ALL GRADED WORK MUST BE DONE IN PENCIL—and a highlighter Graphing Calculator Daily Lessons and Assessments for AP* Calculus AB, A Complete Course Mark Sparks 2012 Page 3 Visit www.batesvilleschools.com and follow the “Schools” link to the “High School” site. Then, click on “Classrooms” and scroll down to Mark Sparks to visit the online website to learn more about me, my philosophy of teaching and more information about the courses that I teach. Also, you will find downloadable documents including the daily lesson handouts, free response practice packets, and unit homework packets in case you misplace yours. I will not give extra copies of the documents once I present them in class. Please detach this page, sign and date it. It requires both your signature and the signature of your parent or guardian. This will need to be returned to Mr. Sparks by the THIRD class meeting of AP Calculus class. Please fill out the information below as it will serve as contact information for me if ever I need to contact you. We have read and understood the requirements, philosophy of teaching, grading policy and materials required. Parent Information: Printed Name:__________________________________________ Phone Number:_______________________________ Cell Phone Number is Preferred Email:___________________________________________________ Student Information: Printed Name:__________________________________________ Phone Number:_______________________________ Cell Phone Number is Preferred Email:___________________________________________________ _______________________________________________ Parent Signature _______________________________________________ Student Signature Daily Lessons and Assessments for AP* Calculus AB, A Complete Course Mark Sparks 2012 Page 4 TABLE OF CONTENTS Unit #1—Limits and Continuity…………………………………………………………………..…...6 Unit #2 – Understanding the Derivative…………………………………………………………….118 Unit #3 – Rules of Differentiation……………………………………………………………………192 Unit #4 – Applications of the Derivative – Part I…………………………………………………...288 Unit #5 – Applications of the Derivative – Part II………………………………………………….373 Unit #6 − Basic Integration and Applications……………………………………………………….471 Unit #7 – Advanced Integration and Applications………………………………………………….569 Daily Lessons and Assessments for AP* Calculus AB, A Complete Course Mark Sparks 2012 Page 5 AP Calculus Day# 1 2 Objectives Course Introduction and Expectations Evaluate, if it exists, the value of a limit from a numerical and graphical approach, including one-sided limits. Evaluate, if it exists, the value of a limit from a numerical and graphical approach, including one-sided limits. Unit #1—Limits and Continuity Note Handouts & Assignments Daily Lessons pages 10 – 17 Daily Lessons pages 10 – 17 Day #1 and 2 HW: #1 – 13 3 Evaluate limits analytically, including direct substitution, cancellation and rationalization, applying the properties of limits. Daily Lessons pages 21 – 29 4 Quiz #1 Day #3 HW: #1 – 22 Study for Quiz #1 Daily Lessons pages 45 – 46 5 Evaluate limits of exponential functions analytically Evaluate limits of trigonometric functions analytically. Day #4 HW: #1 – 7 Daily Lessons pages 48 – 52 6 7 Graphically and analytically, apply the three part definition of continuity to determine if a function is continuous at a point. Understand and apply the intermediate value theorem. Day #5 HW: #1 – 14 Daily Lessons pages 56 – 59 Day #6 HW: #1 – 8 Daily Lessons pages 63 – 65 Day #7 HW: #1 – 5 8 Quiz #2 9 Distinguish between infinite limits and limits at infinity, and use them to identify asymptotes. 10 Review for Unit #1 Test Timed AP Multiple Choice Exercise 11 Daily Lessons pages 77 – 83 Day #9 HW: #1 – 8 Extra Practice on Limits and Multiple Choice Practice Study for Exam #1 Test #1: Unit #1 – Limits and Continuity Daily Lessons and Assessments for AP* Calculus AB, A Complete Course Mark Sparks 2012 Page 6 Numerical, Graphical, and Algebraic Analysis of Functions Given below are tables of values for different functions. Classify each function by type. Sketch a graph of the function. Then, state as many specific properties, including the equation if possible, of each function as you possibly can. 1. x F(x) –5 1 3 –1 73 0 –3 3 –5 5 193 9 –9 x G(x) –6 5 –4 1 –2 –3 0 1 2 5 4 9 2. Daily Lessons and Assessments for AP* Calculus AB, A Complete Course Mark Sparks 2012 Page 7 3. x H(x) –2 –5 –1 0 0 3 1 4 2 3 3 0 4 –5 4. x J(x) Undefined –4 –3 –2 –2 –1 x K(x) –6 –3 3.0156 3.125 –1 3.5 1 0 6 1 13 2 5. 0 4 2 7 4 19 6 67 10 1027 Daily Lessons and Assessments for AP* Calculus AB, A Complete Course Mark Sparks 2012 Page 8 6. x M(x) –11 –6 1.996 1.875 –1 –2 x N(x) –1000 –3.001 –3 –2.999 –1.997 –1.250 Undefined –1.249 0 –6 2 –30 4 –126 6 –510 0 1 0.999 2998 10 –8190 7. 1 1.001 1000 Undefined –3002 –2.003 Daily Lessons and Assessments for AP* Calculus AB, A Complete Course Mark Sparks 2012 Page 9 Understanding the Limit A Numerical and Graphical Approach The equation of the function graphed to the right is f ( x) 2 x 2 5x 3 x2 9 . The coordinates of the hole in the graph are 3, 7 . 6 Pre-calculus Statements Calculus Limit Notation As x → −∞, the graph of f(x) → _________. As x → ∞, the graph of f(x) → _________. As x → –3 from the left, the graph of f(x) → _________. As x → –3 from the right, the graph of f(x) → _________. As x → 3 from the left, the graph of f(x) → _________. As x → 3 from the right, the graph of f(x) → _________. Based on what you have just seen, how might you informally define what the value of a limit represents in terms of the graph? Daily Lessons and Assessments for AP* Calculus AB, A Complete Course Mark Sparks 2012 Page 10 How is the numerical analysis above related in the graph of the function pictured below? Daily Lessons and Assessments for AP* Calculus AB, A Complete Course Mark Sparks 2012 Page 11 How is the numerical analysis above related in the graph of the function pictured below? Daily Lessons and Assessments for AP* Calculus AB, A Complete Course Mark Sparks 2012 Page 12 How is the numerical analysis above related in the graph of the function pictured below? Daily Lessons and Assessments for AP* Calculus AB, A Complete Course Mark Sparks 2012 Page 13 Limit Existence Theorem Limits That Do Not Exist Example #1 Find each of the following from the graph. a) lim f ( x) = b. lim f ( x) = x 2 x 2 c) f(2) = d) Does lim f ( x) exist or not? Why or why not? x 2 Example #2 Find each of the following from the graph. a) lim f ( x) = b. lim f ( x) = x 2 x 2 c) f(2) = d) Does lim f ( x) exist or not? Why or why not? x 2 Example #3 Find each of the following from the graph. a) lim f ( x) = b. lim f ( x) = x 2 x 2 c) f(2) = d) Does lim f ( x) exist or not? Why or why not? x 2 Daily Lessons and Assessments for AP* Calculus AB, A Complete Course Mark Sparks 2012 Page 14 Based on what you have seen so far, does f(a) have to be defined in order for the lim f ( x) to exist? Draw x a and explain two different graphs to justify your reasoning. In both graphs, f(a) should be undefined but in one graph, the limit should exist while in the second one, it should not exist. Daily Lessons and Assessments for AP* Calculus AB, A Complete Course Mark Sparks 2012 Page 15 Daily Lessons and Assessments for AP* Calculus AB, A Complete Course Mark Sparks 2012 Page 16 Daily Lessons and Assessments for AP* Calculus AB, A Complete Course Mark Sparks 2012 Page 17 Daily Lessons and Assessments for AP* Calculus AB, A Complete Course Mark Sparks 2012 Page 18 Name_________________________________________Date____________________Class__________ Day #1 and 2 Homework Below are tables of values for given types of functions. For each table, the type of function represented by the table is given. Use your knowledge of the numerical behavior of each type of function to find the indicated limits. For limits that do not exist, write D.N.E. 1. Exponential Function a) lim H ( x) = x b) lim H ( x) = c) lim H ( x) = x 1 x 2. Rational Function a) lim G( x) = b) d) lim G( x) = e) lim G ( x) = g) lim G ( x) = h) lim G ( x) = x x 2 x1 lim G( x) x 2 x 1 = c) lim G( x) = x 2 f) lim G ( x) = x 1 x 3. Rational Function a) lim H ( x) = b) lim H ( x) = d) lim G( x) = e) lim G ( x) = x x 4 x1 c) lim H ( x) = x 4 x 4 Daily Lessons and Assessments for AP* Calculus AB, A Complete Course Mark Sparks 2012 Page 19 If they exist, determine the indicated values below each graph. For limits that do not exist, write D.N.E. 4. The graph of h(x) is given. 5. The graph of g(x) is given. a) lim h( x) = b) lim h( x) = a) lim g ( x) = b) lim g ( x) = c) lim h( x) = d) h(1) = c) lim g ( x) = d) lim g ( x) = e) h(–2) = f) lim h( x) = e) g(4) = f) lim g ( x) = x 1 x1 x 1 x 2 6. The graph of f(x) is given. x 0 x x 1 x 4 x3 7. The graph of q(x) is given. a) lim f ( x) = b) lim f ( x) = a) lim q( x) = b) lim q( x) = c) lim f ( x) = d) lim f ( x) = c) lim q( x) = d) lim q( x) = e) lim f ( x) = f) lim f ( x) e) q(–3) = f) q(4) = x0 x x 1 x x 1 x 1 x0 x 4 Daily Lessons and Assessments for AP* Calculus AB, A Complete Course Mark Sparks 2012 x 3 x 4 Page 20 Given the graph of the function, g(x), below, determine if the statements are true or false. For statements that are false, explain why. 8. lim g ( x) 2 x 1 9. lim g ( x) exists for every value of c on the interval (–1, 1). x c 10. lim g ( x) does not exist. x 2 Sketch a graph of a function that fits the requirements described below. 11. lim f ( x) 3 x 1 lim f ( x) 1 x 1 1 f(1) = 12. lim x 2 f ( x) lim f ( x ) x 2 f(2) is undefined but lim f ( x) exists. x 2 13. In exercise 11, does lim f ( x) exist? Explain why or why not. x1 Daily Lessons and Assessments for AP* Calculus AB, A Complete Course Mark Sparks 2012 Page 21 Understanding the Limit An Algebraic Approach Consider the function, f ( x) 1 2 x 4 4 , for a moment. The graph of f(x) is pictured below. From 2 the graph, determine the following limits. lim f ( x ) x a Find f(a) using the equation. Find lim f ( x ) from x a the graph. lim f ( x) x 0 lim f ( x) x 2 lim f ( x) x10 When a function is defined and continuous at a value, x = a, how can lim f ( x) be found analytically? x a Find each of the following limits analytically. 2 b. lim 52xx 23 x 3 x 2 1 x 1 d. lim sin 2 a) lim 12 x 2 x 3 x3 c. e. lim x2 lim cos 23 2 f. lim log 8 (11 x) x9 Daily Lessons and Assessments for AP* Calculus AB, A Complete Course Mark Sparks 2012 Page 22 Analytically Finding Limits of Functions at Undefined Values What happens if we try to evaluate the limits below by the direct substitution method that was used in the previous six examples? lim x 2 4x 3 lim x3 x 2 2 x 3 x2 4x 3 x 1 x 2 2x 3 lim x2 4x 3 x 1 x 2 2x 3 Just because a function is undefined at a value of x does not mean that a conclusion cannot be reached about the limit. Consider the rational function above. From the graph of the function pictured to the right, what is the value of each limit below? lim x2 4x 3 x 3 x 2 2 x 3 lim x2 4x 3 2 x 1 x lim 2x 3 x2 4x 3 x 1 x 2 2x 3 _______ _______ _______ The task now is to determine how to find these limits analytically. How was it that we found the discontinuities of a rational function in pre-calculus? We will perform the same algebraic analysis to find the limit of the removable, point discontinuities. Let’s do this Cancellation Process below. lim x 2 4x 3 x3 x 2 2 x 3 Daily Lessons and Assessments for AP* Calculus AB, A Complete Course Mark Sparks 2012 Page 23 Based on our knowledge from pre-calculus, we know that if a rational function has a non-removable infinite discontinuity, graphically a ____________________________ exists. Since the y – values do not approach one specific value from both sides at a __________________________, then the limit does not exist. However, we can determine if the one sided limits approach −∞ or ∞. In order to do this analytically, we will marry the numerical, graphical, and algebraic approaches. For each limit below, determine the sign of the simplified function at the value to the right or the left of x = 1. lim x2 4x 3 x 1 x 2 Value of x to the left of x = 1 lim 2x 3 x2 4x 3 x 1 x Simplified function x 1 x 1 2x 3 Value of x to the right of x = 1 Simplified function x 1 x 1 1.1 0.9 The graph of a function g ( x) 2 x 1 2 is pictured to the right. Often, rationalization can be used x3 to evaluate a limit analytically. Find the following limit. lim x3 x 1 2 x 3 Daily Lessons and Assessments for AP* Calculus AB, A Complete Course Mark Sparks 2012 Page 24 Write the equation of the piece-wise defined function pictured to the right. Equation of Each Piece Constraint of Each Piece f (x) Daily Lessons and Assessments for AP* Calculus AB, A Complete Course Mark Sparks 2012 Page 25 Consider the function below to find each limit. If a limit does not exist, state why. 2 2 x 3x, x 2 G ( x) 1 2 x 1, x 2 a) lim G( x) b) x2 lim G( x) x2 c) lim G ( x) x2 Find each of the following limits analytically. Show your algebraic analysis. a. lim ln x x e 2x b. c. d. e. lim x 5 x 2x 2 5 2 lim sin 2 2 cos lim 53 tan 2 x2 x 6 lim x 2 2 x 4 Daily Lessons and Assessments for AP* Calculus AB, A Complete Course Mark Sparks 2012 Page 26 f. lim x2 5 x 3 x 9 g. 8 x 3 27 lim x 32 2 x 3 h. lim x 2 2x 5 1 x2 i. lim 1 2 x 1 2 x 1 x 1 Daily Lessons and Assessments for AP* Calculus AB, A Complete Course Mark Sparks 2012 Page 27 j. lim 1 1x x2 x 0 k. l. lim x2 x 3x 2 7 x 2 x2 4 2x 5 x 3 x 3 m. lim 2x 5 x 3 x 3 lim f ( x h) f ( x ) . h h 0 If f(x) = 2x2 – 3x + 4, find lim Daily Lessons and Assessments for AP* Calculus AB, A Complete Course Mark Sparks 2012 Page 28 Properties of Limits Suppose lim f ( x) L and lim g ( x) M . Find each of the following limits in terms of L x a x a and M. 1. lim f ( x) g ( x) x a 2. lim f ( x) g ( x) x a f ( x) x a g ( x) 3. lim 4. lim f ( x) g ( x) x a 5. lim c f ( x) x a 6. lim f ( x) p x a 7. lim c x a Daily Lessons and Assessments for AP* Calculus AB, A Complete Course Mark Sparks 2012 Page 29 Graph of f(x) Graph of g(x) Find each of the following limits applying the properties of limits. If a limit does not exist, state why. lim f ( x) g ( x) x 2 2 f ( x) x 6 g ( x) lim lim x 2 2 g ( x) lim 2 f ( x) 3g ( x) x 1 lim 2 f ( x) g ( x) lim f ( x) g ( x) x 3 lim f ( x)2 x 2 x 4 f ( x) x 2 g ( x) lim Daily Lessons and Assessments for AP* Calculus AB, A Complete Course Mark Sparks 2012 lim f ( x) g ( x) x 1 Page 30 Name_________________________________________Date____________________Class__________ Day #3 Homework Find the value of each limit. For a limit that does not exist, state why. 1. 3. lim 3x 2 (2 x 1) x 12 2 lim ( x 6) 3 x 2 2. lim x 3 2 x 2 3x 3 x 1 4. lim x 5 x 6 2 x2 x2 ( x 4) 2 16 x x 0 5. lim 2 tan 6. lim 7. lim x21 8. lim x 23 x 2 6 x 1 x 1 2 x2 x 4 Daily Lessons and Assessments for AP* Calculus AB, A Complete Course Mark Sparks 2012 Page 31 9. lim 5 x4 8 x 2 3 2 x 0 3 x 16 x (2 x) 3 8 x x 0 11. lim 13. f ( x h) f ( x ) if f(x) = 3x2 – 2x h h 0 lim 1 10. lim x 2 x 0 12 x ( x h) 2 2( x h) 3 ( x 2 2 x 3) h h 0 12. lim 2 x 2 4 x , x 2 14. lim f ( x) if f ( x) x 2 4 sin x , x 2 4 Daily Lessons and Assessments for AP* Calculus AB, A Complete Course Mark Sparks 2012 Page 32 3 15. lim e x cos x x 3 17. lim x 3 x 3 x 3 1 19. lim x 2 x 0 12 x x 3 2 x 1 x 1 16. lim 18. 2x 2 9x 9 x 2 9 x 3 20. x 2 2 x, lim 2 x 2 x 2 x, x 2 lim Daily Lessons and Assessments for AP* Calculus AB, A Complete Course Mark Sparks 2012 Page 33 21. If lim f ( x) 2 and lim g ( x) 4 , find each of the following limits. Show your analysis x 3 x 3 applying the properties of limits. 5 f ( x) a. lim x 3 g ( x ) b. lim f ( x) 2 g ( x) c. lim x 3 e. lim 3 f ( x) g ( x) g ( x) x 3 8 d. lim 4 f ( x) x 3 f. lim x 3 x 3 f ( x) g ( x) 12 22. If lim f ( x) 0 and lim g ( x) 3 , find each of the following limits. Show your analysis applying x 4 x 4 the properties of limits. g ( x) a. lim x 4 f ( x ) 1 c. lim g ( x) 3 x 4 b. lim xf ( x) x 4 d. lim g 2 ( x) x 4 Daily Lessons and Assessments for AP* Calculus AB, A Complete Course Mark Sparks 2012 Page 34 Calculus Quiz #1 Answer Key & Rubric Multiple Choice ** 1. D A 2. B E 3. A C 4. E A 5. C B 6. E D 7. C B Free Response Part A – 1 point total ____ 1: lim F ( x) (2) 2 2 2 4 4 8 . x 2 Free Response Part B – 2 points total ( 2 x 1)( x 3) 2(3) 1 ____ 1: lim ( x 3)( x 3) lim 2xx31 33 76 x 3 x 3 ____ 1: Since G(3) is undefined and lim G( x) 7 , then G(3) ≠ lim G( x) . x 3 6 x 3 Free Response Part C – 2 points total ____ 1: ____ 1: lim F ( x) must equal x 2 lim F ( x) = x 2 lim F ( x) in order for lim F ( x) to exist. x 2 x 2 lim F ( x) → 8 = 3(−2) + a → a = 14. x 2 Free Response Part C – 4 points total ____ 1: Vertical asymptote at x = –2 ____ 1: Removable discontinuity at the point (2, 3) ____ 1: Point located at (2, –4) ____ 1: Graph exhibits horizontally asymptotic behavior as x . Daily Lessons and Assessments for AP* Calculus AB, A Complete Course Mark Sparks 2012 Page 35 AP CALCULUS QUIZ #1 Name______________________________________________________Date_____________________ Calculator NOT Permitted 1. 2. 3. 4. 5. 6. 7. MULTIPLE CHOICE For questions 1 and 2, use the graph of the function, H(x), pictured below. 1. Which of the following statements is/are true about the graph of H(x)? I. lim H ( x) H (3) x 3 II. lim H ( x) 6 x III. lim H ( x) x 6 A. I and II only B. II only C. I and III only D. II and III only E. I, II, and III Daily Lessons and Assessments for AP* Calculus AB, A Complete Course Mark Sparks 2012 Page 36 2. Which of the following limit(s) do(es) not exist? I. lim H ( x) II. lim H ( x) x 3 x7 III. lim H ( x) x0 A. I only B. I and II only C. II only D. II and III only E. III only e x ( x 1), x 2 3. If g ( x) , which of the following statements is/are true? cos( x ), x 2 I. g(–2) is undefined. II. lim g ( x) 12 x 2 e A. I and II only B. II only D. I and III only E. I, II, and III III. lim g ( x) exists. x 2 C. II and III only sin x . x 6 3x 4. Find lim A. 3 B. 3 2 x 3 3 2 E. 1 D. −∞ 5. Find lim C. x 1 2 . x 3 A. 4 B. 1 D. –4 E. Limit does not exist. 2 C. 1 Daily Lessons and Assessments for AP* Calculus AB, A Complete Course Mark Sparks 2012 4 Page 37 6. Which one of the following graphs shows that F(c) is defined but the lim F ( x) does not exist? x c A. B. D. E. C. 7. Given above is the graph of a function f(x). The value of lim 2 f ( x) is… x 2 A. 3 B. 6 D. 2 E. 2 2 C. 2 Daily Lessons and Assessments for AP* Calculus AB, A Complete Course Mark Sparks 2012 Page 38 FREE RESPONSE Consider the functions below to answer the following questions. x 2 2 x , x 2 F ( x) 3x a, x 2 G ( x) 2 x 2 5x 3 x2 9 a. Find the value of lim F ( x) . Show your work. x 2 b. Is G(3) = lim G( x) ? Show your work and explain your reasoning. x 3 c. In order for lim F ( x) to exist, what two limits must be equal? Find the value(s) of a for which x 2 this limit exists. Show your work. Daily Lessons and Assessments for AP* Calculus AB, A Complete Course Mark Sparks 2012 Page 39 d. Draw a graph of a function, H(x), that meets the following criteria. lim H ( x) lim H ( x) x2 x 2 lim H ( x) 0 x lim H ( x) 3 x 2 H(2) = –4 Daily Lessons and Assessments for AP* Calculus AB, A Complete Course Mark Sparks 2012 Page 40 AP CALCULUS *QUIZ #1* Name______________________________________________________Date_____________________ Calculator NOT Permitted 1. 2. 3. 4. 5. 6. 7. MULTIPLE CHOICE For questions 1 and 2, use the graph of the function, H(x), pictured below. 1. Which of the following statements is/are true about the graph of H(x)? I. lim H ( x) H (3) x 3 II. lim H ( x) 6 x III. lim H ( x) x 6 A. I and II only B. II and III only C. I and III only D. II only E. I, II, and III Daily Lessons and Assessments for AP* Calculus AB, A Complete Course Mark Sparks 2012 Page 41 2. Which of the following limit(s) do(es) not exist? I. lim H ( x) II. lim H ( x) x7 x0 III. lim H ( x) x 3 A. I only B. II and III only C. II only D. III only E. I and III only e x ( x 1), x 2 3. If g ( x) , which of the following statements is/are true? cos( x ), x 2 I. g(–2) is undefined. II. lim g ( x) 12 x 2 e A. I and II only B. II only D. I and III only E. I, II, and III III. lim g ( x) does not exist. x 2 C. II and III only sin 2 x . x 6 3x 4. Find lim A. 3 B. 3 2 x 3 3 2 E. 1 D. −∞ 5. Find lim C. x 1 2 . x 3 A. 4 B. 1 D. –4 E. Limit does not exist. 4 C. 1 Daily Lessons and Assessments for AP* Calculus AB, A Complete Course Mark Sparks 2012 2 Page 42 6. Which one of the following graphs shows that F(c) is defined and lim F ( x) exists but does not equal x c F(c)? A. B. D. E. C. 7. Given above is the graph of a function f(x). The value of lim 3 f ( x) is… x 2 A. D. 3 3 B. 6 C. 2 E. 3 2 Daily Lessons and Assessments for AP* Calculus AB, A Complete Course Mark Sparks 2012 Page 43 FREE RESPONSE Consider the functions below to answer the following questions. x 2 2 x , x 2 F ( x) 3x a, x 2 G ( x) 2 x 2 5x 3 x2 9 a. Find the value of lim F ( x) . Show your work. x 2 b. Is G(3) = lim G( x) ? Show your work and explain your reasoning. x 3 c. In order for lim F ( x) to exist, what two limits must be equal? Find the value(s) of a for which x 2 this limit exists. Show your work. Daily Lessons and Assessments for AP* Calculus AB, A Complete Course Mark Sparks 2012 Page 44 d. Draw a graph of a function, H(x), that meets the following criteria. lim H ( x) lim H ( x) x2 x 2 lim H ( x) 0 x lim H ( x) 3 x 2 H(2) = –4 Daily Lessons and Assessments for AP* Calculus AB, A Complete Course Mark Sparks 2012 Page 45 Limits of Exponential Functions Graphical and Analytical Connections Consider the four exponential functions graphed below. Find the indicated limits for each function based on the graph. 3 x 1 2 2 x 2 2 f ( x) 2 lim f ( x) lim f ( x) x f ( x) lim f ( x) x x 1 x 1 2 f ( x) 3 lim f ( x) 1 lim f ( x) x lim f ( x) x f ( x) lim f ( x) x x 1 x 3 2 2 lim f ( x) x In order to determine a limit as x approaches −∞ or ∞ for an exponential function, you have to determine what the graph will look like. Based on what we have seen above, what are the three possible results of such a limit for an exponential function? Daily Lessons and Assessments for AP* Calculus AB, A Complete Course Mark Sparks 2012 Page 46 By studying the graphs above, remind yourself of the four rules determining if the function will be a growth or decay function. 1.__________________________________________________________________________________ 2.__________________________________________________________________________________ 3.__________________________________________________________________________________ 4.__________________________________________________________________________________ Determine the limits of each of the following exponential functions. x 3 2. x 2 3 x 3 4. lim 2 x 1 2 lim e x 1 2 6. lim 0.4 x 4 x 3 8. x 1 2 1. lim 2 3. lim 2 5. x 7. lim e 2 x 2 lim 0.4x 4 x x x x 3 lim 1 3 2 x 2 Daily Lessons and Assessments for AP* Calculus AB, A Complete Course Mark Sparks 2012 Page 47 Name_________________________________________Date____________________Class__________ Day #4 Homework Find the limit of each of the following exponential functions. Sketch a graph of each function to aid in your determination of the limit, if necessary.. 1. lim (0.5) x 2 3 x 4. lim (3) x 2 3 x 2 2. lim 2 x 2 3 x 5. x 2 2 x2 lim 1 1 3. x 2 3 x 4 6. lim 2 x 2 2 lim 1 x 1 7. Using the graph of g(x) pictured to the right, find each of the following limits. a. lim g ( x) _________ b. lim g ( x) _________ d. x c. x 1 lim g ( x) _________ x lim g ( x) _________ x 3 Daily Lessons and Assessments for AP* Calculus AB, A Complete Course Mark Sparks 2012 Page 48 Limits of Trigonometric Functions An Analytical Approach We have already looked at how to evaluate limits of trigonometric functions by direct substitution, provided that the function is defined and continuous at θ. Find each of the limits below. sin 3 23 3 lim lim 2 cos 2 Each of the functions above was defined at the value that θ was approaching. However, we have seen that even in the algebraic world, not all functions are undefined at a value, but their limits do exist. The same is true in the trigonometric world. Evaluating Trigonometric Limits by Rewriting the Function Using Identities Let’s consider for a moment the limit below. Try to evaluate this limit by direct substitution. lim 1 cos 0 sin 2 Again, this function is undefined at θ = 0. However, that does not mean that the limit does not exist. In this case, we can often rewrite the function in terms of a single trig ratio using identities in hopes that the new form of the function is not undefined for the approached value of θ. Do this in the space below. Daily Lessons and Assessments for AP* Calculus AB, A Complete Course Mark Sparks 2012 Page 49 Find each of the following limits by rewriting the function in a form that is defined for the approached value of θ. 1. cos tan sin sec cos 4 2 lim 3 tan cos 2. lim 32 3. lim It is important to note that as in all cases of evaluating limits, direct substitution should always be tried FIRST. If that does not yield a value, then a simplification of the function can be tried. Occasionally, it is not even possible to rewrite the function so that it is undefined. There are two special trigonometric limits that can often be employed. Use your graphing calculator to complete the table of values below for the function f ( ) θ – 0.01 – 0.001 – 0.0001 0.0001 0.001 sin . 0.01 sin Based on the values in the table above, what do each of the limits below equal? lim 0 sin _____ lim 0 sin _____ lim 0 sin Daily Lessons and Assessments for AP* Calculus AB, A Complete Course Mark Sparks 2012 _____ Page 50 Use your graphing calculator to complete the table of values below for the function f ( ) θ – 0.01 – 0.001 – 0.0001 0.0001 0.001 sin 3 . 3 0.01 sin 3 3 Based on the values in the table above, what do each of the limits below equal? sin 3 _____ 0 3 lim sin 3 _____ 3 0 sin 3 _____ 0 3 lim lim Based on what you have just observed, what inference can you make about the value of the limit sin c , where c is any constant? 0 c lim Now, in a similar fashion, use your graphing calculator to complete the table of values below for the function f ( ) 1 cos θ . – 0.01 – 0.001 – 0.0001 0.0001 0.001 0.01 1 cos Based on the values in the table above, what do each of the limits below equal? lim 0 1 cos _____ lim 0 1 cos _____ lim 0 1 cos _____ These two special trigonometric functions derived above can often be used to find limits of trigonometric functions that cannot be evaluated by direct substitution nor by rewriting the function using identities. Find each of the following limits. Daily Lessons and Assessments for AP* Calculus AB, A Complete Course Mark Sparks 2012 Page 51 e x cos x x 0 4 2. lim sin 2 x x 0 3x 4. lim 1. lim 3. lim 5. lim 0 0 sin 4 2 sin 5 0 3 tan Daily Lessons and Assessments for AP* Calculus AB, A Complete Course Mark Sparks 2012 Page 52 6. lim 0 2 2 cos2 csc 1 0 csc 8. lim 7. lim 0 9. lim x 0 1 cos sin 2 sin x sin x cos x x2 Daily Lessons and Assessments for AP* Calculus AB, A Complete Course Mark Sparks 2012 Page 53 Name_________________________________________Date____________________Class__________ Day #5 Homework Find the value of each limit. For a limit that does not exist, state why. 2. lim x sin x 2 1. lim cos 1 sin x 0 x 2 2 x 2 3x, x3 3. lim x 38 cos x , x 3 3 4. lim 2 sin 3 0 Daily Lessons and Assessments for AP* Calculus AB, A Complete Course Mark Sparks 2012 Page 54 5. lim sin2 x x 0 2 x x 7. lim sin 2 x x 0 6 x 6. lim 5 x sin 3 x x 0 x 8. lim 2 sin 4 x x 0 3x Daily Lessons and Assessments for AP* Calculus AB, A Complete Course Mark Sparks 2012 Page 55 9. lim cos tan 3 0 11. lim cos cot 10. lim 3 3 cos 0 1 tan sin 0 cos 12. lim 2 13. lim c 27 3 c 3 c 3 14. ( x 3) 3 8 x 1 x 1 lim Daily Lessons and Assessments for AP* Calculus AB, A Complete Course Mark Sparks 2012 Page 56 Limit-Based Continuity Graphical and Analytical Approaches For the function graphed below, fill in the table with the given information. After filling in the table, write three pieces of information that must be true in order for a function, G(x), to be continuous at x = a. 1. 2. 3. x=a Is the function defined? If so, what is its value? What is the value of lim G( x) ? xa What is the value of lim G ( x) ? What is lim G( x) ? Is G(x) continuous at x = a? x a x a x = –6 x = –3 x=0 x=2 x=6 x=8 Daily Lessons and Assessments for AP* Calculus AB, A Complete Course Mark Sparks 2012 Page 57 The graph of the function, G(x), pictured to the right has several x – values at which the function is not continuous. For each of the following x – values, use the three part definition of continuity to determine if the function is continuous or not. 1. x = –8 2. x = –6 3. x = –4 Daily Lessons and Assessments for AP* Calculus AB, A Complete Course Mark Sparks 2012 Page 58 Use the three part definition of continuity to determine if the given functions are continuous at the indicated values of x. 2 x 6, x 2 x 2 at x = –2 4. f ( x ) 3x 2, x e cos(x ), x 2 e x cos x, x 5. g ( x ) x e tan 3 x , x 4 6. Consider the function, f(x), to the right to answer the following questions. a. What two limits must equal in order for f(x) to be continuous at x = –1? at x = π x 1 2, f ( x) mx k , 1 x 3 2, x3 b. What two limits must equal in order for f(x) to be continuous at x = 3? c. Determine the values of m and k so that the function is continuous everywhere. Daily Lessons and Assessments for AP* Calculus AB, A Complete Course Mark Sparks 2012 Page 59 7. Consider the function, g(x), to the right to answer the following questions. a. What two limits must equal in order for g(x) to be continuous at x = –2? kx2 m, g ( x) 4 x 1, kx m, x 2 2 x 3 x3 b. What two limits must equal in order for g(x) to be continuous at x = 3? c. Determine the values of m and k so that the function is continuous everywhere. Daily Lessons and Assessments for AP* Calculus AB, A Complete Course Mark Sparks 2012 Page 60 Name_________________________________________Date____________________Class__________ Day #6 Homework For exercises 1 – 3, determine if the function is continuous at each of the indicated values below. Use the three part definition of continuity to perform your analysis. 1. x = –5 2. x = 1 3. x = –2 Daily Lessons and Assessments for AP* Calculus AB, A Complete Course Mark Sparks 2012 Page 61 4. Write a discussion of continuity for each of the function, g(x), below. Be sure to include in your discussion where g(x) is continuous and where g(x) is discontinuous. For values at which the function is discontinuous, explain using the three part definition of continuity. 5. Use the three part definition of continuity to graphically justify why p(x) is discontinuous at x = 0 and x = 2. 6. For what values of k and m is the function g(x) everywhere continuous? Use limits to set up your work. kx2 m, x 2 g ( x ) eln(2 x 3) , 2 x 3 kx m, x3 Daily Lessons and Assessments for AP* Calculus AB, A Complete Course Mark Sparks 2012 Page 62 Find the value of a that makes each of the functions below everywhere continuous. Write the two limits that must be equal in order for the function to be continuous. 4 x 2 , x 1 7. f ( x) ax 2 1, x 1 x 2 x a, x 2 8. f ( x) ax 3 x 2 , x 2 Daily Lessons and Assessments for AP* Calculus AB, A Complete Course Mark Sparks 2012 Page 63 Intermediate Value Theorem As we study calculus, we will study several different theorems. The first theorem of investigation is the Intermediate Value Theorem. Together, let’s write the theorem. Intermediate Value Theorem Now, investigate the graphs below to determine if the theorem is applicable for these functions on the specified intervals for the values given. ( x 3) 2 4, x 2 f ( x) 1 x 1, x 2 2 ( x 3) 2 4, x 2 f ( x) 1 x 1, x 2 2 Is there a value of c on [−5, 2] such that f(c) = 2? Is there a value of c [−1, 5] such that f(c) = 2? Does the I.V.T. guarantee a value of c such that f(c) = 2 on the interval [–5, 2]? Why or why not? Does the I.V.T. guarantee a value of c such that f(c) = 2 on the interval [–1, 5]? Why or why not? Daily Lessons and Assessments for AP* Calculus AB, A Complete Course Mark Sparks 2012 Page 64 What two conditions must be true to verify the applicability of the Intermediate Value Theorem? 1.__________________________________________________________________________________ 2.__________________________________________________________________________________ For each of the following functions, determine if the I.V.T. is applicable or not and state why or why not. Then, if it is applicable, find the value of c guaranteed to exist by the theorem. 1. f ( x) x 3 on the interval [–1, 3] for f(c) = 2 x2 3 2. f ( x) x 3 on the interval [–4, 1] for f(c) = 2 x2 3 3. p( x) e x 2 cos x on the interval [−2, 1] for p(c) = 5 Daily Lessons and Assessments for AP* Calculus AB, A Complete Course Mark Sparks 2012 Page 65 4. f ( x) x on the interval [–1, 1] for f(c) = x2 1 2 2 x 3 2 on the interval [3, 5] for 5. f ( x) 1 f(c) = –4 Daily Lessons and Assessments for AP* Calculus AB, A Complete Course Mark Sparks 2012 Page 66 Name_________________________________________Date____________________Class__________ Day #7 Homework 1. Determine, using the intermediate value theorem, if the function F(x) = x3 + 2x – 1 has a zero on the interval [0, 1]. Justify your answer and find the indicated zero, if it exists. 2. Determine, using the intermediate value theorem, if the function g(θ) = θ2 – 2 – cosθ has a zero on the interval [0, π]. Justify your answer and find the indicated zero, if it exists. For exercises 3 – 5, first, verify that the I.V.T. is applicable for the given function on the given interval. Then, if it is applicable, find the value of the indicated c, guaranteed by the theorem. 3. f(x) = x2 – 6x + 8 Interval: [0, 3] f(c) = 0 Daily Lessons and Assessments for AP* Calculus AB, A Complete Course Mark Sparks 2012 Page 67 4. g(x) = x3 – x2 + x – 2 Interval: [0, 3] g(c) = 4 x2 x x 1 Interval: 52 ,4 h(c) = 6 5. h( x) Daily Lessons and Assessments for AP* Calculus AB, A Complete Course Mark Sparks 2012 Page 68 AP Calculus Quiz #2 Answer Key & Rubric Multiple Choice ** 1. C D 2. B C 3. D C 4. B A 5. A D 6. E B 7. A C Free Response Part A – 2 points total _____ 1 p = 5 _____ 1 Justification to include that lim f ( x) and lim f ( x) with the shown numerical x 5 x 5 analysis Free Response Part B – 3 points total _____ 1 lim g ( x) exists if and only if lim g ( x) lim g ( x) x x x _____ 1 3 cos a 4 so a = 7 _____ 1 g(x) cannot be continuous at x = π because g(x) is not even defined at x = π. Free Response Part C – 4 points total _____ 1 The intermediate value theorem is applicable because… _____ 1 (1) f(x) is only discontinuous at x = 5, which is not on the interval [6, 8] and _____ 1 (2) f(6) = 15 and f(8) = 19 so f(c) = 10 is between f(6) and f(8). 3 2c 3 10 c 5 _____ 1 10c 50 2c 3 8c 53 c 53 6 5 8 8 Daily Lessons and Assessments for AP* Calculus AB, A Complete Course Mark Sparks 2012 Page 69 AP CALCULUS QUIZ #2 Name______________________________________________________Date_____________________ Calculator NOT Permitted 1. 2. 3. 4. 5. 6. 7. MULTIPLE CHOICE 1. Let f(x) be defined by the function to the right. For what value(s) of x is f(x) NOT continuous? A. 0 only B. 1 only C. 2 only D. 0 and 2 only E. 0, 1, and 2 sin x, 2 x , f ( x) 2 x , x 3, x0 0 x 1 1 x 2 x2 2. The graph of the function g(x) is shown below. Which of the following statements about g(x) is true? A. lim g ( x) lim g ( x) x a x b B. lim g ( x) 2 x a C. lim g ( x) 1 x b D. lim g ( x) 2 x b E. lim g ( x) does not exist xa Daily Lessons and Assessments for AP* Calculus AB, A Complete Course Mark Sparks 2012 Page 70 3. For which of the following does lim f ( x) exist? x4 A. I only B. II only D. I and II only E. I and III only C. III only 4 x sin x 2x x 0 4. Find lim A. 1 2 5 2 B. D. 0 C. 2 E. Limit does not exist 2 x 2 1 . 5. Find lim g ( x) if g ( x) 1 x A. –1 B. 2 C. ∞ D. −∞ E. 1 2 6. If lim f ( x) L , where L is a real number, which of the following must be true? x a I. f(a) = L II. lim f ( x) L x a III. lim f ( x) L x a A. I only B. I and II only C. I and III only D. I, II, and III E. II and III only Daily Lessons and Assessments for AP* Calculus AB, A Complete Course Mark Sparks 2012 Page 71 7. The graph of a function, h(x), is pictured to the right. If g ( x) 7 2 x , then what is lim h( x) 2 g ( x) . x 1 A. 8 B. 2 C. 3 D. 5 E. Limit does not exist Graph of h(x) FREE RESPONSE Consider the functions f(x) and g(x) below to answer the questions that follow. f ( x) 2x 3 x5 3 cos x, g ( x) ax 4, x x a. Find a value of p such that lim f ( x) does not exist. Justify your answer using the Limit Existence x p Theorem. Daily Lessons and Assessments for AP* Calculus AB, A Complete Course Mark Sparks 2012 Page 72 Consider the functions f(x) and g(x) below to answer the questions that follow. f ( x) 2x 3 x5 3 cos x, g ( x) ax 4, x x b. For what value of a does lim g ( x) exist? Does the same value of a make g(x) continuous at x = π? x Show your work and justify your reasoning. c. Is the intermediate value theorem applicable for the function f(x) on the interval [6, 8] if f(c) = 10? Why or why not? If so, find the value(s) of c guaranteed by the theorem. Daily Lessons and Assessments for AP* Calculus AB, A Complete Course Mark Sparks 2012 Page 73 AP CALCULUS *QUIZ #2* Name______________________________________________________Date_____________________ Calculator NOT Permitted 1. 2. 3. 4. 5. 6. 7. MULTIPLE CHOICE 1. Let f(x) be defined by the function to the right. For what value(s) of x is f(x) continuous? A. 0 only B. 1 only C. 2 only D. 0 and 1 only E. 0, 1, and 2 sin x, 2 x , f ( x) 2 x , x 3, x0 0 x 1 1 x 2 x2 2. The graph of the function g(x) is shown below. Which of the following statements about g(x) is true? A. lim g ( x) lim g ( x) x a x b B. lim g ( x) 1 x b C. lim g ( x) 2 x a D. lim g ( x) 2 x b E. lim g ( x) does not exist xa Daily Lessons and Assessments for AP* Calculus AB, A Complete Course Mark Sparks 2012 Page 74 3. For which of the following does lim f ( x) NOT exist? x4 A. I only B. II only D. I and II only E. I and III only C. III only 4 x sin x x x 0 4. Find lim A. 5 D. B. 1 2 5 2 C. 4 E. Limit does not exist 2 x 2 1 . 5. Find lim g ( x) if g ( x) 1 x A. –1 B. 2 C. ∞ D. −∞ E. 1 2 6. If lim f ( x) L , where L is a real number, which of the following must be true? x a I. f(a) = L II. lim f ( x) L x a III. lim f ( x) L x a A. I only B. II and III only C. I and III only D. I, II, and III E. I and II only Daily Lessons and Assessments for AP* Calculus AB, A Complete Course Mark Sparks 2012 Page 75 7. The graph of a function, h(x), is pictured to the right. If g ( x) 7 2 x , then what is lim h( x ) 2 g ( x) . x 1 A. 8 B. 2 C. 3 D. 5 E. Limit does not exist Graph of h(x) FREE RESPONSE Consider the functions f(x) and g(x) below to answer the questions that follow. f ( x) 2x 3 x5 3 cos x, g ( x) ax 4, x x a. Find a value of p such that lim f ( x) does not exist. Justify your answer using the Limit Existence x p Theorem. Daily Lessons and Assessments for AP* Calculus AB, A Complete Course Mark Sparks 2012 Page 76 Consider the functions f(x) and g(x) below to answer the questions that follow. f ( x) 2x 3 x5 3 cos x, g ( x) ax 4, x x b. For what value of a does lim g ( x) exist? Does the same value of a make g(x) continuous at x = π? x Show your work and justify your reasoning. c. Is the intermediate value theorem applicable for the function f(x) on the interval [6, 8] if f(c) = 10? Why or why not? If so, find the value(s) of c guaranteed by the theorem. Daily Lessons and Assessments for AP* Calculus AB, A Complete Course Mark Sparks 2012 Page 77 Infinite Limits and Limits at Infinity In our graphical analysis of limits, we have already seen both an infinite limit and a limit at infinity. Let’s consider the equations and the graphs of the two functions below to find the limits that follow. f ( x) 2x 2 7x 6 g ( x) x2 4 Infinite Limits lim f ( x) x 2 lim f ( x) x 2 lim g ( x) x 1 lim g ( x) x 1 x2 x 1 2x 2 Limits at Infinity lim f ( x) x lim f ( x) x lim g ( x) x lim g ( x) x Daily Lessons and Assessments for AP* Calculus AB, A Complete Course Mark Sparks 2012 Page 78 We have already seen how to find infinite limits by marrying a numerical and analytical approach. For the function, f(x) and g(x), whose graphs appear on the previous page, find the infinite limits below. lim x2 2x 2 7x 6 lim x2 x2 4 x2 x 1 x 1 2 x 2 lim 2x 2 7x 6 x2 4 x2 x 1 x 1 2 x 2 lim Graphically, an infinite limit will always yield a _____________________________________________. In pre-calculus, we discovered through observation that such a graphical property existed when a factor in the equation would not _________________________________________________. From this point forward, this is NOT a viable justification for the existence of a _________________________________. Justification of the Existence of a Vertical Asymptote Using Limits For the function below, find any vertical asymptote(s) that exist. Justify your answer(s) using a limit(s). h( x ) 2x 2 7 x 3 x 2 2x 3 Daily Lessons and Assessments for AP* Calculus AB, A Complete Course Mark Sparks 2012 Page 79 Now, we must develop an analytical procedure by which we can find limits at infinity. Basically, a limit at infinity describes the end behavior of a function. We have spent a great amount of time talking about end behavior of functions. Find each of the following limits at infinity. Give an explanation of your reasoning for each. lim 3x 3 x 4 lim (4 x) 2 ( x 3)( x 1) x x 3 2x x x 3 lim 3 2x will provide us a basis for developing our analytical process by which we x x 3 The third example, lim can find limits at infinity for all types of rational functions. Before we do that, investigate the two functions below both graphically and numerically. f ( x) 3 x g ( x) 1 x2 What does each of these functions have in common algebraically and what do they have in common graphically? Daily Lessons and Assessments for AP* Calculus AB, A Complete Course Mark Sparks 2012 Page 80 In pre-calculus, we learned three rules for determining the existence of horizontal asymptotes of rational functions. When a rational function had a horizontal asymptote, the end behavior was always such that as x →−∞ or ∞, then the graph of f(x) → the horizontal asymptote. We learned three rules for determining the horizontal asymptote, if one existed, for rational functions. We are about to use the idea of a limit and calculus to find out why those rules are such as they are. For each function below, divide every term in both the numerator and the denominator by the highest power of x that appears in the denominator. Then, evaluate the indicated limit. Does the result of each limit make sense based on the graph that is pictured? lim x2 x x 2 5x 6 lim 2x 2 7x 6 x x 2 5x 6 x 2 3x 2 x 1 x lim Daily Lessons and Assessments for AP* Calculus AB, A Complete Course Mark Sparks 2012 Page 81 Let’s see what would happen if our limit at infinity approached −∞. lim 2x 2 7x 6 x x 2 5 x 6 x 2 3x 2 x 1 x lim Based on what we have just seen and what we know graphically about the functions above, would does approaching −∞ make a difference in the result of our limits? Daily Lessons and Assessments for AP* Calculus AB, A Complete Course Mark Sparks 2012 Page 82 Graphically, a limit at infinity can yield a _____________________________________________. In pre-calculus, we discovered through observation that such a graphical property existed by comparing the ____________________________________________________________________________________. From this point forward, this is NOT a viable justification for the existence of a _________________________________. Justification of the Existence of a Horizontal Asymptote Using Limits For the function below, find the horizontal asymptote if it exists. Justify your answer(s) using a limit(s). h( x ) 5 2x 2x 2 3x 2 2 x 3 The algebraic analysis described above to evaluate a limit at infinity can be used to find limits at infinity for any type of rational function, even f ( x) 2x 2 x2 1 , whose graph is pictured to the right. What is the one thing that you notice is different about the graph of this rational function versus the others that we have investigated in the past? Use the graph to find each of the following limits. lim x 2x 2 x2 1 ____________ lim x 2x 2 x2 1 ____________ Daily Lessons and Assessments for AP* Calculus AB, A Complete Course Mark Sparks 2012 Page 83 Perform the same algebraic analysis that we did earlier to find the limits at infinity. The only problem that we will encounter is what to do when x → −∞. lim x 2x 2 x2 1 lim x 2x 2 x2 1 Look at the graph on the previous page to confirm these results. Then, find the limits below. lim x 3x 2 2x 2 1 lim x Daily Lessons and Assessments for AP* Calculus AB, A Complete Course Mark Sparks 2012 x 2 2x 3x 2 2 Page 84 Name_________________________________________Date____________________Class__________ Day #9 Homework For exercises 1 – 3, determine the vertical asymptotes of each function. Justify your answers by using limits, showing your numerical analysis. x2 x 6 1. g ( x) x5 2. f ( x) x5 x2 x 2 3. h( x) 2x 2 x 3 x 2 3x 2 Find each of the following limits at infinity. What do the results show about the existence of a horizontal asymptote? Justify your reasoning. 4. lim 3x 2 5 x 2 x 2 x 2 3x 1 5. lim 3x 5 x 2x 2 3x 6. 2x 2 5 x 3 x 2 lim Daily Lessons and Assessments for AP* Calculus AB, A Complete Course Mark Sparks 2012 Page 85 Find each of the following limits at infinity. What do the results show about the existence of a horizontal asymptote? Justify your reasoning. 7. lim x 2x 1 x2 x 8. lim x 2x 2 x 2x 2 3 Daily Lessons and Assessments for AP* Calculus AB, A Complete Course Mark Sparks 2012 Page 86 AP Calculus Multiple Choice Practice Graphing Calculator NOT Permitted – 20 minutes 1. 2. Daily Lessons and Assessments for AP* Calculus AB, A Complete Course Mark Sparks 2012 Page 87 3. 4. f ( x h) f ( x ) is… h h 0 If f(x) = 3x2 + 2x, then lim (A) 6x + 2 (B) 6x (C) 0 (D) nonexistent (E) 2 5. Daily Lessons and Assessments for AP* Calculus AB, A Complete Course Mark Sparks 2012 Page 88 sin x , 4 g ( x) x 2 , x 2 4 x 6 , 6. x3 x3 x3 For the function above, which of the following would be the reason(s) why the function, g(x), is not continuous at x = 3? I. g(3) is undefined. II. lim g ( x) does not exist. III. lim g ( x) g (3). x3 x 3 (A) III only (B) II only (C) I and II only (D) I only (E) II and III only 7. 3 2x is… x 2 x 2 lim (A) 0 (B) ∞ (C) −∞ (D) 2 (E) ½ 8. Daily Lessons and Assessments for AP* Calculus AB, A Complete Course Mark Sparks 2012 Page 89 AP Calculus Free Response Practice #1 Calculator Permitted Consider the function h( x) 2 x sin x to answer the following questions. x 1 a. Find lim h( x) (2 x 2). Show your analysis. x 2 b. Identify the vertical asymptote(s), if any exist, of h(x) and justify the existence by writing a limit. Daily Lessons and Assessments for AP* Calculus AB, A Complete Course Mark Sparks 2012 Page 90 c. Identify the horizontal asymptote(s), if any exist, of h(x) and justify the existence by writing a limit. d. Explain why the Intermediate Value Theorem guarantees a value of c on the interval [1.5, 2.5] such that h(c) = –4. Then, find c. Daily Lessons and Assessments for AP* Calculus AB, A Complete Course Mark Sparks 2012 Page 91 AP Calculus Free Response Practice #2 Calculator NOT Permitted ax 3, x 3 f ( x ) x 2 3 x, 3 x 2 bx 5, x 2 Graph of g(x) Equation of f(x) Pictured above is the graph of a function g(x) and the equation of a piece-wise defined function f(x). Answer the following questions. a. Find lim 2 g ( x) f ( x) cos x. Show your work applying the properties of limits. x 1 Daily Lessons and Assessments for AP* Calculus AB, A Complete Course Mark Sparks 2012 Page 92 b. On its domain, what is one value of x at which g(x) is discontinuous? Use the three part definition of continuity to explain why g(x) is discontinuous at this value. c. For what value(s) of a and b, if they exist, would the function f(x) be continuous everywhere? Justify your answer. Daily Lessons and Assessments for AP* Calculus AB, A Complete Course Mark Sparks 2012 Page 93 AP Calculus Extra Practice on Limits and Multiple Choice Practice For questions 1 – 5, refer to the graph of f(x) to the right. Find the value of each indicated limit. If a limit does not exist, give a reason. lim f ( x) lim 3 f ( x) 1. x 3 x 5 lim 1 f ( x) cosx x 1 2 2. lim f ( x) 3. x 4 lim f ( x) 4. x lim f ( x) 5. x 3 For questions 6 – 11, find the value of each limit analytically. If a limit does not exist, state why. x3 2 x 2 3 x 6. lim x x 0 8. lim x2 4 2 x 3 x x 6 10. lim x x3 2 x 2 3 x x 2 3 x3 3 tan x x 0 x sec x 7. lim 9. lim ln( x 2) x 2 11. lim 5 x 5 x Daily Lessons and Assessments for AP* Calculus AB, A Complete Course Mark Sparks 2012 Page 94 For question 12 – 16, use the equation g(x) below and the graph of the function f(x). Graph of f(x) 3 x 3 , x 2 g ( x) cos x , 2 x 2 2 2 ax 2 x, x 2 12. Is g(x) continuous at x = –2. [Base your response on the three part definition of continuity.] ` 13. For what value(s) of a is g(x) continuous at x = 2? 14. For what value(s) of b is the function f(x) discontinuous? At which of these values does lim f ( x) xb exist? Explain your reasoning. 15. Find lim [ g ( x) 2 f ( x)] . x 2 16. Which of the following limits do(es) not exist? Give a reason for your answers. lim f ( x) lim f ( x) lim f ( x) x1 x4 x 0 Daily Lessons and Assessments for AP* Calculus AB, A Complete Course Mark Sparks 2012 Page 95 17. Find the values of k and m so that the function below is continuous on the interval , . x 2 kx 3, f ( x) 2 x 3, 3 2m, x 2 2 x 3 x3 4x 3 x 0 7 x 1 18. lim A. ∞ B. −∞ C. 0 D. 4 E. –3 B. −∞ C. 0 D. 2 E. 3 B. 0 C. 1 D. 3 E. 2 7 9x 2 1 x 13 3 x 1 19. lim A. ∞ 20. lim x3 8 x2 x 2 4 A. 4 x 3, x 2 21. The function G ( x) 5, x 2 is not continuous at x = 2 because… 3x 7, x 2 A. G(2) is not defined C. lim G( x) G(2) B. lim G ( x) does not exist x 2 x 2 D. Only reasons B and C 22. lim 3x 2 7 x 3 2 x 2 x 3 3x 2 5 A. ∞ E. All of the above reasons. B. −∞ C. 1 D. 7 Daily Lessons and Assessments for AP* Calculus AB, A Complete Course Mark Sparks 2012 2 E. 3 2 Page 96 23. 2x 5 1 x2 x 2 lim A. 1 C. ∞ B. 0 D. −∞ E. Does Not Exist D. −∞ E. ∞ f ( x h) f ( x ) . h h 0 24. If f ( x) 3x 2 5x , then find lim A. 3x – 5 B. 6x – 5 C. 6x D. 0 E. Does not exist 25. 2 5x lim x 2 x 2 B. –5 A. 5 26. The function f ( x) A. C. 2x 2 x 3 x 2 4x 5 C. 0 has a vertical asymptote at x = –5 because… lim f ( x) B. lim f ( x) D. lim f ( x) 5 x 5 x 5 lim x 5 f ( x) x E. f(x) does not have a vertical asymptote at x = –5 3x 5, x 3 27. Consider the function H ( x) 2 . Which of the following statements is/are true? x 2 x, x 3 I. lim H ( x) 4 . x 3 II. lim H ( x) exists. III. H(x) is continuous at x = 3. x 3 A. I only B. II only D. I, II and III E. None of these statements is true Daily Lessons and Assessments for AP* Calculus AB, A Complete Course Mark Sparks 2012 C. I and II only Page 97 AP Calculus Test #1 Answer Key & Rubrics Raw Score to Percentage Conversion Multiple Choice – Calculator ** 1. B D 2. D E 3. A B 4. A C 5. B E 6. C A 7. E A Free Response – Calculator Permitted Part A – 2 points total ____ 1 A sample graph is pictured to the right. The graph must contain a hole at (3, –2) and a point at (3, 4). ____ 1 The function is not continuous at x = 3 because conditions I and II exist, but condition III does not. That is, h(3) is defined. lim h( x) exists x 3 but lim h( x) h(3) . x 3 Part B – 3 points total ____ 1 Condition I: f(7) is defined and its value is f(7) = 3 2 7 =6. ____ 1 Condition II: lim f ( x) 3 2 7 6 and lim f ( x) 8 7 3 6 and since these two x7 x7 one sided limits are equal, then lim f ( x) exists. x7 ____ 1 Condition III: Since lim f ( x) f (7) 6 , then f(x) is continuous at x = 7. x7 Part C – 4 points total ____ 1 If g(x) is continuous at x = , then lim k sin x x 2 lim e ln( x 2) . 4 x 4 x 4 ____ 1 When k sin 4 4 4 2 , then k = 3.067. 2 ____ 1 If g(x) is continuous at x = e, then lim e ln( x2) lim m ln e x . xe ____ 1 When e 2 me , then m = 1.736. xe Daily Lessons and Assessments for AP* Calculus AB, A Complete Course Mark Sparks 2012 Page 98 AP Calculus Test #1 Answer Key & Rubrics Raw Score to Percentage Conversion Multiple Choice – Non Calculator 1. D 2. D 3. A 4. A 5. C 6. B 7. D C E E C B C A Free Response – Calculator NOT Permitted Part A – 2 points total ____ 1 Rewrites as lim 2 x2 7 x 3 2 lim cos x 2 x 1 x x 12 ____ 1 Correctly evaluates the limit as: x 1 2( 1) 2 7( 1) 3 ( 1) ( 1) 12 2 2 cos( ) 2 2 (1) 2 10 5 Part B – 2 points total ____ 1 One of the two numerical analyses to the right is shown ____ 1 f(x) has a vertical asymptote at x = 4 because lim f ( x) or lim f ( x) x 4 x 4 Part C – 2 points total ____ 1 The limit analysis to the right is shown ____ 1 Since lim f(x) = 2, then y = 2 is a horizontal x asymptote. 2 7x 32 x x 1 1 12 x x2 lim 200 2 1 0 0 Part D – 3 points total ____ 1 The limit analysis to the right is shown 3x 2 2 x ____ 1 lim x x 2 2x ____ 1 Since the limit is not approaching a real number value, then there is no horizontal asymptote. In order for a horizontal asymptote to exist as x , then lim f ( x) c , where c is a real number. 3 x2 2xx x lim x x2 22x x2 x lim x 3x 2 1 2x 1 x Daily Lessons and Assessments for AP* Calculus AB, A Complete Course Mark Sparks 2012 Page 99 AP CALCULUS AB TEST #1 Unit #1 – Limits and Continuity Name______________________________________________________Date_____________________ A GRAPHING CALCULATOR IS REQUIRED FOR SOME PROBLEMS OR PARTS OF PROBLEMS IN THIS PART OF THE EXAMINATION. (1) The exact numerical value of the correct answer does not always appear among the choices given. When this happens, select from among the choices the number that best approximates the exact numerical value. (2) Unless otherwise specified, the domain of a function f is assumed to be the set of all real numbers x for which f(x) is a real number. MULTIPLE CHOICE x2 a2 is… x a x 4 a 4 1. If x ≠ a, then lim 1 a2 1 B. 2a 2 1 C. 6a 2 D. 0 E. nonexistent A. Daily Lessons and Assessments for AP* Calculus AB, A Complete Course Mark Sparks 2012 Page 100 2. Consider the function g ( x) I. x cos x . Which of the following statement(s) is/are true? x3 lim g ( x) does not exist because lim g ( x) . x 3 x 3 II. The graph of g(x) has a horizontal asymptote at y = 1 because lim g ( x) 1 x III. The intermediate value theorem guarantees a value of c for g(x) on the interval [–4, 0] if g(c) = –1. A. I and II only B. II and III only C. I only D. II only E. I, II, and III 3. If f(x) = 2x2 + 1, then find lim x 0 f ( x) f (0) . x2 A. 2 B. Does not exist C. 1 D. 0 E. 4. Find lim x 3x 2 2 x 5 x x 2 3x 5 . A. 23 B. C. 0 D. 23 E. 3 Daily Lessons and Assessments for AP* Calculus AB, A Complete Course Mark Sparks 2012 Page 101 5. If lim f ( x) 2 and lim g ( x) 2 , what is the value of lim [3 f ( x) g 2 ( x)] ? xa x a x a A. 0 B. –2 C. –4 D. 2 E. 8 x 2 6. Let f ( x) 4x - 7 x3 be a function. Which of the following statements are true about f ? x3 II. f is continuous at x 3 I. lim f ( x) exists x3 III. lim f ( x) 3 x 1 A. I only B. I and II only C. I and III only D. II only E. III only 5 . x 0 2 g ( x) 7. The graph of a function, g(x), is pictured to the right. Find lim A. 5 2 B. Does not exist C. 1 2 E. 5 2 D. 0 Daily Lessons and Assessments for AP* Calculus AB, A Complete Course Mark Sparks 2012 Page 102 FREE RESPONSE In order for a function, f(x), to be continuous at a value, x = c, three conditions must be true about the function. Keep these conditions in mind as you answer the following questions. a. Draw a graph of a function that satisfies the following conditions. Then, state if the function is continuous at x = 3 or not. If it is not, give the reason(s), based on the three part definition of continuity, why it is not continuous. h(3) = 4 lim h( x) 2 x3 lim h( x) 2 x3 3 2 x , x 7 b. Is the function f ( x) continuous at x = 7? Justify your reasoning based on the 8 x 3, x 7 three part definition of continuity. Daily Lessons and Assessments for AP* Calculus AB, A Complete Course Mark Sparks 2012 Page 103 k sin x x 2 , 0 x 4 x e be continuous on the c. For what values of k and m will the function g ( x) x 2, 4 x m ln e , e x 2 interval (0, 2π)? Use limits to justify your work. Round your answers to the nearest thousandth. Daily Lessons and Assessments for AP* Calculus AB, A Complete Course Mark Sparks 2012 Page 104 AP CALCULUS AB TEST #1 Unit #1 – Limits and Continuity Name______________________________________________________Date_____________________ A GRAPHING CALCULATOR IS NOT ALLOWED FOR THIS SECTION OF THE EXAM. (1) The exact numerical value of the correct answer does not always appear among the choices given. When this happens, select from among the choices the number that best approximates the exact numerical value. (2) Unless otherwise specified, the domain of a function f is assumed to be the set of all real numbers x for which f(x) is a real number. MULTIPLE CHOICE Use the graph of F(x) pictured to the right to answer questions 8 – 9. 8. At which of the following values of c does lim F ( x) NOT exist? x c I. c = 0 II. c = 1 III. c = 4 A. I only B. II only C. III only D. I and II only E. I, II, and III Daily Lessons and Assessments for AP* Calculus AB, A Complete Course Mark Sparks 2012 Page 105 9. F(x) is discontinuous at x = 4 for which of the following reasons? I. F(4) is undefined II. lim F ( x) does not exist. III. lim F ( x) ≠ F(4) x 4 x 4 A. I and III only B. II only C. II and III only D. III only E. I, II, and III x 2 cos(x), 0 x 2 10. If f ( x) , then the value of lim f ( x) is… x2 log 2 ( x 14), 2 x 4 A. 4 B. 2 C. 0 D. –4 E. nonexistent 3 x 5 . x 4 x 2 16 11. Find lim 1 A. 48 B. ∞ C. 0 D. 16 1 E. 72 Daily Lessons and Assessments for AP* Calculus AB, A Complete Course Mark Sparks 2012 Page 106 12. The graph of a function, g(x) is pictured to the right. Find the value of lim [2 2 g ( x)] . x 1 A. –3 B. –1 C. –4 D. 6 E. Does not exist 2x2 7x 3 . x2 9 x3 13. Find lim A. 0 B. C. – D. ½ E. 2 cos 4 x tan 4 x . x0 6x 14. Find lim A. 1 B. 0 C. ∞ D. 2 3 E. Does not exist Daily Lessons and Assessments for AP* Calculus AB, A Complete Course Mark Sparks 2012 Page 107 FREE RESPONSE Consider the rational function f ( x) 2x 2 7 x 3 x 2 x 12 to answer the following questions. a. Find lim f ( x) 2 cos x . Show your analysis applying the properties of limits. x 1 b. Does f(x) have any vertical asymptotes? If so, what are they? Show your analysis, justifying your answer using limits. c. Does f(x) have any horizontal asymptotes? If so, what are they? Show your analysis, justifying your answer using limits. Daily Lessons and Assessments for AP* Calculus AB, A Complete Course Mark Sparks 2012 Page 108 d. Find lim 3x 2 2 x . What does this result say about the existence of horizontally asymptotic behavior x2 2x as x ? Show your work and justify your reasoning. x Daily Lessons and Assessments for AP* Calculus AB, A Complete Course Mark Sparks 2012 Page 109 AP CALCULUS AB *TEST #1* Unit #1 – Limits and Continuity Name______________________________________________________Date_____________________ A GRAPHING CALCULATOR IS REQUIRED FOR SOME PROBLEMS OR PARTS OF PROBLEMS IN THIS PART OF THE EXAMINATION. (1) The exact numerical value of the correct answer does not always appear among the choices given. When this happens, select from among the choices the number that best approximates the exact numerical value. (2) Unless otherwise specified, the domain of a function f is assumed to be the set of all real numbers x for which f(x) is a real number. MULTIPLE CHOICE x2 a2 is… x a x 4 a 4 1. If x ≠ a, then lim 1 a2 B. 0 1 C. 6a 2 1 D. 2a 2 E. nonexistent A. Daily Lessons and Assessments for AP* Calculus AB, A Complete Course Mark Sparks 2012 Page 110 2. Consider the function g ( x) I. x cos x . Which of the following statement(s) is/are true? x3 lim g ( x) does not exist because lim g ( x) . x 3 x 3 II. The graph of g(x) has a horizontal asymptote at y = 1 because lim g ( x) 1 x III. The intermediate value theorem does not guarantees a value of c for g(x) on the interval [–5, 0] if g(c) = –1. A. I only B. II and III only C. I and II only D. II only E. I, II, and III 3. If f(x) = 2x2 + 1, then find lim x 0 f ( x) f (0) . x2 A. 0 B. 2 C. 1 D. E. Does not exist 3x 2 2 x5 x . x x 2 3x5 4. Find lim A. B. 0 C. 23 D. 23 E. 3 Daily Lessons and Assessments for AP* Calculus AB, A Complete Course Mark Sparks 2012 Page 111 5. If lim f ( x) 2 and lim g ( x) 2 , what is the value of lim [ f 2 ( x) 2 g ( x)] ? xa x a x a A. 0 B. –4 C. –2 D. 2 E. 8 x 2 6. Let f ( x) 4x - 7 x3 be a function. Which of the following statements is/are true about f ? x3 II. f is continuous at x 3 I. lim f ( x) exists x3 III. lim f ( x) 3 x 1 A. I, II and III B. I and II only C. I and III only D. II only E. III only 5 . x 3 2 g ( x) 7. The graph of a function, g(x), is pictured to the right. Find lim A. 5 2 B. Does not exist C. 1 2 E. 5 2 D. 0 Daily Lessons and Assessments for AP* Calculus AB, A Complete Course Mark Sparks 2012 Page 112 FREE RESPONSE In order for a function, f(x), to be continuous at a value, x = c, three conditions must be true about the function. Keep these conditions in mind as you answer the following questions. a. Draw a graph of a function that satisfies the following conditions. Then, state if the function is continuous at x = 3 or not. If it is not, give the reason(s), based on the three part definition of continuity, why it is not continuous. h(3) = 4 lim h( x) 2 x3 lim h( x) 2 x3 3 2 x , x 7 b. Is the function f ( x) continuous at x = 7? Justify your reasoning based on the 8 x 3, x 7 three part definition of continuity. Daily Lessons and Assessments for AP* Calculus AB, A Complete Course Mark Sparks 2012 Page 113 k sin x x 2 , 0 x 4 x e be continuous on the c. For what values of k and m will the function g ( x) x 2, 4 x m ln e , e x 2 interval (0, 2π)? Use limits to justify your work. Round your answers to the nearest thousandth. Daily Lessons and Assessments for AP* Calculus AB, A Complete Course Mark Sparks 2012 Page 114 AP CALCULUS AB *TEST #1* Unit #1 – Limits and Continuity Name______________________________________________________Date_____________________ A GRAPHING CALCULATOR IS NOT ALLOWED FOR THIS SECTION OF THE EXAM. (1) The exact numerical value of the correct answer does not always appear among the choices given. When this happens, select from among the choices the number that best approximates the exact numerical value. (2) Unless otherwise specified, the domain of a function f is assumed to be the set of all real numbers x for which f(x) is a real number. MULTIPLE CHOICE Use the graph of F(x) pictured to the right to answer questions 8 – 9. 8. At which of the following values of c does lim F ( x) exist? x c I. c = 0 II. c = 1 III. c = 4 A. I only B. II only C. III only D. I and II only E. The limit does not exist at any of these values of c. Daily Lessons and Assessments for AP* Calculus AB, A Complete Course Mark Sparks 2012 Page 115 9. F(x) is discontinuous at x = 1 for which of the following reasons? I. F(1) is undefined II. lim F ( x) does not exist. III. lim F ( x) ≠ F(1) x1 x1 A. I and III only B. II only C. II and III only D. III only E. I, II, and III 0 x2 ln x, 10. If f ( x) 2 , then the value of lim f ( x) is… x2 x ln 2 , 2 x 4 A. ln 16 B. ln 8 C. ln 2 D. 4 E. nonexistent 11. Find lim x 4 3 x 5 . x 2 16 A. 0 B. ∞ 1 C. 48 D. 16 1 E. 72 Daily Lessons and Assessments for AP* Calculus AB, A Complete Course Mark Sparks 2012 Page 116 12. The graph of a function, g(x) is pictured to the right. Find the value of lim [2 3g ( x)] . x 1 A. –3 B. 8 C. –1 D. 6 E. Does not exist 13. Find lim x 3 2x 2 7x 3 x2 9 . A. 0 B. ½ C. – D. E. 2 cos 4 x tan 4 x . x0 6x 14. Find lim A. 2 3 B. 0 C. ∞ D. 1 E. Does not exist Daily Lessons and Assessments for AP* Calculus AB, A Complete Course Mark Sparks 2012 Page 117 FREE RESPONSE Consider the rational function f ( x) 2x 2 7 x 3 x 2 x 12 to answer the following questions. a. Find lim f ( x) 2 cos x . Show your analysis applying the properties of limits. x 1 b. Does f(x) have any vertical asymptotes? If so, what are they? Show your analysis, justifying your answer using limits. c. Does f(x) have any horizontal asymptotes? If so, what are they? Show your analysis, justifying your answer using limits. Daily Lessons and Assessments for AP* Calculus AB, A Complete Course Mark Sparks 2012 Page 118 d. Find lim 3x 2 2 x . What does this result say about the existence of horizontally asymptotic behavior x2 2x as x ? Show your work and justify your reasoning. x Daily Lessons and Assessments for AP* Calculus AB, A Complete Course Mark Sparks 2012 Page 119 AP Calculus UNIT #2 – Conceptualizing the Derivative Day# 12 Objective Find an equation of the derivative of a function as a limit of the difference quotient. 13 Estimate the value of the derivative of a function at a point graphically and numerically and use the value of the derivative to find an equation of a tangent line drawn to the graph of a function. Analytically find the derivative of a polynomial, sine, or cosine function, and use it to find the equation of a tangent line. Analytically find the first derivative of a polynomial, sine, or cosine function and use it to find intervals of increasing, decreasing, and relative maximums/minimums for the graph of the function. 14 15 17 Solidify the concept of the derivative being the tangent line and learn to approximate the value of a function using the equation of the tangent line. Quiz #3 18 Review for Unit #2 Test Test #2: Unit #2—Conceptualizing the Derivative 16 Note Handouts & Assignments Daily Lesson pages 121 – 126 Day #12 HW: #1 – 9 Daily Lessons pages 129 – 132 Day #13 HW: #1 – 15 Daily Lessons pages 136 – 140 Day #14 HW: #1 – 19 Daily Lessons pages 145 – 150 Day #15 HW: #1 – 5 Daily Lessons pages 153 – 157 Day #16 HW: #1 – 18 Study for Unit #2 Test Daily Lessons and Assessments for AP* Calculus AB, A Complete Course Mark Sparks 2012 Page 120 The Difference Quotient A First Look at the Derivative Today we are introduced to the concept with which we will spend our greatest amount of time investigating in Calculus AB—the derivative. Let’s draw a picture together. What does the expression f ( x h) f ( x ) represent? What does this expression simplify to? ( x h) x As h, the distance between the x – values, x and (x + h), approaches zero, what happens to the secant line? f ( x h) f ( x ) represent? h h 0 What does the limit lim f ( x h) f ( x ) . h h 0 Suppose f ( x) x 2 4 x 1 . Find lim Your result to the previous limit is defined to be the derivative, f ' ( x) , of the function f(x). Now, let’s see what this derivative represents in terms of the graph of f(x). Daily Lessons and Assessments for AP* Calculus AB, A Complete Course Mark Sparks 2012 Page 121 f ( x h) f ( x ) for f ( x) x 2 4 x 1 is a function in terms of x. The graph of f(x) h h 0 is pictured below. Complete the chart for the indicated x – values and f ' ( x) . Your result of lim x – value Value of f ' ( x) 2 x 4 –4 –2 –1 Now, use a ruler and draw a tangent line to the graph of f(x) on the grid above at x = –4, x = –2, and x = –1. By investigating the graph, what does it appear that the derivative function f ' ( x) 2 x 4 represents in terms of the graph at given values of x? Definition of the Derivative and What It Represents Graphically Find the equation of the tangent line to f(x) at each of the points below. Then, draw the graphs of the tangent lines on the grid above where f(x) is graphed. Equation of the tangent line at x = –4 Equation of the tangent line at x = –2 Equation of the tangent line at x = –1 Daily Lessons and Assessments for AP* Calculus AB, A Complete Course Mark Sparks 2012 Page 122 When you hear “DERIVATIVE,” you think “SLOPE OF THE TANGENT LINE.” When you hear “SLOPE OF THE TANGENT LINE,” you think “DERIVATIVE.” Now that we understand what the derivative of a function represents graphically, let’s practice using the f ( x h) f ( x ) limit of the difference quotient, lim , to find f ' ( x) for each of the functions below. h h 0 f ( x) 2 x 3 f ( x) 1 x 2 2 x 3 3 2 Notice that f ' ( x) for f ( x) 2 x 3 was different than f ' ( x) for f ( x) 1 x 2 2 x 3 . How are they 3 2 different and why do you suppose this is so? Daily Lessons and Assessments for AP* Calculus AB, A Complete Course Mark Sparks 2012 Page 123 f ( x h) f ( x ) for the functions given below to find and use f ' ( x) . h h 0 Find lim f ( x) x 2 f ( x) Find the equation of the line tangent to the graph of f ( x) x 2 at x = 7. Find the equation of the line tangent to the graph of 3 f ( x) at x = 1. x2 3 x2 Using a graphing calculator, graph each of the functions above and the equation of the tangent line that you found to verify your work. Daily Lessons and Assessments for AP* Calculus AB, A Complete Course Mark Sparks 2012 Page 124 Over the course of this lesson so far, you have found derivatives of several functions and evaluated that derivative at certain x – values. Look back at your work and complete the table below. Equation of Function, f(x) Equation of Derivative, f ' ( x) Value of f ' ( x) at the Indicated value of x f ( x) x 2 4 x 1 x = –1 f ( x) x 2 x=7 Find the Value of the Limit f ( x) f (a) , where a is the value of x. lim xa x a f ( x) f (a) represents? xa x a What inference can you make that explains what the limit lim Daily Lessons and Assessments for AP* Calculus AB, A Complete Course Mark Sparks 2012 Page 125 Complete the table below, stating what each of the indicated limits finds in terms of the derivative of a function, f(x). Definition of the Derivative f ( x h) f ( x ) h h 0 Alternate Form of the Definition of the Derivative f ( x) f (a) xa x a lim lim Daily Lessons and Assessments for AP* Calculus AB, A Complete Course Mark Sparks 2012 Page 126 Name_________________________________________Date____________________Class__________ Day #12 Homework f ( x h) f ( x ) for each of the functions below. Then, find the equation of the tangent line to h h 0 the graph of f(x) at the given value of x. 3. f ( x) 3 x 1. f ( x) x 3 2 x Find lim 2. Find the equation of the line tangent to the graph 3 of f ( x) x 2 x at x = –1. 4. Find the equation of the line tangent to the graph of f ( x) 3 x at x = –6. Daily Lessons and Assessments for AP* Calculus AB, A Complete Course Mark Sparks 2012 Page 127 For problems 5 – 9, use the function f ( x) x . x2 f ( x h) f ( x ) . h h 0 5. Find f ' ( x) by finding lim 6. Find the slope of the tangent line drawn to the graph of f(x) at x = –2. 7. Find the slope of the tangent line drawn to the graph of f(x) at x = –1. 8. Find the equation of the tangent line drawn to the graph of f(x) at x = –1. f ( x) f (a) , where a = –1. xa x a 9. Find lim Daily Lessons and Assessments for AP* Calculus AB, A Complete Course Mark Sparks 2012 Page 128 Understanding the Derivative from a Graphical and Numerical Approach So far, our understanding of the derivative is that it represents the slope of the tangent line drawn to a curve at a point. Complete the table below, estimating the value of f ' ( x) at the indicated x – values by drawing a tangent line and estimating its slope. x– Value Estimation of Derivative Is the function Increasing, Decreasing or at a Relative Maximum or Relative Minimum Equation of the tangent line at this value of x. –7 –6 –4 –2 –1 1 3 5 7 Daily Lessons and Assessments for AP* Calculus AB, A Complete Course Mark Sparks 2012 Page 129 Based on what you observed in the table on the previous page, what inferences can you make about the value of the derivative, f ' ( x) , and the behavior of the graph of the function, f(x)? Numerically, the value of the derivative at a point can be estimated by finding the slope of the secant line passing through two points on the graph on either side of the point for which the derivative is being estimated. x– Value x –3 0 1 4 6 10 f(x) 2 1 –3 0 –7 2 Estimation of Derivative Is the function Increasing, Decreasing or at a Relative Maximum or Relative Minimum Equation of the tangent line at this value of x. 0 1 4 6 Daily Lessons and Assessments for AP* Calculus AB, A Complete Course Mark Sparks 2012 Page 130 The graph of a function, g(x), is pictured to the right. Identify the following characteristics about the graph of the derivative, g ' ( x) . Give a reason for your answers. The interval(s) where g ' ( x) < 0 The interval(s) where g ' ( x) > 0 The value(s) of x where g ' ( x) = 0 Definition of the Normal Line Pictured to the right is the graph of f ( x) 1 ( x 1) 2 4 . 2 Draw the tangent line to the graph of f(x) at x = 1. Then, estimate the value of f ' (1) . Find the equation of the tangent line to the graph of f(x) at x = 1. The normal line is the line that is perpendicular to the tangent line at the point of tangency. Draw this line and find the equation of the normal line. Daily Lessons and Assessments for AP* Calculus AB, A Complete Course Mark Sparks 2012 Page 131 The graph of the derivative, h' ( x) , of a function h(x) is pictured below. Identify the following characteristics about the graph of h(x) and give a reason for your responses. The interval(s) where h(x) is increasing The interval(s) where h(x) is decreasing The value(s) of x where h(x) has a relative maximum. The value(s) of x where h(x) has a relative minimum. If h(–1) = ½, what is the equation of the tangent line drawn to the graph of h(x) at x = –1? If h(2) = –3, what is the equation of the normal line drawn to the graph of h(x) at x = 2? Daily Lessons and Assessments for AP* Calculus AB, A Complete Course Mark Sparks 2012 Page 132 Name_________________________________________Date____________________Class__________ Day #13 Homework 1. The line defined by the equation 2 y 3 2 ( x 3) is tangent to the graph of g(x) at x = –3. What is 3 g ( x) g (3) the value of lim ? Show your work and explain your reasoning. x3 x 3 Use the graph of f(x) pictured to the right to perform the actions in exercises 2 – 6. Give written explanations for your choices. 2. Label a point, A, on the graph of y = f(x) where the derivative is negative. 3. Label a point, B, on the graph of y = f(x) where the value of the function is negative. 4. Label a point, C, on the graph of y = f(x) where the derivative is greatest in value. 5. Label a point, D, on the graph of y = f(x) where the derivative is zero. 6. Label two different points, E and F, on the graph of y = f(x) where the values of the derivative are opposites. Daily Lessons and Assessments for AP* Calculus AB, A Complete Course Mark Sparks 2012 Page 133 7. Match the points on the graph of g(x) with the value of g ' ( x) in the table. Value of g ' ( x) –3 Point on g(x) –1 0 ½ 1 2 8. The function to the right is such that h(4) = 25 and h ' (4) = 1.5. Find the coordinates of A, B, and C. For exercises 9 – 11, use the function f ( x) 9. Find f ' ( x) . 1 . x 1 10. Find the equation of the tangent line drawn to the graph of f(x) at x = 0. 11. Find the equation of the normal line drawn to the graph of f(x) at x = 0. Daily Lessons and Assessments for AP* Calculus AB, A Complete Course Mark Sparks 2012 Page 134 12. Given below are graphs of four functions—f(x), g(x), h(x), and p(x). Below those graphs are graphs of their derivatives. Label the graphs below as f ' ( x) , g ' ( x) , h ' ( x) , and p' ( x) . The table below represents values on the graph of a cubic polynomial function, h(x). Use the table to complete exercises 22 – 24. x h(x) –3 –24 –2 0 –1 8 0 6 1 0 2 –4 4 18 13. Two of the zeros of h(x) are listed in the table. Between which two values of x does the Intermediate Value Theorem guarantee that a third value of x exists such that h(x) = 0? Explain your reasoning. 14. Estimate the value of h ' (1.5) . Based on this value, describe the behavior of h(x) at x = 1.5. Justify your reasoning. 15. Estimate the value of h ' (1.75) . Based on this value, describe the behavior of h(x) at x = –1.75. Justify your reasoning. Daily Lessons and Assessments for AP* Calculus AB, A Complete Course Mark Sparks 2012 Page 135 Analytically Finding the Derivative of Polynomial, Polynomial Type, Sine, and Cosine Functions Consider the function f(x) = 3. What does the graph of this function look like? If a tangent line were drawn to f(x) at any value of x, what would the slope of that tangent line be? Based on this though process, if f(x) = c, where c is any constant, then f ' ( x) ___________ . Shown below are 6 different polynomial, or polynomial–type, functions. Watch as I find the derivative of each function. See if you can figure out the algorithm that I am using for each function. Function, f(x) Derivative, f ' ( x) f ( x) 3 x 2 2 x 3 f ( x) 5x 3 2 x 2 3x 1 f ( x) 6 3 x 3 6 x 4 f ( x) 2 x 1 3x 2 2 f ( x) 6 x 3 4 x 2 1 1 f ( x) 6 x 2 3x 2 Based on what you have seen in the table above, you should now be able to infer how to complete the following Power Rule for Differentiation. d n x __________________ dx Daily Lessons and Assessments for AP* Calculus AB, A Complete Course Mark Sparks 2012 Page 136 In order to apply the Power Rule for Differentiation, the equation must be written in “polynomial form.” To what do you suppose “polynomial form” refers? Find f ' ( x) for each of the following functions. Leave your answers with no negative or rational exponents and as single rational functions, when applicable. f ( x) 2 x2 4x3 f ( x) ( x 3)( x 2)(2 x 1) f ( x) 3x 3 x2 f ( x) 3x 4 3x 2 2 x x f ( x) x 3 5x 2 x5 3 1 f ( x) 4 x 4 2 x 4 Daily Lessons and Assessments for AP* Calculus AB, A Complete Course Mark Sparks 2012 Page 137 Remember two trigonometric identities that we will use to find the derivatives of the sine and cosine functions. cos(a + b) = _________________________________________ sin(a + b) = _________________________________________ f ( x h) f ( x ) to find f ' ( x) for each of the following functions. Your results will show the h h 0 derivative of the sine and cosine functions. Use lim f(x) = sin x d sin x __________________ dx f(x) = cos x d cos x __________________ dx Daily Lessons and Assessments for AP* Calculus AB, A Complete Course Mark Sparks 2012 Page 138 For each of the following functions, find the equation of the tangent line to the graph of the function at the given point. f ( x) (2 x 1)( x 1) 2 when x = –1 g ( ) 2 3 cos when θ = π f ( ) 4 sin when θ = 2 h( x ) 2x x Daily Lessons and Assessments for AP* Calculus AB, A Complete Course Mark Sparks 2012 3 when x = 2 Page 139 Given the equation of a function, how might you determine the value(s) at which the function has a horizontal tangent? Explain your reasoning. At what value(s) of x will the function f ( x) x 3 x have a horizontal tangent? At what value(s) of θ at which the function f ( ) sin has a horizontal tangent on the interval [0, 2π)? Daily Lessons and Assessments for AP* Calculus AB, A Complete Course Mark Sparks 2012 Page 140 Name_________________________________________Date____________________Class__________ Day #14 Homework For exercises 1 – 12, find the derivative of each function. Leave your answers with no negative or rational exponents and as single rational functions, when applicable. 1. f ( x) 5 2 x 2 3x 3 2 x 3 3x 2 2 x 2. h( x) x 3 2x5 3. h( x) x7 5. f ( ) 3 2 cos 7. g ( ) 2 sin 4. g ( x) x8 3 6. h( x) x 2 3 8. p( x) 2 x 2 x Daily Lessons and Assessments for AP* Calculus AB, A Complete Course Mark Sparks 2012 Page 141 9. g ( x) ( x 3)(2 x 1) 2 11. f ( x) 3x 10. h( x) x 2 2x 2 x3 12. h( x) 6 x 3 cos x 3x 13. For what value(s) of x will the slope of the tangent line to the graph of h( x) 4 x be –2? Find the equation of the line tangent to h(x) at this/these x – values. Show your work. Daily Lessons and Assessments for AP* Calculus AB, A Complete Course Mark Sparks 2012 Page 142 14. Find the equation of the line tangent to the graph of g ( x) 2 4 3 when x = 1. x 15. The line defined by the equation 1 x 3 2( y 3) is the line tangent to the graph of a function 2 f(x) when x = a. What is the value of f ' (a) ? Show your work and explain your reasoning. 16. The line defined by the equation y 3 2 ( x 3) is the line tangent to the graph of a function 3 f(x) at the point (–3, 3). What is the equation of the normal line when x = –3. Explain your reasoning. 17. Determine the value(s) of x at which the function f ( x) x 4 8x 2 2 has a horizontal tangent. 18. Determine the value(s) of θ at which the function f ( ) 3 2 cos has a horizontal tangent on Daily Lessons and Assessments for AP* Calculus AB, A Complete Course Page 143 Mark Sparks 2012 the interval [0, 2π). 19. For what value(s) of k is the line y = 4x – 9 tangent to the graph of f(x) = x2 – kx? Connections between F(x) and F’(x) for Polynomial and Trigonometric Functions Daily Lessons and Assessments for AP* Calculus AB, A Complete Course Mark Sparks 2012 Page 144 F’(x) F(x) Is = 0 Is > 0 Is < 0 Changes from positive to negative Changes from negative to positive Possible Graph of f ' ( x) Pictured below is the graph of a function f(x). Answer the following questions about the derivative. Daily Lessons and Assessments for AP* Calculus AB, A Complete Course Mark Sparks 2012 Page 145 1. Approximate the value of f ' (4) . 2. At what value(s) of x is f ' ( x) = 0. Justify your answer. Graph of f(x) 3. On what open interval(s) is f ' ( x) < 0? Justify your answer. 4. On what open interval(s) is f ' ( x) > 0? Justify your answer. 5. At what value(s) of x does the graph of f ' ( x) go from being below the x – axis to above the x – axis? Justify your answer. 6. At what value(s) of x does the graph of f ' ( x) go from being above the x – axis to below the x – axis? Justify your answer. Daily Lessons and Assessments for AP* Calculus AB, A Complete Course Mark Sparks 2012 Page 146 Pictured below is the graph of f ' ( x) on the interval [–3, 4]. Answer the following questions about f(x). 1. On what open interval(s) is the graph of f(x) increasing? Justify your reasoning. Graph of f ' ( x) 2. On what open interval(s) is the graph of f(x) decreasing? Justify your answer. 3. At what value(s) of x does the graph of f(x) have a horizontal tangent? Justify your answer. 4. At what value(s) of x does the graph of f(x) have a relative maximum? Justify your answer. 5. At what value(s) of x does the graph of f(x) have a relative minimum? Justify your answer. 6. What is the slope of the tangent line to the graph of f(x) at x = 0? Justify your reasoning. 7. What is the slope of the normal line to the graph of f(x) at x = 4? Justify your reasoning. Daily Lessons and Assessments for AP* Calculus AB, A Complete Course Mark Sparks 2012 Page 147 For each of the given functions, determine the interval(s) on which f(x) is increasing and/or decreasing. Find all coordinates of the relative extrema. Unless otherwise noted, perform the analysis on all values on , . Provide justification for your answers. 1. f ( x) x3 6 x 1 Daily Lessons and Assessments for AP* Calculus AB, A Complete Course Mark Sparks 2012 Page 148 2. f(x) = 3x5 – 5x3 Daily Lessons and Assessments for AP* Calculus AB, A Complete Course Mark Sparks 2012 Page 149 3. f(θ) = θ + 2sinθ on (0, 2π) Daily Lessons and Assessments for AP* Calculus AB, A Complete Course Mark Sparks 2012 Page 150 Name_________________________________________Date____________________Class__________ Day #15 Homework For exercises 1 – 3, determine on what intervals the given function is increasing or decreasing. Also, identify the coordinates of any relative extrema of the function. Show your work and justify your reasoning. 1. f(x) = 2x3 + 3x2 – 12x 2. g(x) = x3 – 6x2 + 15 Daily Lessons and Assessments for AP* Calculus AB, A Complete Course Mark Sparks 2012 Page 151 3. h(x) = (x + 2)2(x – 1) 4. Pictured to the right is the graph of f ' ( x) . On what interval(s) is the graph of f(x) increasing or decreasing? Justify your reasoning. 5. Pictured to the right is the graph of f ' ( x) . At what value(s) of x does the graph of f(x) have a relative maximum/minimum? Justify your reasoning. Daily Lessons and Assessments for AP* Calculus AB, A Complete Course Mark Sparks 2012 Page 152 Solidifying the Concept of the Derivative as the Tangent Line Pictured to the right is the graph of a quadratic function, g ( x) 12 ( x 2)2 4 . 1. Find g ' (4) and explain what this value represents in terms of the graph of the function g(x). 2. Find the equation of the tangent line drawn to the graph of g(x) at x = –4. Sketch a graph of this tangent line on the grid with the graph of g(x) above. 3. Using the equation of the tangent line, find the value of y when x = –3.9. Then, find the value of g(–3.9). 4. What do you notice about the values of these two results from question 3? What does this imply about how the equation of the tangent line might be used? Daily Lessons and Assessments for AP* Calculus AB, A Complete Course Mark Sparks 2012 Page 153 Pictured to the right is the graph of the function g ( x) 2 x x 2 . Use the graph and the equation to answer questions 5 – 9. 5. Based on the graph, at what value(s) does the graph of g(x) have a horizontal tangent? Give a reason. Show an algebraic analysis that supports your answer. 6. On what interval(s) is g ' ( x) < 0? Give a reason for your answer. 7. On what interval(s) is g ' ( x) > 0? Give a reason for your answer. 8. For what value(s) of x is the slope of the tangent line equal to 2? Show your work. 9. Find an equation of the tangent line drawn to the graph of g(x) when x = 4. Then, draw the tangent line on the grid above. Daily Lessons and Assessments for AP* Calculus AB, A Complete Course Mark Sparks 2012 Page 154 The table of values below represents values on the graph of the derivative, h ' ( x) , of a polynomial function h(x). The zeros indicated in the table are the only zeros of the graph of h ' ( x) . Use the table to answer questions 10 – 15. x h’(x) −8 11 −5 5 −2 0 0 −1 3 −3 5 −1 7 0 10 −3 12 −9 10. On what interval(s) is the function h(x) increasing and decreasing? Give reasons for your answers. 11. At what x – value(s) does the graph of h(x) have a relative maximum? Justify your answer. 12. At what x – value(s) does the graph of h(x) have a relative minimum? Justify your answer. 13. If h(3) = 2, what is the equation of the tangent line to the graph of h(x) at x = 3? What is the equation of the normal line to the graph of h(x) at x = 3? 14. Find the tangent line approximation of h(3.1). 15. Find the value of each of the following limits: lim h( x) x lim h( x) x Daily Lessons and Assessments for AP* Calculus AB, A Complete Course Mark Sparks 2012 Page 155 The derivative of a polynomial function, f(x), is given by the equation f ' ( x) x(2 x)( x 3) . Use this equation to answer questions 16 – 20. 16. On what intervals is f(x) increasing? Decreasing? Justify your answers. 17. At what value(s) of x does the graph of f(x) reach a relative minimum? Justify your answers. 18. At what value(s) of x does the graph of f(x) reach a relative maximum? Justify your answers. 19. If f(4) = –1, what is the equation of the tangent line drawn to the graph of f(x) at x = 4? 20. Approximate the value of f(4.1). Explain why this is a good approximation of the true value of f(4.1). Daily Lessons and Assessments for AP* Calculus AB, A Complete Course Mark Sparks 2012 Page 156 Pictured to the right is a graph of p' ( x) , the derivative of a polynomial function, p(x). Use the graph to answer the questions 21 – 25. 21. On what interval(s) is the graph of p(x) decreasing? Justify your answer. 22. On what interval(s) is the graph of p(x) increasing? Justify your answer. 23. At what value(s) of x does the graph of p(x) reach a relative maximum? Justify your answer. 24. At what value(s) of x does the graph of p(x) reach a relative minimum? Justify your answer. 25. Approximate the value of p(1.8) using the tangent line approximation if p(2) = –3. Daily Lessons and Assessments for AP* Calculus AB, A Complete Course Mark Sparks 2012 Page 157 Name_________________________________________Date____________________Class__________ Day #16 Homework 1. If g ' ( x) ( x 3) 2 ( x 1) , determine on what intervals the graph of g(x) is increasing or decreasing and identify the value(s) of x at which g(x) has a relative maximum or minimum. Justify your reasoning and show your work. For exercises 2 – 4, use the graph of t function, h(x), pictured to the right. Use the graph to identify the following. Provide written justification. 2. On what interval(s) is h ' ( x) < 0? 3. On what interval(s) is h ' ( x) > 0? 4. At what value(s) of x does h ' ( x) change from positive to negative? From negative to positive? Daily Lessons and Assessments for AP* Calculus AB, A Complete Course Mark Sparks 2012 Page 158 Consider the quadratic function f ( x) 12 x 2 x 4 . 5. Sketch an accurate graph of the function. 6. Find f ' ( x) and use it to find the absolute maximum of the graph of f(x). 7. Estimate the value of f ' (0) and explain what this value represents in terms of the graph of f(x). 8. Find the equation of the tangent line to the graph of f(x) at x = 0. Draw a graph of this line. 9. Sketch a graph of the normal line to the tangent line at x = 0. What is the equation of this line? 10. Use the equation of the tangent line to approximate f(0.1). Then, find f(0.1) using the equation of f(x). Is the approximation an under or over approximation of the actual value of f(0.1)? Based on the graph of f(x), why do you suppose this is true? Daily Lessons and Assessments for AP* Calculus AB, A Complete Course Mark Sparks 2012 Page 159 2 sin( x h) 2 sin x give the derivative? Find the limit. h h 0 11. For what function does lim ( x h) 5 x 5 . h h 0 12. Find lim 14. If f ( x) 3x x x xh . h h 0 13. Find lim , what is the slope of the normal line to the graph of f(x) when x = 4? 15. If 2x – 3 = 5(y + 1) is the equation of the normal line to the graph of f(x) when x = a, find the value of f ' (a) . Show your work and explain your reasoning. Daily Lessons and Assessments for AP* Calculus AB, A Complete Course Mark Sparks 2012 Page 160 16. On the interval [0, 2π), find the coordinates of the relative minimum(s) of f ( ) 3 2 sin . The derivative of a function f(x) is f ' ( x) (3 x) 2 ( x 5) . Use this derivative for exercises 17 and 18. 17. At what value(s) of x does the graph of f(x) have a relative maximum? Justify your answer. 18. Use the equation of the tangent line to approximate the value of f(2.1) if f(2) = –3. Daily Lessons and Assessments for AP* Calculus AB, A Complete Course Mark Sparks 2012 Page 161 AP Calculus Quiz #3 Answer Key & Rubric Multiple Choice 1. E 2. A 3. C 4. C 5. A 6. A 7. B C B B A E C D Free Response Part A – 3 points total ____ 1 Performs the sign analysis pictured to the right ____ 1 f(x) has a relative maximum at x = 0 b/c f ' ( x) changes from positive to negative. ____ 1 f(x) does not have a relative minimum b/c f ' ( x) never changes from negative to positive Free Response Part B – 2 points total ____ 1 f(x) is increasing on the interval (−∞, 0) b/c f ' ( x) > 0. ____ 1 f(x) is decreasing on the intervals (0, 3) and (3, ∞) b/c f ' ( x) < 0. Free Response Part C – 4 points total ____ 1 Finds the slope of the tangent line: f ' (1) 2(1)(1 3) 2 8 ____ 1 Correct equation of the tangent line: y + 3 = –8(x – 1) or y = –8x + 5 ____ 1 Find that f(1.1) ≈ –8(1.1) + 5 ≈ –8.8 + 5 ≈ –3.8 ____ 1 Since f(x) is concave up at x = 1, then the graph of the tangent line is below the graph of f(x) so the tangent line approximation is an under estimation of the actual value. Daily Lessons and Assessments for AP* Calculus AB, A Complete Course Mark Sparks 2012 Page 162 AP CALCULUS QUIZ #3 Name______________________________________________________Date_____________________ Calculator NOT Permitted 1. 2. 3. 4. 5. 6. 7. MULTIPLE CHOICE 1. If g(x) = –3x3 + 5x – 3, then the slope of the line that is perpendicular to the tangent line at x = 1 would be… A. 4 B. –4 C. 2 D. –¼ E. ¼ 2. The equation x – ½ = –2(y + 4) represents the equation of the tangent line to the graph of g(x) when x = –1. What is the value of g ' (1) ? A. –½ B. –2 C. 4 D. ½ E. Cannot be determined. Daily Lessons and Assessments for AP* Calculus AB, A Complete Course Mark Sparks 2012 Page 163 3. If the line tangent to the graph of the function f at the point (1, 7) also passes through the point (–2, –2), then what is the value of f ' (1) ? A. –5 B. 1 C. 3 D. 7 E. Undefined 4. The graph of the derivative of a function f is shown to the right. The graph has horizontal tangent lines at x = –1, x = 1, and x = 3. At which of the following values of x does f have a relative maximum? A. –2 only B. 1 only C. 4 only D. –1 and 3 only E. –2, 1, and 4 5. Pictured to the right is the graph of h' ( x) , the derivative of a function, h(x). Which of the following statements is/are true? I. The graph of h(x) has a horizontal tangent when x = 3. II. The graph of h(x) is increasing on the interval (−∞, 2). III. The graph of h(x) is decreasing on the interval (−∞, 3). A. I only B. II only C. I and II only D. I and III only E. I, II, and III Daily Lessons and Assessments for AP* Calculus AB, A Complete Course Mark Sparks 2012 Page 164 sin( x h) sin x h h0 6. Find lim A. cos x B. sin x C. –sin x D. –cos x E. Limit does not exist 7. The graph of a quartic function, p(x), is pictured to the right On which of the following intervals is p' ( x) < 0? I. (0, 2) II. (−∞, 0) III. (4, ∞) IV. (2, 4) A. II and IV only B. I and III only C. III only D. IV only E. I only FREE RESPONSE The derivative of a polynomial function, f(x), is represented by the equation f ' ( x) 2 x( x 3) 2 . Additionally, f(1) = –3 and the graph of f(x) is concave up at x = 1. Use this information to answer the following questions. a. At what value(s) of x does the graph of f(x) have a relative maximum? A relative minimum? Justify your reasoning. Daily Lessons and Assessments for AP* Calculus AB, A Complete Course Mark Sparks 2012 Page 165 b. On what interval(s) is the graph of f(x) increasing? Decreasing? Justify your reasoning. c. What is the tangent line approximation of f(1.1)? Is this estimate greater or less than the actual value of f(1.1)? Give a reason for your answer. Daily Lessons and Assessments for AP* Calculus AB, A Complete Course Mark Sparks 2012 Page 166 AP CALCULUS *QUIZ #3* Name______________________________________________________Date_____________________ Calculator NOT Permitted 1. 2. 3. 4. 5. 6. 7. MULTIPLE CHOICE 2x 1 , then the slope of the line that is perpendicular to the tangent line at x = 1 x would be… 1. If g(x) = A. 4 B. –4 C. –2 D. –½ E. ½ 2. The equation y – ½ = –2(x + 4) represents the equation of the tangent line to the graph of g(x) when x = –1. What is the value of g ' (1) ? A. –½ B. –2 C. 4 D. ½ E. Cannot be determined. Daily Lessons and Assessments for AP* Calculus AB, A Complete Course Mark Sparks 2012 Page 167 3. If the line tangent to the graph of the function f at the point (1, 7) also passes through the point (3, 5), then what is the value of f ' (1) ? A. –5 B. –1 C. 3 D. 7 E. Undefined 4. The graph of the derivative of a function f is shown to the right. The graph has horizontal tangent lines at x = –1, x = 1, and x = 3. At which of the following values of x does f have a relative minimum? A. –2 only B. 1 only C. 4 only D. –1 and 3 only E. –2, 1, and 4 5. Pictured to the right is the graph of h' ( x) , the derivative of a function, h(x). Which of the following statements is/are true? I. The graph of h(x) has a horizontal tangent when x = 3. II. The graph of h(x) is decreasing on the interval (−∞, 3). III. The graph of h(x) is increasing when x = 4. A. I only B. II only C. I and II only D. I, II, and III E. I and III only Daily Lessons and Assessments for AP* Calculus AB, A Complete Course Mark Sparks 2012 Page 168 cos( x h) cos x h h0 6. Find lim A. cos x B. sin x C. –sin x D. –cos x E. Limit does not exist 7. The graph of a quartic function, p(x), is pictured to the right On which of the following intervals is p' ( x) > 0? I. (0, 2) II. (−∞, 0) III. (4, ∞) IV. (2, 4) A. I and III only B. II and III only C. III only D. II and IV only E. I only FREE RESPONSE The derivative of a polynomial function, f(x), is represented by the equation f ' ( x) 2 x( x 3) 2 . Additionally, f(1) = –3 and the graph of f(x) is concave up at x = 1. Use this information to answer the following questions. a. At what value(s) of x does the graph of f(x) have a relative maximum? A relative minimum? Justify your reasoning. Daily Lessons and Assessments for AP* Calculus AB, A Complete Course Mark Sparks 2012 Page 169 b. On what interval(s) is the graph of f(x) increasing? Decreasing? Justify your reasoning. c. What is the tangent line approximation of f(1.1)? Is this estimate greater or less than the actual value of f(1.1)? Give a reason for your answer. Daily Lessons and Assessments for AP* Calculus AB, A Complete Course Mark Sparks 2012 Page 170 AP Calculus Test #2 Answer Key Raw Score to Percentage Conversion Multiple Choice Calculator Permitted * 1. D A 2. D C 3. E D 4. A E 5. B A 6. E B 7. A D Calculator Permitted Free Response Part A – 2 points total ____ 1 Correctly finds the derivative of 2 x to be x 2 and correctly finds the derivative of 3 cos x to be 3sin x 1 ____ 1 Simplifies algebraically to show that f ' ( x) 1 3 x sin x x Calculator Permitted Free Response Part B – 2 points total ____ 1 At x = 4, the graph of f(x) is decreasing so the tangent line drawn to f has a negative slope. ____ 1 The slope of the normal line is the opposite reciprocal of the slope of the tangent line at a point of tangency. Thus, the normal line drawn to f at x = 4 will have a positive slope. Calculator Permitted Free Response Part C – 3 points total ____ 1 The graph of f ' ( x) is drawn as shown to the right. ____ 1 The graph of f(x) has a relative maximum at x = 3.325 because f ' ( x) changes from negative to positive. ____ 1 The graph of f(x) has a relative minimum at x = 6.148 because f ' ( x) changes from negative to positive. Calculator Permitted Free Response Part D – 2 points total ____ 1 f(x) is increasing on the interval 0 < x < 3.325 and 6.148 < x < 2π since the graph of f ' ( x) > 0. ____ 1 f(x) is increasing on the interval 3.325 < x < 6.148 since the graph of f ' ( x) < 0. Daily Lessons and Assessments for AP* Calculus AB, A Complete Course Mark Sparks 2012 Page 171 Calculator NOT Permitted * 8. A E 9. C E 10. B E 11. C B 12. C D 13. E C 14. C A Raw Score to Percentage Conversion Calculator NOT Permitted Free Response Part A – 3 points total ____ 1 Creates an f ' ( x) sign chart correctly using the zeros of f ' ( x) as shown below ____ 1 f(x) is increasing on 0,2 because f ' ( x) > 0 f ' ( x) 6 x 3x 2 0 3x(2 x) 0 3x 0 2 x 0 x 0 and x 2 ____ 1 f(x) is decreasing on ,0 and 2, because f ' ( x) < 0 Calculator NOT Permitted Free Response Part B – 2 points total ____ 1 f(x) has a relative minimum at the point (0, 0) because f ' ( x) changes from negative to positive at x = 0. ____ 1 f(x) has a relative maximum at the point (2, 4) because f ' ( x) changes from positive to negative at x = 2. Calculator NOT Permitted Free Response Part C – 2 points total ____ 1 Rewrites h(x) as a polynomial in standard form: h(x) = (2 – x2)(3x – x3) = 6x2 – 3x4 – 2x3 + x5 = x5 – 3x4 – 2x3 + 6x2 ____ 1 Correctly finds h ' ( x) 5x 4 12 x 3 6 x 2 12 x Calculator NOT Permitted Free Response Part D – 2 points total ____ 1 Finds f ' ( x) 6 x 3x 2 and g ' ( x) 2 x ____ 1 Shows that the answer to Lillian’s question is NO because 6 x 3x 2 2 x 12 x 2 6 x 3 which is not the same as h ' ( x) found in part C. Daily Lessons and Assessments for AP* Calculus AB, A Complete Course Mark Sparks 2012 Page 172 AP CALCULUS TEST #2 Unit #2 – Conceptualizing the Derivative Name______________________________________________________Date_____________________ A GRAPHING CALCULATOR IS REQUIRED FOR SOME PROBLEMS OR PARTS OF PROBLEMS IN THIS PART OF THE EXAMINATION. (1) The exact numerical value of the correct answer does not always appear among the choices given. When this happens, select from among the choices the number that best approximates the exact numerical value. (2) Unless otherwise specified, the domain of a function f is assumed to be the set of all real numbers x for which f(x) is a real number. Multiple Choice − Calculator Permitted Section 1. If f ( x) 3x( x 2) 2 , then what is the slope of the tangent line to the graph when x = –1? A. 2 B. –3 C. 1 D. 3 E. –2 Daily Lessons and Assessments for AP* Calculus AB, A Complete Course Mark Sparks 2012 Page 173 xh x . h h 0 2. Find lim A. x B. – x C. x 2 D. E. 2 x 1 2 x 3. If g ' (1) 3 , then which of the following could be the equation for g(x)? I. g ( x) 2 x 2 7 x 3 II. g ( x) 4 x 5x III. g ( x) x3 2 x 2 4 x x2 A. I only B. I and II only C. II only D. II and III only E. I, II and III 4. The graph of a polynomial function, f(x), is pictured to the right. Which of the following statements is/are true about f ' ( x) ? Graph of f(x) I. f ' ( x) < 0 on the interval (–1, 2). II. f ' ( x) changes from negative to positive when x = –1. III. There are two values of x such that f ' ( x) = 0. A. I and III only B. I only C. III only D. I and II only E. I, II, and III Daily Lessons and Assessments for AP* Calculus AB, A Complete Course Mark Sparks 2012 Page 174 5. Which of the following would represent f ' ( x) if f ( x) A. 3x 1 x B. 3x 2 2 x C. 3 x 1 2 x D. 1 x x2 2x ? x E. 4 x 3 4 x 6. The graph pictured to the right is the graph of f ' ( x) , the derivative of a polynomial function, f(x). Which of the following statements is/are true? I. f(x) is increasing on the interval (–5, 2). II. f(x) has a relative minimum when x = –5. III. The slope of the normal line drawn to f(x) at x = –4 is 1 . 3 A. II only B. I and II only D. I, II, and III E. II and III only C. III only 7. If g ' ( x) 3x( x 2) 2 , then the graph of g(x) has a relative maximum at what value(s) of x? A. 0 only B. –2 and 0 only C. –2 only D. –2 and 2 only 3 E. g(x) never reaches a relative maximum Daily Lessons and Assessments for AP* Calculus AB, A Complete Course Mark Sparks 2012 Page 175 Free Response Consider the function f ( x) 2 x 3 cos x on the interval 0 < x < 2π. The graph of f(x) is shown to the right. Answer the following questions rounding all values of x to three decimal places. a. Show, algebraically, that f ' ( x) 1 3 x sin x . Make sure you show each step of your work. x b. Based on the graph of f(x), will the slope of the normal line drawn to the graph of f at x = 4 be positive or negative? Daily Lessons and Assessments for AP* Calculus AB, A Complete Course Mark Sparks 2012 Page 176 c. Using your graphing calculator, sketch a graph of f ' ( x) on the axes below on the interval 0 < x < 2π. Then, determine the value(s) of x at which the graph of f(x) reaches a relative maximum or minimum. Justify your answers. d. Based on the graph of f ' ( x) , on what open interval(s) within the interval 0 < x < 2π is f(x) increasing? Decreasing? Justify your answers. Daily Lessons and Assessments for AP* Calculus AB, A Complete Course Mark Sparks 2012 Page 177 AP CALCULUS *TEST #2* Unit #2 – Conceptualizing the Derivative Name______________________________________________________Date_____________________ A GRAPHING CALCULATOR IS NOT ALLOWED FOR THIS SECTION OF THE EXAM. (1) The exact numerical value of the correct answer does not always appear among the choices given. When this happens, select from among the choices the number that best approximates the exact numerical value. (2) Unless otherwise specified, the domain of a function f is assumed to be the set of all real numbers x for which f(x) is a real number. Multiple Choice − Calculator Not Permitted Section The derivative, g ' , of a polynomial function g is continuous and has exactly two zeros. Selected values of g ' are given in the table below. Use the table to answer questions 1 – 2. 8. On which of the following intervals is the graph of g(x) increasing? I. x < −2 A. I only B. II only II. −2 < x < 2 C. III only III. x > 2 D. I and III only E. I and II only 9. At what value(s) of x does the graph of g(x) reach a relative maximum? A. 2 only B. −2 and 2 only D. 0 and 2 only E. g(x) does not have a relative maximum. Daily Lessons and Assessments for AP* Calculus AB, A Complete Course Mark Sparks 2012 C. −2 only Page 178 10. The graph of a function, h(x) is shown to the right. Which of the following conclusions can be made about the derivative of h, h ' ( x) ? A. h ' ( x) > 0 when x = −3. B. h ' ( x) = 0 when x = −2. C. h ' ( x) < 0 when x = 2 D. Both A and B are valid conclusions. E. Both B and C are valid conclusions. 11. Let f be the function defined by f ( x) 4 x3 5x 3 . Which of the following is an equation of the line tangent to the graph of f at the point where x = –1? A. y = 7x – 3 B. y = 7x + 5 C. y = 7x + 11 D. y = –5x – 1 E. y = –5x – 5 12. The function f is defined on the closed interval [0, 8]. The graph of its derivative, f ' , is pictured below. If f(3) = 5, then what is the equation of the tangent line to the graph of f when x = 3? A. y = 2 B. y = 5 C. y – 5 = 2(x – 3) D. y + 5 = 2(x – 3) E. y + 5 = 2(x + 3) 3x 13. If h(x) = 3 , then what is the slope of the normal line to the graph of h when x = –8? x A. 1 4 B. –4 C. –1 D. 12 E. 1 Daily Lessons and Assessments for AP* Calculus AB, A Complete Course Mark Sparks 2012 Page 179 14. The graph of f ( ) 2 sin , for 0 < θ < 2π, has a relative maximum at… A. only 3 B. 4 only 3 C. 2 only 3 D. 2 and 4 3 3 E. 5 and 7 6 6 Free Response Consider the function f(x) = 3x2 – x3. Determine each of the following properties of the graph of f(x). a. Determine the interval(s) where f(x) is increasing or decreasing. Justify your answers. b. Determine the coordinates of any relative maximums or minimums of f(x). Justify your answers. Daily Lessons and Assessments for AP* Calculus AB, A Complete Course Mark Sparks 2012 Page 180 c. If g(x) = (2 − x2), then what is the derivative of the function h( x) f ( x) g ( x) . Show your work. d. Lillian questions if the derivative of the product of two functions is equivalent to the product of the derivatives of the two functions. In other words, if h( x) f ( x) g ( x) then is h ' ( x) f ' ( x) g ' ( x) ? Using f(x) and g(x), show and tell her the answer to her question. Daily Lessons and Assessments for AP* Calculus AB, A Complete Course Mark Sparks 2012 Page 181 AP CALCULUS *TEST #2* Unit #2 – Conceptualizing the Derivative Name______________________________________________________Date_____________________ A GRAPHING CALCULATOR IS REQUIRED FOR SOME PROBLEMS OR PARTS OF PROBLEMS IN THIS PART OF THE EXAMINATION. (1) The exact numerical value of the correct answer does not always appear among the choices given. When this happens, select from among the choices the number that best approximates the exact numerical value. (2) Unless otherwise specified, the domain of a function f is assumed to be the set of all real numbers x for which f(x) is a real number. Multiple Choice − Calculator Permitted Section 1. If f ( x) 2 x( x 2)( x 3) , then what is the slope of the tangent line to the graph when x = −1? A. −10 B. –1 C. 1 D. 3 E. 12 Daily Lessons and Assessments for AP* Calculus AB, A Complete Course Mark Sparks 2012 Page 182 2. Find lim 3 xh 3 x h 0 A. C. E. 3 h x2 1 3 3 x2 . 3 B. – x 2 3 D. x2 3 3 3 x2 3. If g ' (1) 3 , then which of the following could be the equation for g(x)? I. g ( x) 2 x 2 7 x 3 II. g ( x) 4 x 5x III. g ( x) x3 2 x 2 4 x x2 A. I only B. I and II only C. II only D. II and III only E. I, II and III 4. The graph of a polynomial function, f(x), is pictured to the right. Which of the following statements is/are true about f ' ( x) ? Graph of f(x) I. f ' ( x) < 0 on the interval (–1, 2). II. f ' ( x) changes from positive to negative when x = –1. III. There are two values of x such that f ' ( x) = 0. A. I and III only B. I only C. III only D. I and II only E. I, II, and III Daily Lessons and Assessments for AP* Calculus AB, A Complete Course Mark Sparks 2012 Page 183 5. Which of the following equations would represent f ' ( x) if f ( x) A. 3x 2 2 x B. 3x 1 x C. 3 x 1 2 x D. 1 x x2 2x ? x E. 4 x 3 4 x 6. The graph pictured to the right is the graph of f ' ( x) , the derivative of a polynomial function, f(x). Which of the following statements is/are true? I. f(x) is increasing on the interval (–5, –2) and (–2, 2). II. f(x) has a relative minimum when x = –5. III. The slope of the normal line drawn to f(x) at x = –4 is undefined. A. II only B. I and II only D. I, II, and III E. II and III only C. III only 7. If g ' ( x) 3x( x 2)( x 3) , then the graph of g(x) has a relative maximum at what value(s) of x? A. 0 only B. –2 and 0 only C. –2 only D. –2 and 3 only E. g(x) never reaches a relative maximum Daily Lessons and Assessments for AP* Calculus AB, A Complete Course Mark Sparks 2012 Page 184 Free Response Consider the function f ( x) 2 x 3 cos x on the interval 0 < x < 2π. The graph of f(x) is shown to the right. Answer the following questions rounding all values of x to three decimal places. a. Show, algebraically, that f ' ( x) 1 3 x sin x . Make sure you show each step of your work. x b. Based on the graph of f(x), will the slope of the normal line drawn to the graph of f at x = 4 be positive or negative? Daily Lessons and Assessments for AP* Calculus AB, A Complete Course Mark Sparks 2012 Page 185 c. Using your graphing calculator, sketch a graph of f ' ( x) on the axes below on the interval 0 < x < 2π. Then, determine the value(s) of x at which the graph of f(x) reaches a relative maximum or minimum. Justify your answers. d. Based on the graph of f ' ( x) , on what open interval(s) within the interval 0 < x < 2π is f(x) increasing? Decreasing? Justify your answers. Daily Lessons and Assessments for AP* Calculus AB, A Complete Course Mark Sparks 2012 Page 186 AP CALCULUS AB *TEST #2* Unit #2 – Conceptualizing the Derivative Name______________________________________________________Date_____________________ A GRAPHING CALCULATOR IS NOT ALLOWED FOR THIS SECTION OF THE EXAM. (1) The exact numerical value of the correct answer does not always appear among the choices given. When this happens, select from among the choices the number that best approximates the exact numerical value. (2) Unless otherwise specified, the domain of a function f is assumed to be the set of all real numbers x for which f(x) is a real number. Multiple Choice − Calculator Not Permitted Section The derivative, g ' , of a polynomial function g is continuous and has exactly two zeros. Selected values of g ' are given in the table below. Use the table to answer questions 1 – 2. 8. On which of the following intervals is the graph of g(x) decreasing? I. x < −2 A. I only B. II only II. −2 < x < 2 C. III only III. x > 2 D. I and III only E. I and II only 9. At what value(s) of x does the graph of g(x) reach a relative maximum? A. −2 only B. −2 and 2 only D. 0 and 2 only E. g(x) does not have a relative maximum. Daily Lessons and Assessments for AP* Calculus AB, A Complete Course Mark Sparks 2012 C. 2 only Page 187 10. The graph of a function, h(x) is shown to the right. Which of the following conclusions can be made about the derivative of h, h ' ( x) ? A. h ' ( x) > 0 when x = −3. B. h ' ( x) = 0 when x = −2. C. h ' ( x) > 0 when x = −1 D. Both A and B are valid conclusions. E. Both B and C are valid conclusions. 11. Let f be the function defined by f ( x) 4 x3 5x 3 . Which of the following is an equation of the line tangent to the graph of f at the point where x = –1? A. y = 7x – 3 B. y = 7x + 11 C. y = –5x – 5 D. y = –5x – 1 E. y = 7x + 5 12. The function f is defined on the closed interval [0, 8]. The graph of its derivative, f ' , is pictured below. If f(3) = −5, then what is the equation of the tangent line to the graph of f when x = 3? A. y = 2 B. y = 5 C. y – 5 = 2(x – 3) D. y + 5 = 2(x – 3) E. y – 5 = 2(x + 3) 3x 13. If h(x) = 3 , then what is the slope of the normal line to the graph of h when x = 8? x A. 1 4 B. –4 C. –1 D. 12 E. 1 Daily Lessons and Assessments for AP* Calculus AB, A Complete Course Mark Sparks 2012 Page 188 14. The graph of f ( ) 2 sin , for 0 < θ < 2π, has a relative minimum at… A. 4 only 3 B. only 3 C. 2 only 3 D. 2 and 4 3 3 E. 5 and 7 6 6 Free Response Consider the function f(x) = 3x2 – x3. Determine each of the following properties of the graph of f(x). a. Determine the interval(s) where f(x) is increasing or decreasing. Justify your answers. b. Determine the coordinates of any relative maximums or minimums of f(x). Justify your answers. Daily Lessons and Assessments for AP* Calculus AB, A Complete Course Mark Sparks 2012 Page 189 c. If g(x) = (2 − x2), then what is the derivative of the function h( x) f ( x) g ( x) . Show your work. d. Lillian questions if the derivative of the product of two functions is equivalent to the product of the derivatives of the two functions. In other words, if h( x) f ( x) g ( x) then is h ' ( x) f ' ( x) g ' ( x) ? Using f(x) and g(x), show and tell her the answer to her question. Daily Lessons and Assessments for AP* Calculus AB, A Complete Course Mark Sparks 2012 Page 190 EXTRA Free Response Pictured to the right is the graph of the first derivative, f ' ( x) , of a polynomial function f(x), such that f(–1) = 2. a. Approximate the value of f(–0.9) using the equation of the tangent line drawn to the graph of f(x) when x = –1. b. On what interval(s) is the graph of f(x) increasing or decreasing? Give a reason for your answer. c. At what value(s) of x does the graph of f(x) have a relative maximum? Justify your answer. d. At what value(s) of x does the graph of f(x) have a relative minimum? Justify your answer. Daily Lessons and Assessments for AP* Calculus AB, A Complete Course Mark Sparks 2012 Page 191 Calculator Permitted Free Response Part A – 3 points total ____ 1 Uses the point (–1, 2) as the point of tangency and f ' (1) 3 as the slope of the tangent line ____ 1 Finds the equation of the tangent line: y – 2 = –3(x + 1) or y = –3x – 1 ____ 1 Uses the equation of the tangent line to approximate f (0.9) 3(0.9) 1 1.7 Calculator Permitted Free Response Part B – 2 points total ____ 1 f(x) is increasing on the interval (−∞,−2) because f ' ( x) > 0 ____ 1 f(x) is decreasing on the interval (−2, 1) and (1, ∞) because f ' ( x) < 0 Calculator Permitted Free Response Part C – 2 points total ____ 1 f(x) has a relative maximum when f ' ( x) changes from positive to negative. ____ 1 Thus, the graph of f(x) has a relative maximum when x = –2. Calculator Permitted Free Response Part D – 2 points total ____ 1 f(x) has a relative minimum when f ' ( x) changes from negative to positive. ____ 1 Thus, the graph of f(x) does not have a relative minimum Daily Lessons and Assessments for AP* Calculus AB, A Complete Course Mark Sparks 2012 Page 192 AP Calculus UNIT #3 – Rules of Differentiation Day# 19 Objective Apply the product rule of differentiation analytically, graphically, and numerically. 20 Apply the quotient rule of differentiation analytically, graphically, and numerically and derive the differentiation rules for tangent, cotangent, secant, and cosecant. Apply the chain rule of differentiation analytically, graphically, and numerically and derive the differentiation rule for an inverse function. 21 22 Complete in class problems in groups to review for quiz. 23 Quiz #4 Analytically determine the derivative of exponential functions whose bases are e and natural logarithmic functions. Note Handouts & Assignments Daily Lessons pages 192 – 196 Day #19 HW: #1 – 12 Daily Lessons pages 200 – 205 Day #20 HW: #1 – 12 Daily Lessons pages 210 – 214 Day #21 HW: #1 – 15 Study for Quiz #4 Daily Lessons pages 219 – 220 Daily Lessons pages 232 – 235 Day #23 HW: #1 – 11 24 Numerically determine and interpret the value of the derivative of a function using the graphing calculator. Understand the relationship between differentiability and continuity. Daily Lessons pages 238 – 241 Day #24 HW: #1 – 11 25 26 Find the derivative of inverse functions. Daily Lessons pages 245 – 247 Quiz #5 Day #25 HW: #1 – 4 and 2006 AB #6 Parts a and b Study for Quiz #5 Daily Lessons pages 250 – 254 Study for Unit #3 Test 27 Test #3: Unit #3 Test – Rules of Differentiation Daily Lessons and Assessments for AP* Calculus AB, A Complete Course Mark Sparks 2012 Page 191 Rules for Differentiation Finding the Derivative of a Product of Two Functions Rewrite the function f(x) = (2x – 3)(x2 – 2x + 1) as a cubic function. Then, find f ' ( x). What does this equation of f ' ( x) represent, again? Two men, Isaac Newton and Gottfried Leibniz, are credited for developing the study of calculus. In 1673, Leibniz published an article in which he derived what we know today as the Product Rule of Differentiation. Let’s write this rule together in the box below. Product Rule of Differentiation To show that this rule works, let’s apply this rule to the function f(x) = (2x – 3)(x2 – 2x + 1) that we rewrote and differentiated as a polynomial above. Students often wonder why this rule is so important if we could just rewrite as a polynomial and easily differentiate it. The answer to that question is simple. If it is possible to rewrite as a polynomial, always do so. But in the case of the function g ( x) x 2 sin x , there is no way to rewrite as a polynomial. Apply the product rule to find the slope of the normal line to the graph of g ( x) x 2 sin x when x = π. Daily Lessons and Assessments for AP* Calculus AB, A Complete Course Mark Sparks 2012 Page 192 Use the product rule to find the derivative of each of the following functions. f ( x) 2 x 2 3 x x 2 3 g ( x) x x 2 3 x 2 f ( x) x 3 sin x h( x) (3x 2) cos x g ( x) 3 sin h( x) sin x cos x Find the equation of the line tangent to the graph of g (t ) t 2 cos t when t = . 6 Daily Lessons and Assessments for AP* Calculus AB, A Complete Course Mark Sparks 2012 Page 193 There is a very valuable lesson that we must learn when we are introduced to the product rule. On page 193, you were asked to find f ' ( x) by applying the product rule to the function f ( x) x x 2 3x 2 . In the space below, write the result that you obtained. Given the function f ( x) x x 2 3x 2 . Rewrite the function in polynomial form. Then, find f ' ( x) . What is the lesson to be learned from the algebraic analysis above? If g ( x) (2 x 3)( x 1)( x 3) , what is the slope of the normal line to the graph of g(x) when x = 2? Daily Lessons and Assessments for AP* Calculus AB, A Complete Course Mark Sparks 2012 Page 194 Below are graphs of two functions—f(x) and g(x). Let P( x) f ( x) g ( x) and let R( x) x 2 g ( x) . Use the graphs to answer the questions that follow. Graph of f(x) Graph of g(x) If g ' (4) 2 , what is the value of P ' (4) ? If R ' (2) = 20, what is the value of g ' (2) ? Find the equation of the line tangent to the graph of P(x) when x = –4. Find the equation of the line tangent to the graph of R(x) when x = –2. Daily Lessons and Assessments for AP* Calculus AB, A Complete Course Mark Sparks 2012 Page 195 Let f(x) and g(x) be differentiable functions such that the following values are true. x f(x) g(x) f ' ( x) g ' ( x) 4 1 7 2 –3 3 –2 –3 –4 2 –1 2 –2 1 –1 Estimate the value of f ' (3.5) . If q( x) 2 f ( x) 4 g ( x) , what is the value of q ' (4) ? If p( x) 2 f ( x) g ( x) , what is the value of p ' (3) ? Find the equation of the line tangent to the graph of v( x) x 3 f ( x) when x = –1. If k ( x) 2 f ( x) 33 g ( x) , what is the value of k ' (3) ? Daily Lessons and Assessments for AP* Calculus AB, A Complete Course Mark Sparks 2012 Page 196 Name_________________________________________Date____________________Class__________ Day #19 Homework In the table below, a function is given. Show the algebraic analysis that leads to the derivative of the function. Find the derivative by the specified method. 1. f ( x) x 2 2 x x 3 Rewrite f(x) as a polynomial first. Then apply the power rule to find f ' ( x) . 2. f ( x) x 2 2 x x 3 Apply the product rule to find f ' ( x) . For exercises 3 – 5, find the derivative of each function. 3. f ( x) x 2 2 x 2 2 x 5. f ( x) 3 x x 2 4 4. f ( x) x 3 3x 2 x 2 3x 5 Daily Lessons and Assessments for AP* Calculus AB, A Complete Course Mark Sparks 2012 Page 197 Find the slope of the normal line drawn to the graph of each function at the indicated value of x. 6. g ( x) x sin x when x = π 7. h( x) sin x(sin x cos x) when x = 4 For each of the functions below, find the equation of the tangent line drawn to the graph of g(x) at the indicated value of x. 8. g ( x) x 2 x 2 4 when x = 4 9. g ( x) x 2 cos x when x = Daily Lessons and Assessments for AP* Calculus AB, A Complete Course Mark Sparks 2012 2 Page 198 Use the table below to complete exercises 10 – 12. x f(x) f ’(x) g(x) g’(x) –2 1 –1 2 4 –1 3 –2 1 1 0 –1 2 –2 –3 10. If H ( x) 2 f ( x) g ( x) , what is the equation of the tangent line when x = –1? 11. If J ( x) g ( x) sin x , what is the value of J ' (0) ? 12. If K ( x) 4 x f ( x)2 g ( x) 2, what is the slope of the normal line when x = –2? Daily Lessons and Assessments for AP* Calculus AB, A Complete Course Mark Sparks 2012 Page 199 Rules for Differentiation Finding the Derivative of a Quotient of Two Functions Rewrite the function f ( x) 2 x 3 3x 2 2 x2 as a function in polynomial form. Then, find f ' ( x). Just as Leibniz was the first to publish a proof of the Product Rule for Differentiation, Isaac Newton was the first to publish a proof of the Quotient Rule of Differentiation using the limit definition of the derivative. Let’s write this rule together in the box below. Quotient Rule of Differentiation To show that this rule works, let’s apply this rule to the function f ( x) 2 x 3 3x 2 2 x2 that we rewrote and differentiated as a polynomial-form above. Find the equation of the tangent line drawn to the graph of g ( x) 2 x 1 when x = –2. x3 Daily Lessons and Assessments for AP* Calculus AB, A Complete Course Mark Sparks 2012 Page 200 We will now use the quotient rule to derive the derivative formulas for the remaining trigonometric functions. Rewrite each function in terms of sine and/or cosine and differentiate using the Quotient Rule. f ( ) tan f ( ) cot f ( ) sec f ( ) csc Find the equation of the normal line drawn to the graph of f ( ) 3 when . cos Daily Lessons and Assessments for AP* Calculus AB, A Complete Course Mark Sparks 2012 Page 201 Find the derivative of each of the functions below by applying the quotient rule. f ( x) x 2 2x x2 h( ) sin 1 cos g ( x) f ( x) Daily Lessons and Assessments for AP* Calculus AB, A Complete Course Mark Sparks 2012 tan x x2 3 1 x x5 Page 202 Show, using the quotient rule, that if f ( x) x 2 3x 2 x2 1 , then f ' ( x) 3 ( x 1) 2 . Similar to the Product Rule, there is a very valuable lesson that we must learn when we are introduced to the quotient rule. In the box below, first factor and simplify the function, f ( x) x 2 3x 2 x2 1 , from above. Then, differentiate using the quotient rule What is the lesson to be learned from the algebraic analysis above? Daily Lessons and Assessments for AP* Calculus AB, A Complete Course Mark Sparks 2012 Page 203 Below are graphs of two functions—f(x) and g(x). Let P( x) f ( x) sin x and let R( x) . Use the graphs g ( x) f ( x) to answer the questions that follow. Graph of f(x) Graph of g(x) Find P ' (5) . Find R ' (0) Find the equation of the line tangent to the graph of P(x) when x = 5. Find the equation of the line tangent to the graph of R(x) when x = 0. Daily Lessons and Assessments for AP* Calculus AB, A Complete Course Mark Sparks 2012 Page 204 Let f(x) and g(x) be differentiable functions such that the following values are true. x f(x) g(x) f ' ( x) g ' ( x) 4 1 7 8 –2 3 –5 –3 –4 6 2 2 –1 9 –1 Estimate the value of g ' (2.5) . If p( x) g ( x) , what is the value of p ' (4) ? What f ( x) does this value say about the graph of p(x) when x = 4? Give a reason for your answer. f ( x) , what is the value of q ' (2) ? If q( x) 2 x 2 g ( x) Find the equation of the line tangent to the graph of v( x) 3x when x = 3. g ( x) Daily Lessons and Assessments for AP* Calculus AB, A Complete Course Mark Sparks 2012 Page 205 Name_________________________________________Date____________________Class__________ Day #20 Homework For exercises 1 and 2, show the algebraic analysis that leads to the derivative of the function. Find the derivative by the specified method. 1. f ( x) 2 x 3 3x 2 3 x2 Rewrite f(x) in a polynomialform first. Then apply the power rule to find f ' ( x) . 2. f ( x) 2 x 3 3x 2 3 x2 Apply the quotient rule to find f ' ( x) . 3. Find the equation of the line tangent to the graph of g ( x) 2 x 2 3x when x = –1. 3x 1 Daily Lessons and Assessments for AP* Calculus AB, A Complete Course Mark Sparks 2012 Page 206 Find the derivative of each of the following functions. 4. h( x) 6. g ( ) x 2 x 1 cos 3 5. h( x) x x 1 7. f ( ) 3(1 sin ) 2 cos Daily Lessons and Assessments for AP* Calculus AB, A Complete Course Mark Sparks 2012 Page 207 Use the table below to complete exercises 8 – 10. 8. If H ( x) x f(x) f ’(x) g(x) g’(x) –2 1 –1 2 4 –1 3 –2 1 1 0 –1 2 –2 –3 2 f ( x) , what is the equation of g ( x) the tangent line when x = –1? 10. If K ( x) 9. If J ( x) 3x cos x , what is the value of f ( x) J ' (0) ? 4 x f ( x) , what is the slope of the normal line when x = –2? 3 g ( x) Daily Lessons and Assessments for AP* Calculus AB, A Complete Course Mark Sparks 2012 Page 208 11. If f ( ) csc sin , show that f ' ( ) cot 2 cos . 12. Find the equation of the line tangent to the graph of f ( ) tan sin when . 4 Daily Lessons and Assessments for AP* Calculus AB, A Complete Course Mark Sparks 2012 Page 209 Rules for Differentiation Finding the Derivative of a Composite Function Rewrite the function f ( x) (2 x 3) 3 as a function in polynomial form. Then, find f ' ( x). Leibniz was the first of the two great calculus developers to use the Chain Rule to differentiate composite functions. Let’s write this rule together in the box below. Chain Rule of Differentiation of Composite Functions To show that this rule works, let’s apply this rule to the function f ( x) (2 x 3) 3 that we rewrote and differentiated as a polynomial-form above. Find the slope of the normal line to the graph of f ( ) sin 2 when 3 . 4 Daily Lessons and Assessments for AP* Calculus AB, A Complete Course Mark Sparks 2012 Page 210 Find the derivative of each of the following functions by applying the chain rule. 3 g ( x) 2 x 5 h( x) 3 ( x 2) 2 F ( x) 5 x 2 2 x G( x) cos 2 3x h( x) sin 2 (2 x 1) f ( x) 3x 2 2 3 Daily Lessons and Assessments for AP* Calculus AB, A Complete Course Mark Sparks 2012 Page 211 Now that you know “THE BIG THREE” rules of differentiation—product, quotient, and chain— let’s see how the three can be incorporated with each other. Find the derivative of each of the following functions. f ( x) 5 x x 3 2x 1 g ( x) sin x3 Daily Lessons and Assessments for AP* Calculus AB, A Complete Course Mark Sparks 2012 Page 212 h( x ) 2x 5 x3 Given the graph of H(x) pictured to the right, find the equation of the tangent line to the graph of P( x) H ( x) when x = –4. Daily Lessons and Assessments for AP* Calculus AB, A Complete Course Mark Sparks 2012 Page 213 Let f(x) and g(x) be differentiable functions such that the following values are true. x f(x) g(x) f ' ( x) g ' ( x) 4 1 7 8 –2 3 –5 4 –4 6 2 2 –1 0 –1 Is the graph of h( x) f ( g ( x)) increasing, decreasing or at a relative maximum or minimum when x = 3? Give a reason for your answer. If p( x) g (2 x) , what is the value of p ' (1) ? If q( x) f ( x) g ( x) , what is the value of q ' (4) ? What does this value say about the graph of q(x) when x = 4? Give a reason for your answer. Daily Lessons and Assessments for AP* Calculus AB, A Complete Course Mark Sparks 2012 Page 214 Name_________________________________________Date____________________Class__________ Day #21 Homework In exercises 1 – 6, find the derivative of each of the following functions. x5 1. f ( x) x2 2 3 3. h( x) x 2 3x 1 2. f ( x) 2x 3 x2 4. g ( x) 3 9 x 2 4 Daily Lessons and Assessments for AP* Calculus AB, A Complete Course Mark Sparks 2012 Page 215 5. f ( x) x 1 x 2 6. p( x) x x2 1 For exercises 7 and 8, find the value of the derivative of the function at the given point. 7. g ( ) 1 sin 2 2 when 8. f ( ) sin 2 cos 2 when 4 Daily Lessons and Assessments for AP* Calculus AB, A Complete Course Mark Sparks 2012 4 Page 216 The graph of the function f ( x) 25 x 2 is pictured to the right. Use this function to complete exercises 9 – 11. 9. Find the values of f (3) and f ' (3) . 10. Find the equation of the line tangent to the graph of f(x) when x = 3 and graph this line on the grid with f(x). 11. Find the equation of the normal line to the graph of f(x) when x = 3 and graph this line on the grid with f(x). 12. At what value(s) of x does the graph of h( x) 1 x 2 2 x 1 have a horizontal tangent? Show your 2 work and give a reason for your answer. Daily Lessons and Assessments for AP* Calculus AB, A Complete Course Mark Sparks 2012 Page 217 Use the table below to complete exercises 13 – 14. x f(x) f ’(x) g(x) g’(x) –2 1 –1 2 4 –1 3 –2 1 1 0 –1 2 –2 –3 13. If H ( x) f ( x) g ( x) , is the graph of H(x) increasing or decreasing when x = –1? Give a reason for your answer. 14. If P( x) 2 f ( x) g ( x) 3 , what is the value of P ' (0) ? 2 15. Find the equation of the normal line to the graph of h( x) tan(3x) when x . 12 Daily Lessons and Assessments for AP* Calculus AB, A Complete Course Mark Sparks 2012 Page 218 Problems to Discuss before Quiz #4 Problem #1 Find the following limit. Explain the reasoning that you used to arrive at your answer. cos 3( x h) cos 3x h h 0 lim Problem #2 Find the equation of the tangent line to the graph of the given function when x . f ( x) 3x cos x Daily Lessons and Assessments for AP* Calculus AB, A Complete Course Mark Sparks 2012 3 Page 219 Problem #3 Find the equation of the normal line to the graph of the function below when x = –2. f ( x) 3 3 x 2 Problem #4 At what point on the graph of the function f ( x) 3x 2 is the normal line perpendicular to the line defined by the equation y 1 x 3 ? 4 Daily Lessons and Assessments for AP* Calculus AB, A Complete Course Mark Sparks 2012 Page 220 AP Calculus Quiz #4 Answer Key & Rubrics Multiple Choice ** 1. D A 2. A B 3. C B 4. E C 5. C A 6. D E 7. A D Free Response Part A – 3 points total ____ 1 Correctly finds f ' ( x) 1 1 1 → f ' (4) → The slope of the tangent line 2x 1 2(4) 1 3 ____ 1 Correctly finds f (4) 2(4) 1 3 → The point of tangency ____ 1 Equation of the tangent line: y 3 1 ( x 4) 3 Free Response Part B – 3 points total 6 ____ 1 Correctly finds g ' ( x) sin x → g ' sin 1 → The slope of the tangent line which 6 2 Therefore the slope of the normal line would be 2. 6 ____ 1 Correctly finds g cos 6 3 → The point of tangency 2 ____ 1 Equation of the normal line: y 3 2 x 2 6 Free Response Part C – 3 points total 1 ____ 1 Correctly differentiates to find h' ( x) sin x 2 x 1 cos x 1 (2 x 1) 2 (2) 2 ____ 1 Correctly evaluates h' 3 3 1 2 2 ____ 1 Since h' 3 0 , then the graph of h(x) is increasing at x = 3 2 Daily Lessons and Assessments for AP* Calculus AB, A Complete Course Mark Sparks 2012 Page 221 AP CALCULUS QUIZ #4 Name______________________________________________________Date_____________________ Calculator NOT Permitted 1. 2. 3. 4. 5. 6. 7. MULTIPLE CHOICE 1. Find lim tan 3( x h) tan 3x . h 0 h A. sec 3x tan 3x B. sec2 3x D. 3sec2 3x E. Does not exist C. tan 3x 2. If f ( x) 2 x sin x , then what is the slope of the tangent line to the graph of f(x) when x = ? 2 A. 2 B. –2 C. 0 D. 12 E. 12 Daily Lessons and Assessments for AP* Calculus AB, A Complete Course Mark Sparks 2012 Page 222 3. When x = −1, then the graph of f ( x) 2 x x 2 … A. is undefined. B. is increasing. C. is decreasing. D. has a vertical tangent. E. has a horizontal tangent. 4. What is the slope of the normal line to the graph of the function g ( x) x 2 2 x when x = 4? A. 3 8 B. 8 2 C. 3 8 D. 3 8 8 E. 3 3x 2 x 5. If f ( x) 2 , then f ' ( x) is 3x x A. 1 B. 6 x 1 6x 1 C. 6 (3x 1) 2 2x2 D. ( x 2 x) 2 36 x 3 2 x E. ( x 2 x) 2 Daily Lessons and Assessments for AP* Calculus AB, A Complete Course Mark Sparks 2012 Page 223 6. If f ( ) 2 3 cos , what is f ' ( ) ? A. f ' ( ) C. f ' ( ) E. f ' ( ) 1 3sin B. f ' ( ) D. f ' ( ) 1 3 cos 2 3 sin 1 3 sin 1 3 sin 7. The graph of f(x) and a table of values for g(x) and g ' ( x) are given below. If h(x) = f ( x) g ( x) , what is the slope of the line tangent to the graph of h(x) when x = –3? x g(x) g ' ( x) –3 4 –2 0 –1 –3 3 2 –4 5 4 1 A. 1 8 13 B. 2 6 8 2 D. 1 2 8 C. E. 6 Daily Lessons and Assessments for AP* Calculus AB, A Complete Course Mark Sparks 2012 Page 224 FREE RESPONSE Consider the functions f ( x) 2 x 1 and g(x) = cos x to answer the following questions. a. Find the equation of the tangent line to the graph of f(x) when x = 4. b. Find the equation of the normal line to the graph of g(x) when x = 6 Daily Lessons and Assessments for AP* Calculus AB, A Complete Course Mark Sparks 2012 Page 225 c. Suppose that h(x) = g(x) ∙ f(x). Is h(x) increasing or decreasing at x = 3 ? Show your work and 2 give a reason for your answer. Daily Lessons and Assessments for AP* Calculus AB, A Complete Course Mark Sparks 2012 Page 226 AP CALCULUS *QUIZ #4* Name______________________________________________________Date_____________________ Calculator NOT Permitted 1. 2. 3. 4. 5. 6. 7. MULTIPLE CHOICE 1. Find lim sec 2( x h) sec 2 x . h 0 h A. 2sec 2x tan 2x B. sec 2x D. sec 2x tan 2x E. Does not exist C. –2sec2 2x 2. If f ( x) 2 x cos x , then what is the slope of the tangent line to the graph of f(x) when x = ? 2 A. π B. –π C. 0 D. –2 E. 12 Daily Lessons and Assessments for AP* Calculus AB, A Complete Course Mark Sparks 2012 Page 227 3. When x = 2, the graph of f ( x) 2 x 4 x 3 … A. is undefined. B. is increasing. C. is decreasing. D. has a vertical tangent. E. has a horizontal tangent. 4. What is the slope of the normal line to the graph of the function g ( x) x 2 2 when x = 4? A. B. C. D. E. 4 12 14 4 14 4 4 14 14 2 3x 2 x 5. If f ( x) 2 , then f ' ( x) is 3x x A. 6 (3x 1) 2 B. 6 x 1 6x 1 C. 1 D. 2x2 ( x 2 x) 2 36 x 3 2 x E. ( x 2 x) 2 Daily Lessons and Assessments for AP* Calculus AB, A Complete Course Mark Sparks 2012 Page 228 6. If f ( ) 2 3 sin , what is f ' ( ) ? A. f ' ( ) 1 3 sin B. f ' ( ) C. f ' ( ) D. f ' ( ) E. f ' ( ) 1 3 cos 2 3 cos 1 3 cos 1 3 cos 7. The graph of f(x) and a table of values for g(x) and g ' ( x) are given below. If h(x) = f ( x) g ( x) , what is the slope of the line tangent to the graph of h(x) when x = 3? x g(x) g ' ( x) –3 4 –2 0 –1 –3 3 2 –4 5 4 1 1 6 B. 1 2 8 A. 8 2 13 D. 2 6 C. E. 6 Daily Lessons and Assessments for AP* Calculus AB, A Complete Course Mark Sparks 2012 Page 229 FREE RESPONSE Consider the functions f ( x) 2 x 1 and g(x) = cos x to answer the following questions. a. Find the equation of the tangent line to the graph of f(x) when x = 4. b. Find the equation of the normal line to the graph of g(x) when x = 6 Daily Lessons and Assessments for AP* Calculus AB, A Complete Course Mark Sparks 2012 Page 230 c. Suppose that h(x) = g(x) ∙ f(x). Is h(x) increasing or decreasing at x = 3 ? Show your work and 2 give a reason for your answer. Daily Lessons and Assessments for AP* Calculus AB, A Complete Course Mark Sparks 2012 Page 231 Rules for Differentiation Finding the Derivative of the Natural Exponential and Logarithmic Functions Differentiation Rule for Natural Exponential Functions Find the derivative of each of the following functions. f ( x) e sin x f ( x) e 2 x 3 f ( x) 3e 2 x f ( x) (2 x 3)e 3x f ( x) x 2 e 2 x f ( x) e 2 x 6 Daily Lessons and Assessments for AP* Calculus AB, A Complete Course Mark Sparks 2012 Page 232 f ( x) e5x 3x 2 Differentiation Rule for Natural Logarithmic Functions Find the derivative of each of the following functions. f ( x) ln(2 x 3) f ( x) ln 3x 2 2 x f ( x) ln(cos x) f ( x) ln 2 x 4 Daily Lessons and Assessments for AP* Calculus AB, A Complete Course Mark Sparks 2012 Page 233 Finding Values of Derivatives Using the Graphing Calculator For each of the functions below, find the value of f ' ( x) at the indicated value of x using the graphing calculator. Then, determine if the function is increasing, decreasing, has a horizontal tangent or has a vertical tangent. Give a reason for your answer. Function Value of f ' (a) 1. a = –2 Is f(x) increasing or decreasing, or does f(x) have a horizontal or a vertical tangent? f ( x) 3e x sin x 2. a=1 f ( x) 3e x sin x a= 3. f ( x) 3 ln(cos x) x2 4. f ( x) a=π ln(cos x) x2 5. a=0 f ( x) e tan(0.34 x) 6. a=1 f ( x) 5 sin 2 (ln x) Daily Lessons and Assessments for AP* Calculus AB, A Complete Course Mark Sparks 2012 Page 234 When the value of the derivative of a function is positive, we say that the function is increasing. When the value of the derivative of a function is negative, we say that the function is decreasing. When speaking of quantities increasing or decreasing, they do so at a certain rate. We already understand the derivative to be the SLOPE OF THE TANGENT LINE. Slope is a rate. Therefore, the derivative of a function actually represents the RATE AT WHICH A FUNCTION IS CHANGING. 7. a. b. The number of people entering a concert can be modeled by the function f (t ) 560e sin t , where t represents the number of hours after the gates are open. Find the values of f 1 and f ' 1 . Using correct units, explain what each value represents in the 2 2 context of this problem. How many people have entered the concert 2 hours after the gates are opened? Is the number of people entering increasing or decreasing at this time? Justify your answer. After being poured into a cup, coffee cools so that its temperature, T(t), is represented by the 8. function T (t ) 70 110e 2 , where t is measured in minutes and T(t) is measured in degrees Fahrenheit. What is the temperature of the coffee 5 minutes after it has been poured into the cup? t a. Is the temperature decreasing faster 1 minute after it is poured or 3 minutes after it is poured? Give a reason for your answer. b. Daily Lessons and Assessments for AP* Calculus AB, A Complete Course Mark Sparks 2012 Page 235 Name_________________________________________Date____________________Class__________ Day #23 Homework In exercises 1 – 10, find the derivative of the function. Express your answer in simplest factored form. 1. F ( x) x 3e 2 x 2. P( x) e 2 x 3. H ( x) e x ln x 4. g ( x) 2 x 2 3 e x 5. J ( x) ln e 2 x 1 2 6. F ( x) ln(3 2 x) Daily Lessons and Assessments for AP* Calculus AB, A Complete Course Mark Sparks 2012 Page 236 7. K ( x) ln 5x 2 8. F ( x) x 2 e 4 x 9. T ( x) ln x 10. P( x) x2 e2x x3 11. Find the equation of the tangent line to the graph of y ln x when x = 1. 4x Daily Lessons and Assessments for AP* Calculus AB, A Complete Course Mark Sparks 2012 Page 237 The Relationship between Continuity and Differentiability In this lesson, our goal is to establish a relationship between a function being continuous at a value of x and a function being differentiable at the same value. In other words, if a function is continuous at a particular value of x, does that imply that it is also differentiable. Or, if a function is differentiable, does that mean that it must also be continuous. Let’s investigate three functions. Consider the function f ( x) x 2 4 at x = 2. Answer the questions that follow. On the grid to the right, sketch a graph of f(x) from your graphing calculator. Based on the graph, if f(x) continuous at x = 2? Explain your reasoning. Find the value of f ' (2) to determine if f(x) is differentiable at x = 2. Daily Lessons and Assessments for AP* Calculus AB, A Complete Course Mark Sparks 2012 Page 238 1 Consider the function f ( x) x 3 2 at x = 0. Answer the questions that follow. On the grid to the right, sketch a graph of f(x) from your graphing calculator. Based on the graph, if f(x) continuous at x = 0? Explain your reasoning. Find the value of f ' (0) to determine if f(x) is differentiable at x = 0. 2 Consider the function f ( x) x 3 2 at x = 0. Answer the questions that follow. On the grid to the right, sketch a graph of f(x) from your graphing calculator. Based on the graph, if f(x) continuous at x = 0? Explain your reasoning. Find the value of f ' (0) to determine if f(x) is differentiable at x = 0. Daily Lessons and Assessments for AP* Calculus AB, A Complete Course Mark Sparks 2012 Page 239 Based on what we have seen, does continuity imply differentiability or does differentiability imply continuity? In order for a function to be differentiable at a value of x, then two things must be true: 1.___________________________________________________________________________________ 2.___________________________________________________________________________________ x 1, 0 x 3 to answer the following questions. 5 x, 3 x 5 Consider the function g ( x) Is g(x) continuous at x = 3? Show the complete analysis. Is g(x) differentiable at x = 3? Show the complete analysis. Daily Lessons and Assessments for AP* Calculus AB, A Complete Course Mark Sparks 2012 Page 240 For what values of k and m will the function below be both continuous and differentiable at x = 3? k x 1, 0 x 3 h( x ) mx 2, 3 x 5 For what values of a and b will the function below be differentiable at x = 1? 2 3ax 2bx 1, f ( x) 4 2 ax 4bx 3x, x 1 x 1 Daily Lessons and Assessments for AP* Calculus AB, A Complete Course Mark Sparks 2012 Page 241 Name_________________________________________Date____________________Class__________ Day #24 Homework Use the graph of H(x), pictured to the right, to complete exercises 1 – 4. 1. Graphically, identify a value of x at which the function is continuous but not differentiable. Give a reason for your answer. 2. Write an equation of H(x) and show analytically that H(x) is, in fact, continuous at the x – value that you identified in exercise 1. Show and explain your work. 3. Show analytically that H(x) is, in fact, not differentiable at the x – value that you identified in exercise 1. Show and explain your work. 4. Given the graph of H(x) pictured above, find the equation of the tangent line to the graph of P( x) H ( x) when x = 3. Daily Lessons and Assessments for AP* Calculus AB, A Complete Course Mark Sparks 2012 Page 242 A continuous function on the interval −4 < x < 5, h(x), is described in the table below. Use the information to complete exercises 5 – 8. x h(x) –4 –5 –2 –4 –1 –2 0 –4 < x < 0 1 Increasing & Concave Up 1 –1 3 0<x<3 3<x<5 5 –2 Decreasing & Concave Up Increasing & Concave Up 0 5. Sketch a graph of h(x). 6. Estimate the value of h ' (2) . Does this value support the claim that h(x) is increasing on the interval –4 < x < 0? Give a reason for your answer. 7. There are three x – values in the domain of h at which h(x) is not differentiable. What are these three values and give a reason for why h(x) is not differentiable at these values. 8. On what interval(s) of x is h ' ( x) 0 ? Give a reason for your answer. Daily Lessons and Assessments for AP* Calculus AB, A Complete Course Mark Sparks 2012 Page 243 9. At what value(s) of x will the graph of f(x) = 2e2x – 3x have a tangent line whose slope is 1? 10. The graph of x – 2y = 9 is parallel to the normal line to the graph of f(x) when x = 5. What is the value of f ' (5) ? Justify your answer. 3 x, x 1 f ( x ) 11. Let f be defined by the function . 2 ax bx, x 1 a. If the function is continuous at x = 1, what is the relationship between a and b? Explain your reasoning using limits. b. Find the unique values of a and b that will make f both continuous and differentiable at x = 1. Show your analysis using limits. Daily Lessons and Assessments for AP* Calculus AB, A Complete Course Mark Sparks 2012 Page 244 Derivatives of Inverse Functions From earlier courses, let’s take just a moment and remember what inverse functions are. Given a function, f(x), the inverse function, f 1( x), is numerically defined to be __________________________ ____________________________________________________________________________________. Graphical Representation of the Inverse Analytical Representation of the Inverse Consider the two functions, f(x) and g(x), represented numerically below. Answer the questions that follow. x f(x) g(x) Complete the table of values below. x f −1 (x) Find the value of f 1 f (1) . −2 3 1 1 2 −2 x Find the value of f g 1 (2) . Complete the table of values below. g−1(x) Find the value of g 1 f 1(2) . Find the value of f 1 g g 1 (1) . Daily Lessons and Assessments for AP* Calculus AB, A Complete Course Mark Sparks 2012 Page 245 Given below is the relationship that exists when a composite function is formed using a function and its inverse. Use the chain rule to differentiate both sides of the equation and find the formula for the derivative of f 1( x) . Finding a Formula for the Derivative of an Inverse Differentiate both sides of the equation below. f [ f 1 ( x)] x Suppose that f(x) = 3x + 2 and f ' (2) 3 . What is the value of [ f 1(4)] ' ? Given to the right is a table of values for f, g, f ' , and g ' . Use the values in the table to find each indicated value in the boxes below. Find the value of [ f 1(3)] ' . x f g f' g' −2 1 2 0 3 0 −4 −3 −1 2 1 3 −2 2 1 3 1 1 −3 −2 Find the value of [ g 1(2)] ' . Daily Lessons and Assessments for AP* Calculus AB, A Complete Course Mark Sparks 2012 Page 246 The functions, f and g, are differentiable functions and selected values of them and their first derivatives, f ' and g ' , are shown in the table below. Use the table of values to answer the questions that follow. x f g f' g' −2 1 2 0 3 0 −4 −3 −1 2 1 3 −2 2 1 3 1 1 −3 −2 Find the value of [ g 1(1)] ' . Then, find the equation of the line tangent to the graph of g −1 when x = 1. Estimate the value of g ' (1) . Based on this value, what conclusion can be reached about the graph of g when x = −1? Explain your reasoning. Estimate the value of g ' ' (1). Based on this value, what conclusion can be reached about the graph of g ' when x = −1? Explain your reasoning. Daily Lessons and Assessments for AP* Calculus AB, A Complete Course Mark Sparks 2012 Page 247 Name_________________________________________Date____________________Class__________ Day #25 Homework For questions 1 – 4, refer to the table of values to the right. 1. Find the value of [ f 1 (2)] ' . x f f' g g' −3 −1 2 1 1 1 2 3 0 −2 2 −3 2 5 −2 −6 2 2. Find the value of [ g 1 (1)] ' 3. Find the equation of the line tangent to the graph of g 1 when x = −2. 4. If h(x) = g(f(x)), what is the equation of the line normal to the graph of h when x = 2? Daily Lessons and Assessments for AP* Calculus AB, A Complete Course Mark Sparks 2012 Page 248 Daily Lessons and Assessments for AP* Calculus AB, A Complete Course Mark Sparks 2012 Page 249 Unit #3 Problems for Review 2 1. Find the value of f ' (0) if f(x) = x 3 + 1. Then, sketch a graph of the function f(x) and use the graph to explain your analytical result for f ' (0) . 2. Find the value of f ' (2) if f ( x) 4 2 x . Then, sketch a graph of the function f(x) and use the graph to explain your analytical result for f ' (2) . 3. Find each of the indicated limits below. (Hint, remember the definition and alternate definition of a limit and what each tells you about a function.) a. lim ln( x h) ln x h 0 h b. lim sin x sin 2 x 2 x d. lim 3 2( x h) 3 3 2 x 3 h 0 h cos 4 h cos 4 h 0 h c. lim 2 4. The equation of the normal line to the graph of y = e2x when dy 2 is… dx A. y 12 x 1 B. y 12 x 1 C. y = 2x + 1 D. y 12 x ln22 2 E. y 2 x ln22 2 Daily Lessons and Assessments for AP* Calculus AB, A Complete Course Mark Sparks 2012 Page 250 5. Given that f(x) = x2ex, what is an approximate value of f(1.1) if you use the equation of the tangent line to the graph of f at x = 1? A. 3.534 B. 3.635 C. 7.055 D. 8.155 E. 10.244 6. If f ( x) 5 cos ( x) , then f ' 2 is … 2 A. 0 B. 23 C. 32 D. 56 E. 56 7. For what value(s) of k does the graph of g(x) = ke2x + 3x have a normal line whose slope is 15 when x = 1? Daily Lessons and Assessments for AP* Calculus AB, A Complete Course Mark Sparks 2012 Page 251 The graph of the first derivative, f ' ( x) , of a polynomial function, f(x), is pictured to the right. Use the graph to answer questions 8 – 11. 8. What type of polynomial function is f(x)? Give a reason for your answer. 9. At what value(s) of x does the graph of f(x) have a horizontal Graph of f ' ( x) tangent? Give a reason for your answer. 10. On what interval(s) of x would the graph of f(x) be increasing? Give a reason for your answer. 11. On what interval(s) of x would the graph of f(x) be decreasing? Give a reason for your answer. 12. Find two values on the interval (0, 2π) where the slope of the tangent to the graph of f(x) = cos 2x is equal to 3. Daily Lessons and Assessments for AP* Calculus AB, A Complete Course Mark Sparks 2012 Page 252 13. Consider the piece-wise defined function below to answer the questions that follow. ax 2 bx 2, x 2 f ( x) x2 ax b, a. If a = –3 and b = 4, will f(x) be continuous at x = 2? Justify your answer. b. If a = –3 and b = 4, will f(x) be differentiable at x = 2? Justify your answer. c. For what value(s) of a and b will f(x) be both continuous and differentiable at x = 2? Show your work. Daily Lessons and Assessments for AP* Calculus AB, A Complete Course Mark Sparks 2012 Page 253 14. A rodeo performer spins a lasso in a circle perpendicular to the ground. The height from the ground of the knot, measured in units of feet, in the lasso is modeled by the function H (t ) 3 cos 5 t 5 , 3 where t is the time measured in seconds after the lasso begins to spin. a. Find the value of H(0.75). Using correct units, explain what this value represents in the context of this problem. b. Find the value of H ' (0.75) . Using correct units, explain what this value represents in the context of this problem. Daily Lessons and Assessments for AP* Calculus AB, A Complete Course Mark Sparks 2012 Page 254 AP Calculus Quiz #5 Answer Key & Rubric Multiple Choice ** 1. E B 2. D C 3. D D 4. D B 5. B E 6. C B 7. A E Free Response Part A – 2 points total f ( 2) f (1) ____ 1 f ' (1.5) 21 513 2 ____ 1 f ' (1.5) is best approximated by finding the slope of a secant line passing through two points on the graph of f(x) that lie on either side of x = 1.5 and the slope of the secant line should be approximately the same as the secant line is closely parallel to the tangent line. Free Response Part B – 2 points total ____ 1 Finds B' ( x) g ' ( x) 2 g ( x) and evaluates B' (1) g ' (1) 2 g (1) 3 to find the slope of the tangent line 2 3 3 ( x 1) ____ 1 Equation of the tangent line using B(1) and B' (1) : y 3 2 3 Free Response Part C – 3 points total ____ 1 Correctly finds A' ( x) 2 x ln( f ( x)) x 2 ____ 1 A' (2) 2(2) ln( f (2)) (2) 2 f '( x ) f ( x) f '( 2) 4 ln(5) 4 3 4 ln 5 12 f (2) 5 5 ____ 1 Since A' (2) 0 , then the graph of A(x) is increasing when x = 2. Free Response Part D – 2 points total ____ 1 Correctly finds [ g 1 (3)] ' 1 1 1 3 g ' [ g 1 (3)] g '(1) ____ 1 Correct equation of the normal line: y – 1 = 3(x – 3) Daily Lessons and Assessments for AP* Calculus AB, A Complete Course Mark Sparks 2012 Page 255 AP CALCULUS QUIZ #5 Name______________________________________________________Date_____________________ Calculator Permitted 1. 2. 3. 4. 5. 6. 7. MULTIPLE CHOICE 1. Which of the following statements can be made about the graph of the function h( x) ln(cos x) when x= . tan x 2 A. The graph of h(x) is increasing. B. The graph of h(x) is decreasing. C. The graph of h(x) has a vertical tangent. D. The graph of h(x) has a horizontal tangent. E. No conclusion can be made about the graph of h(x). Daily Lessons and Assessments for AP* Calculus AB, A Complete Course Mark Sparks 2012 Page 256 2. Consider the graph of f(x) to the right to determine which of the following statements is/are true. I. f ' ( x) = 0 when x = 3. II. f ' (2.5) 0 . III. On the interval 4,5 there are three values of x at which f(x) is not differentiable. A. I only B. I and II only C. III only D. II and III only E. I, II and III 3. Let f(7) = 0, f ' (7) = 14, g(7) = 1 and g ' (7) = 1 . Find h ' (7) if h(x) = 7 f ( x) . g ( x) A. 98 B. –14 C. –2 D. 14 E. Cannot be determined 4. The graph to the right shows data of a function, H(t), which shows the relationship between temperature in C (y-axis) and the time in hours (x-axis). What does the value of H ' (6) represent? A. H ' (6) represents the temperature after 6 hours measured in C. B. H ' (6) represents the rate at which the temperature is changing after 6 hours measured in C. C. H ' (6) represents the temperature after 6 hours measured in C per hour D. H ' (6) represents the rate at which the temperature is changing after 6 hours measured in C per hour. E. H ' (6) represents the amount of temperature change over the first 6 hours measured in C Daily Lessons and Assessments for AP* Calculus AB, A Complete Course Mark Sparks 2012 Page 257 5. The graph of h( x) 2 x 4 2 is pictured below. What is the value of [h 1 (6)] ' ? A. 1 4 B. 2 C. 4 D. 1 2 E. 0 6. Find y ' if y x 2 e x . A. 2 xe x B. x x 2e x C. xe x x 2 x D. 2 x e E. 2 x e 7. The function f is pictured to the right. At which of the following values of x is f defined and continuous but not differentiable. I. x = 2 II. x = 3 III. x = 5 A. II only B. I only C. II and III only D. I and II only E. III only Daily Lessons and Assessments for AP* Calculus AB, A Complete Course Mark Sparks 2012 Page 258 FREE RESPONSE The table below shows values of differentiable functions, f(x) and g(x), and their derivatives at selected values of x. Use the table of values below to answer each of the questions below. x f (x) f ' ( x) g(x) g ' ( x) 0 3 −1 2 5 1 3 2 3 −3 2 5 3 1 −2 3 10 4 0 −1 a. Approximate the value of f ' (1.5) ? Explain why your answer is a good approximation of f ' (1.5) . b. If B(x) = g (x) , what is the equation of the tangent line drawn to B(x) when x = 1? Daily Lessons and Assessments for AP* Calculus AB, A Complete Course Mark Sparks 2012 Page 259 c. If A( x) x 2 ln f ( x) , what is the value of A' (2) ? What does this result say about the behavior of the graph of A(x) when x = 2? Give a reason for your answer. d. Find the value of [ g 1 (3)] ' . Then, find the equation of the line normal to the graph of g−1(x) at x = 3. Daily Lessons and Assessments for AP* Calculus AB, A Complete Course Mark Sparks 2012 Page 260 AP CALCULUS *QUIZ #5* Name______________________________________________________Date_____________________ Calculator Permitted 1. 2. 3. 4. 5. 6. 7. MULTIPLE CHOICE 1. Which of the following statements can be made about the graph of the function h( x) ln(cos x) when x= . tan x 4 A. The graph of h(x) is increasing. B. The graph of h(x) is decreasing. C. The graph of h(x) has a vertical tangent. D. The graph of h(x) has a horizontal tangent. E. No conclusion can be made about the graph of h(x). Daily Lessons and Assessments for AP* Calculus AB, A Complete Course Mark Sparks 2012 Page 261 2. Consider the graph of f(x) to the right to determine which of the following statements is/are true. I. f ' ( x) = 0 when x = 3. II. f ' (2.5) 0 . III. On the interval 4,5 there are three values of x at which f(x) is not differentiable. A. II and III only B. I and II only C. III only D. I only E. I, II and III 3. Let f(7) = 0, f ' (7) = 14, g(7) = 1 and g ' (7) = 1 . Find h ' (7) if h(x) = 7 f ( x) . g ( x) A. 98 B. –2 C. –14 D. 14 E. Cannot be determined 4. The graph to the right shows data of a function, H(t), which shows the relationship between temperature in C (y-axis) and the time in hours (x-axis). What does the value of H ' (6) represent? A. H ' (6) represents the temperature after 6 hours measured in C. B. H ' (6) represents the rate at which the temperature is changing after 6 hours measured in C per hour. C. H ' (6) represents the temperature after 6 hours measured in C per hour D. H ' (6) represents the rate at which the temperature is changing after 6 hours measured in C. E. H ' (6) represents the amount of temperature change over the first 6 hours measured in C Daily Lessons and Assessments for AP* Calculus AB, A Complete Course Mark Sparks 2012 Page 262 5. The graph of h( x) 2 x 4 2 is pictured below. What is the value of [h 1 (4)] ' ? A. 1 4 B. −3 C. 4 D. 1 3 E. 1 6. Find y ' if y x 2 e x . A. 2 xe x B. xe x x 2 C. x x 2e x x D. 2 x e E. 2 x e 7. The function f is pictured to the right. At which of the following values of x is f defined, continuous, and differentiable. I. x = 2 II. x = 3 III. x = 5 A. II only B. I only C. II and III only D. I and II only E. III only Daily Lessons and Assessments for AP* Calculus AB, A Complete Course Mark Sparks 2012 Page 263 FREE RESPONSE The table below shows values of differentiable functions, f(x) and g(x), and their derivatives at selected values of x. Use the table of values below to answer each of the questions below. x f (x) f ' ( x) g(x) g ' ( x) 0 3 −1 2 5 1 3 2 3 −3 2 5 3 1 −2 3 10 4 0 −1 a. Approximate the value of f ' (1.5) ? Explain why your answer is a good approximation of f ' (1.5) . b. If B(x) = g (x) , what is the equation of the tangent line drawn to B(x) when x = 1? Daily Lessons and Assessments for AP* Calculus AB, A Complete Course Mark Sparks 2012 Page 264 c. If A( x) x 2 ln f ( x) , what is the value of A' (2) ? What does this result say about the behavior of the graph of A(x) when x = 2? Give a reason for your answer. d. Find the value of [ g 1 (3)] ' . Then, find the equation of the line normal to the graph of g−1(x) at x = 3. Daily Lessons and Assessments for AP* Calculus AB, A Complete Course Mark Sparks 2012 Page 265 AP Calculus Test #3 Free Response Rubrics Multiple Choice Calculator Permitted 1. 2. 3. 4. 5. 6. 7. E E E A D D C ** D A A C B E D Raw Score to Percentage Conversion Calculator Not Permitted 1. 2. 3. 4. 5. 6. 7. A C D C A E C ** C D A E C D B Calculator Permitted Free Response Part A – 2 points total _____ 1 Correctly finds d(t) = 0 when t = 12 meters by either setting the equation equal to 0 and solving algebraically or by looking at the graph or table of values on the graphing calculator. _____ 1 The tank will be completely empty by 6:30 a.m. Calculator Permitted Free Response Part B – 2 points total 2 3.190 12 2 _____ 1 Finds the value of the value of d(0) = 61 0 6 , which is the initial depth of the 12 _____ 1 Finds the value of d(3.25) = 6 1 3.25 water and finds that at 1:45, the depth of the water in the tank is d(0) – d(3.25) = 2.810 meters Calculator Permitted Free Response Part C – 2 points total _____ 1 Correctly finds d ' (2) 0.833 meters per hour _____ 1 d ' (2) represents the rate at which the depth of the liquid waste is decreasing 2 hours after the drainage valve is opened. Calculator Permitted Free Response Part D – 3 points total _____ 1 Correctly finds d ' (0.5) 0.958 meters per hour _____ 1 Compares d ' (2) 0.833 and d ' (0.5) 0.958 to determine that water is draining out of the tank faster after 30 minutes _____ 1 Justification: Since d ' (0.5) is “more negative” than d ' (2) , the water is draining faster when t = 0.5. [DO NOT award point if student says d ' (0.5) > d ' (2) .] Daily Lessons and Assessments for AP* Calculus AB, A Complete Course Mark Sparks 2012 Page 266 Calculator NOT Permitted Free Response Part A – 3 points total ____ 1 f(1) is defined and f(1) = 1. ____ 1 lim f ( x) exists because lim f ( x) = lim f ( x) 1 . x 1 x 1 x1 ____ 1 f(x) is continuous at x = 1 because f(1) = lim f ( x) . x 1 Calculator NOT Permitted Free Response Part B – 3 points total ____ 1 From part a), it is known that f(x) is continuous at x = 1. ____ 1 lim f ' ( x) does not exist because lim f ' ( x) = 0 lim f ' ( x) = 5 x 1 x 1 x 1 ____ 1 f(x) is not differentiable at x = 1 because lim f ' ( x) does not exist. x 1 Calculator NOT Permitted Free Response Part C – 3 points total 2 2 x, x 1 ____ 1 Finds and uses f ' ( x) 2 x k , x 1 ____ 1 k = –2 ____ 1 p = 2 Daily Lessons and Assessments for AP* Calculus AB, A Complete Course Mark Sparks 2012 Page 267 AP CALCULUS AB TEST #3 Unit #3 – Rules of Differentiation Name______________________________________________________Date_____________________ A GRAPHING CALCULATOR IS REQUIRED FOR SOME PROBLEMS OR PARTS OF PROBLEMS IN THIS PART OF THE EXAMINATION. (1) The exact numerical value of the correct answer does not always appear among the choices given. When this happens, select from among the choices the number that best approximates the exact numerical value. (2) Unless otherwise specified, the domain of a function f is assumed to be the set of all real numbers x for which f(x) is a real number. Multiple Choice x 2 , x3 is … 6 x 9, x 3 1. At x = 3, the function f ( x) A. undefined. B. continuous but not differentiable. C. differentiable but not continuous. D. neither continuous nor differentiable. E. both continuous and differentiable. Daily Lessons and Assessments for AP* Calculus AB, A Complete Course Mark Sparks 2012 Page 268 2. What is the point on the graph of f ( x) 2 x 1 at which the normal line is parallel to the line y = –3x + 6? 5) A. (2, B. (0, 1) C. (1, 3 ) D. (4, –6) E. (4, 3) 3. What is an equation of the tangent line to the graph of y = cos(2x) at x = ? 4 A. y 1 x 4 B. y 1 2x 4 C. y 2x 4 D. y x 4 E. y 2x 4 4. If f(x) is the function given by f(x) = e3x + 1, at what value of x is the slope of the tangent line equal to 2? A. –0.135 B. 0 C. 0.231 D. –0.366 E. 0.693 Daily Lessons and Assessments for AP* Calculus AB, A Complete Course Mark Sparks 2012 Page 269 5. If f ( x) A. B. e2x , what is f ' x ? 2x e 2 x (1 2 x ) 2x2 e 2 x ( 2 x 1) x2 C. 1 D. e 2 x ( 2 x 1) E. e 2x2 2x For questions 6 and 7, use the graph of the function g(x), shown to the right. 6. At which of the following values of x is g ' ( x) 0 ? I. x = –2 II. x = 2 III. x = 3 A. I and III only B. I only C. II and III only D. III only E. I and II only 7. Which of the following statements is/are true? I. If y = ax + b is the equation of any normal line drawn to the graph of g on the interval ,2 , then a < 0. II. g ' ( x) 0 on the intervals 2,2 and 3, III. g ' ( x) is undefined at x = –2 and x = 2. A. I only B. II only C. II and III only D. I and II only E. I and III only Daily Lessons and Assessments for AP* Calculus AB, A Complete Course Mark Sparks 2012 Page 270 Free Response A storage tank at a factory collects liquid waste and can be drained by opening a drainage valve at the base of the tank. The depth, d(t), of fluid in the tank t hours after the valve is opened is given by the formula below. 12 2 meters d (t ) 6 1 t a. Daily production in the factory ends at 5:30 p.m. If the valve is opened at 6:30 p.m. and remains open, at what time will the tank be completely empty? Explain how you arrived at your answer. b. At 10:30 a.m., the storage tank is completely full and the valve is left open until 1:45 p.m. How many meters of liquid waste have emptied from the tank by 1:45 p.m.? c. What is the value of d ' (2) . Using correct units of measure, explain what this value represents. Daily Lessons and Assessments for AP* Calculus AB, A Complete Course Mark Sparks 2012 Page 271 d. Is waste draining faster after 30 minutes or after 2 hours? Justify your answer. Daily Lessons and Assessments for AP* Calculus AB, A Complete Course Mark Sparks 2012 Page 272 AP CALCULUS AB TEST #3 Unit #3 – Rules of Differentiation Name______________________________________________________Date_____________________ A GRAPHING CALCULATOR IS NOT ALLOWED FOR THIS SECTION OF THE EXAM. (1) The exact numerical value of the correct answer does not always appear among the choices given. When this happens, select from among the choices the number that best approximates the exact numerical value. (2) Unless otherwise specified, the domain of a function f is assumed to be the set of all real numbers x for which f(x) is a real number. Multiple Choice 8. Find f ' (0) if f ( x) ln x 4 e 3x . A. 2 5 B. C. D. 1 5 1 4 2 5 E. nonexistent Daily Lessons and Assessments for AP* Calculus AB, A Complete Course Mark Sparks 2012 Page 273 4 x3 2 find dy at x = 1. 9. If y 2x 5 1 dx A. –28 B. –13 C. –52 D. 13 E. 52 10. If the line x + 7y = 29 is an equation of the normal line to the graph of f(x) at the point (1, 4), then what is the value of f ' (1) ? A. 7 1 B. 1 7 C. 29 D. 7 E. –7 7 11. The lim tan 3( x h) tan 3x is h0 h A. 0 2 B. sec (3x) 2 C. 3 sec (3x) D. 3 cot(3x) E. nonexistent Daily Lessons and Assessments for AP* Calculus AB, A Complete Course Mark Sparks 2012 Page 274 12. Given that f(x) and g(x) are differentiable functions and the following function values are true, find h' (a) if h(x) = f(g(x)). f(a) = –4 g(a) = c g(c) = 10 f(c) = 15 f ' (a) = 8 g ' (a) = b g ' (c) = 5 f ' (c ) = 6 A. 6b B. 8b C. –4b D. 80 E. 15b 13. The piecewise function, f(x), pictured to the right consists of segments of linear functions. Which of the following statements is/are true? I. f ' (4) 2 . II. f ' ( x) < 0 on the intervals (–1, 1) and (2, ∞). III. f(x) is continuous at x = 1, but not differentiable at x = 1. A. I only B. II only C. I and II only D. II and III only E. I, II and III 14. If H(x) = A. f (x) and f(3) = 10 and f ' (3) = 4. What is the value of H ' (3) ? 1 4 1 2 10 C. 2 10 B. D. 2 E. 2 10 Daily Lessons and Assessments for AP* Calculus AB, A Complete Course Mark Sparks 2012 Page 275 Free Response 2 x x , 2 Let f(x) be defined by the function f ( x) 2 x 1 x kx p, x 1 , where k and p are constants. a. If k = 3 and p = –3, is f(x) continuous at x = 1? Justify your answer using limits. b. If k = 3 and p = –3, is f(x) differentiable at x = 1? Justify your answer using limits. c. For what values of k and p is the function f(x) both continuous and differentiable at x = 1? Daily Lessons and Assessments for AP* Calculus AB, A Complete Course Mark Sparks 2012 Page 276 AP CALCULUS AB *TEST #3* Unit #3 – Rules of Differentiation Name______________________________________________________Date_____________________ A GRAPHING CALCULATOR IS REQUIRED FOR SOME PROBLEMS OR PARTS OF PROBLEMS IN THIS PART OF THE EXAMINATION. (1) The exact numerical value of the correct answer does not always appear among the choices given. When this happens, select from among the choices the number that best approximates the exact numerical value. (2) Unless otherwise specified, the domain of a function f is assumed to be the set of all real numbers x for which f(x) is a real number. Multiple Choice x 2 , x3 is … 6 x 9, x 3 1. At x = 3, the function f ( x) A. undefined. B. continuous but not differentiable. C. differentiable but not continuous. D. both continuous and differentiable. E. neither continuous nor differentiable. Daily Lessons and Assessments for AP* Calculus AB, A Complete Course Mark Sparks 2012 Page 277 2. What is the point on the graph of f ( x) 2 x 1 at which the normal line is parallel to the line y = –3x + 6? A. (4, 3) B. (0, 1) C. (1, 3 ) D. (4, –6) E. (2, 5) 3. What is an equation of the tangent line to the graph of y = cos(2x) at x = ? 4 A. y 2x 4 B. y 1 2x 4 C. y 2x 4 D. y x 4 E. y 1 x 4 4. If f(x) is the function given by f(x) = e3x + 1, at what value of x is the slope of the tangent line equal to 2? A. 0 B. 0.231 C. –0.135 D. –0.366 E. 0.693 Daily Lessons and Assessments for AP* Calculus AB, A Complete Course Mark Sparks 2012 Page 278 5. If f ( x) A. B. e2x , what is f ' x ? 2x e 2 x (1 2 x ) 2x2 e 2 x ( 2 x 1) 2x2 C. 1 D. e 2 x ( 2 x 1) E. e x2 2x For questions 6 and 7, use the graph of the function, g(x), shown to the right. 6. At which of the following values of x is g ' ( x) undefined? I. x = –2 II. x = 2 III. x = 3 A. I and III only B. I only C. II and III only D. III only E. I and II only 7. Which of the following statements is/are true? I. If y = ax + b is the equation of any normal line drawn to the graph of g on the interval 3, , then a < 0. II. g ' ( x) 0 on the intervals ,2 and 2,3 . III. g ' ( x) 0 only when x = 3. A. I only B. II only C. II and III only D. I, II, and III E. I and II only Daily Lessons and Assessments for AP* Calculus AB, A Complete Course Mark Sparks 2012 Page 279 Free Response A storage tank at a factory collects liquid waste and can be drained by opening a drainage valve at the base of the tank. The depth, d(t), of fluid in the tank t hours after the valve is opened is given by the formula below. 12 2 meters d (t ) 6 1 t a. Daily production in the factory ends at 5:30 p.m. If the valve is opened at 6:30 p.m. and remains open, at what time will the tank be completely empty? Explain how you arrived at your answer. b. At 10:30 a.m., the storage tank is completely full and the valve is left open until 1:45 p.m. How many meters of liquid waste have emptied from the tank by 1:45 p.m.? c. What is the value of d ' (2) . Using correct units of measure, explain what this value represents. Daily Lessons and Assessments for AP* Calculus AB, A Complete Course Mark Sparks 2012 Page 280 d. Is waste draining faster after 30 minutes or after 2 hours? Justify your answer. Daily Lessons and Assessments for AP* Calculus AB, A Complete Course Mark Sparks 2012 Page 281 AP CALCULUS AB *TEST #3* Unit #3 – Rules of Differentiation Name______________________________________________________Date_____________________ A GRAPHING CALCULATOR IS NOT ALLOWED FOR THIS SECTION OF THE EXAM. (1) The exact numerical value of the correct answer does not always appear among the choices given. When this happens, select from among the choices the number that best approximates the exact numerical value. (2) Unless otherwise specified, the domain of a function f is assumed to be the set of all real numbers x for which f(x) is a real number. Multiple Choice 8. Find f ' (0) if f ( x) ln x 4 e 3x . A. 1 4 B. 1 5 C. 2 5 D. 2 5 E. nonexistent Daily Lessons and Assessments for AP* Calculus AB, A Complete Course Mark Sparks 2012 Page 282 4 x3 2 find dy at x = 1. 9. Find y 2x 5 1 dx A. –28 B. –13 C. 13 D. –52 E. 52 10. If the line x + 7y = 29 is an equation of the normal line to the graph of f(x) at the point (1, 4), then what is the value of f ' (1) ? A. 7 B. 17 C. 1 7 D. 297 E. –7 tan 3( x h) tan 3x is h h0 11. The lim A. 0 B. 3 cot(3x) C. nonexistent D. sec 2 (3x) E. 3sec 2 (3x) Daily Lessons and Assessments for AP* Calculus AB, A Complete Course Mark Sparks 2012 Page 283 12. Given that f(x) and g(x) are differentiable functions and the following function values are true, find h' (a) if h(x) = f(g(x)). f(a) = –4 g(a) = c g(c) = 10 f(c) = 15 f ' (a) = 8 g ' (a) = b g ' (c) = 5 f ' (c ) = 6 A. 8b B. –4b C. 6b D. 80 E. 15b 13. The piecewise function, f(x), pictured to the right consists of segments of linear functions. Which of the following statements is/are true? I. f ' (4) 1. II. f ' ( x) > 0 on the intervals (–∞,–3) and (1, 2). III. f(x) is continuous at x = 1, but not differentiable at x = 1. A. I and II only B. II only C. I only D. II and III only E. I, II and III 14. If H(x) = f (x) and f(3) = 10 and f ' (3) = 4. What is the value of H ' (3) ? A. 1 4 2 10 C. 1 2 10 B. D. 2 E. 2 10 Daily Lessons and Assessments for AP* Calculus AB, A Complete Course Mark Sparks 2012 Page 284 Free Response 2 x x , 2 Let f(x) be defined by the function f ( x) 2 x 1 x kx p, x 1 , where k and p are constants. a. If k = 3 and p = –3, is f(x) continuous at x = 1? Justify your answer using limits. b. If k = 3 and p = –3, is f(x) differentiable at x = 1? Justify your answer using limits. c. For what values of k and p is the function f(x) both continuous and differentiable at x = 1? Daily Lessons and Assessments for AP* Calculus AB, A Complete Course Mark Sparks 2012 Page 285 UNIT #4 – Applications of the Derivative – Part I AP Calculus Day # 28 Objective Differentiate equations that are implicitly defined. 2008 AB #6 (Form B) Parts a, b, and c 29 & 30 31 Solve related rate problems that are in context. Small group discussions talking about related rate problems from Day #29 homework assignment. Note Handouts & Assignments Daily Lessons pages 286 – 290 Day #28 HW: #1 – 8 and 2000 AB #5 Parts a, b, and c Daily Lessons pages 294 – 300 Day #29 & 30 HW: #1 – 21 2002 AB #5 Parts a, b, and c 2002 (Form B) AB #6 Parts a, b, and c 2005 AB #5 (Form B) Parts a, b, and c 32 33 34 35 Test #4: Implicit Differentiation and Related Rates This test only covers two main topics. Thus, it will not be graded using the adjusted curve as other tests in this class are. Use the first and second derivative of a function to analytically determine intervals of increasing/decreasing, concave up/down, coordinates of relative extrema, and coordinates of points of inflection. Use the first and second derivative of a function to analytically determine intervals of increasing/decreasing, concave up/down, coordinates of relative extrema, and coordinates of points of inflection. Understand the graphical and numerical connections between the graphs of F (x) , F ' ( x) , and F ' ' ( x) . Study for Test #4 Free Responses Due Daily Lessons pages 322 – 327 Day #33 & 34 HW: #1 – 3, and 8 – 11 Daily Lessons pages 328 – 329 Day #33 & 34 HW: #4 – 7 Daily Lessons pages 335 – 339 37 2001 AB #4 Parts a, b, c, and d 2006 AB #2 (Form B) Parts a, b, and c 2007 AB #6 Parts a, b, and c Quiz #6 Day #35 HW: #1 – 17 Daily Lessons pages 343 – 345 Test #4 Extra Practice Problems Pages 346 – 348 Study for Unit #4 Test 38 Test #5: Relationships between f ( x ) , f ' ( x ) , and f ' ' ( x ) Unit Free Responses Due 36 Daily Lessons and Assessments for AP* Calculus AB, A Complete Course Mark Sparks 2012 Page 285 Implicit Differentiation Much of our algebraic study of mathematics has dealt with functions. In pre-calculus, we talked about two different types of equations that relate x and y—explicit and implicit. Explicit equations in x and y are such that y is isolated on one side of the equation, and it equals an expression that is totally in terms of x. When I look at an explicitly defined equation, I can EXPLICITLY tell if it represents a function or not. In other words, when I see y 25 x 2 , I know that for every x, there is only one y, making this equation a function. When I see y 25 x 2 , I know that for every x there are two y values, making this equation not a function. Implicit equations are very different. Typically, they do not have y isolated on one side of the equation. Often, there is a power on the y term(s) in the equation and both y’s and x’s may appear throughout the equation. For example, the equation, x2 + y2 – 2x + 4y +16 = 0, is that of a circle, if you will remember. Calculus can even be applied to implicitly defined equations. In this lesson, we will see how to differentiate those equations that are implicitly defined. We will utilize an alternate notation for the derivative. Instead of f ' ( x) , we will use dy . As calculus was dx developed by two different men, a blend of their notations has been accepted. Let’s think about how we differentiate y 25 x 2 . Then, let’s differentiate the implicit form of this equation, x2 + y2 = 25. Daily Lessons and Assessments for AP* Calculus AB, A Complete Course Mark Sparks 2012 Page 286 Consider the graph of the circle to the right. Find the equation of the circle in implicit form below. Now, implicitly differentiate the equation of the circle in the space below. Complete the table below finding the value of dy at each of the indicated points. Then, draw the graphical dx representation, the tangent line, on the graph at each indicated point. (0, 2) (3, 3) (8, –2) (3, –7) (6, 2) (–2 , –2) (6, –6) Daily Lessons and Assessments for AP* Calculus AB, A Complete Course Mark Sparks 2012 Page 287 Find dy for each of the following implicitly defined equations. dx y 2 2x 3 y 2x e y x 2 y 2 2 xy 3 y 2 2 x 5 x 3 3 2 y 3x 2 y Daily Lessons and Assessments for AP* Calculus AB, A Complete Course Mark Sparks 2012 Page 288 For what values of x will the curve x 3 y 3 4 xy 1 have a horizontal tangent? Show your work and explain your thinking. In terms of y, describe the values of x for which the curve x 3 y 3 4 xy 1 will have a vertical tangent? Show your work and explain your thinking. 2 Given the curve y 2 2 y 2 x 1 , find d 2y . dx 2 Given the curve x 2 y 2 1 , find d 2y . Daily Lessons and Assessments for AP* Calculus AB, A Complete Course Mark Sparks 2012 dx Page 289 Daily Lessons and Assessments for AP* Calculus AB, A Complete Course Mark Sparks 2012 Page 290 Name_________________________________________Date____________________Class__________ Day #28 Homework For exercises 1 – 4, find dy by implicit differentiation. dx 1. 2 x 3 y 2 3 y 2. x 3 xy y 2 4 3. x 2 y y 2 x 2 4. 2 sin x cos y 1 Daily Lessons and Assessments for AP* Calculus AB, A Complete Course Mark Sparks 2012 Page 291 5. Find the coordinates of the points where the curve 4 x 2 y 2 8x 4 y 4 0 has a horizontal tangent. 6. Find the coordinates of the points where the curve 4 x 2 y 2 8x 4 y 4 0 has a vertical tangent. 7. Find (a) the equation of the tangent line and (b) the equation of the normal line drawn to the curve 2 2 x 3 y 3 5 at the point (8, 1). Daily Lessons and Assessments for AP* Calculus AB, A Complete Course Mark Sparks 2012 Page 292 2 8. Find d 2y given the curve y 2 x 2 2 x . dx Daily Lessons and Assessments for AP* Calculus AB, A Complete Course Mark Sparks 2012 Page 293 Related Rates Implicitly differentiate the following formulas with respect to time. State what each rate in the differential equation represents 1. A 4r 2 Surface Area of a Sphere 2. V 4 r 3 3 Volume of a Sphere 3. a c 2 b 2 , where c is is constant 4. V r h , where r is 2 constant Volume of a Cylinder 5. cos x 15 6. V 1 r 2 h 3 Volume of a Cone Daily Lessons and Assessments for AP* Calculus AB, A Complete Course Mark Sparks 2012 Page 294 Air is leaking out of an inflated balloon in the shape of a sphere at a rate of 230π cubic centimeters per minute. At the instant when the radius is 4 centimeters, what is the rate of change of the radius of the balloon? 1. Identify all of the variables involved in the problem. 2. Identify which, if any, of the variables in the problem that remain constant. 3. Identify the rate(s) that are given and the rate that you wish to find. 4. Write an equation, often a geometric formula or trigonometric equation, that relates all of the variables in the problem for which a rate is given or for which a rate is to be determined. Substitute any value that represents a variable that is constant throughout the problem. It is important to keep in mind that you can have only one more variable than you have rates. You may have to make a substitution that relates one variable in terms of another. 5. Implicitly differentiate both sides of the equation with respect to time. 6. Substitute all instantaneous rates and values of the variable and solve for the remaining rate or variable. Daily Lessons and Assessments for AP* Calculus AB, A Complete Course Mark Sparks 2012 Page 295 A stone is dropped into a calm body of water, causing ripples in the form of concentric circles. The radius of the outer ripple is increasing at a rate of 1 foot per second. When the radius is 4 feet, at what rate is the total area of the disturbed water changing? 1. Identify all of the variables involved in the problem. 2. Identify which, if any, of the variables in the problem that remain constant. 3. Identify the rate(s) that are given and the rate that you wish to find. 4. Write an equation, often a geometric formula or trigonometric equation, that relates all of the variables in the problem for which a rate is given or for which a rate is to be determined. . Substitute any value that represents a variable that is constant throughout the problem. It is important to keep in mind that you can have only one more variable than you have rates. You may have to make a substitution that relates one variable in terms of another. 5. Implicitly differentiate both sides of the equation with respect to time. 6. Substitute all instantaneous rates and values of the variable and solve for the remaining rate or variable. Daily Lessons and Assessments for AP* Calculus AB, A Complete Course Mark Sparks 2012 Page 296 Water is leaking out of a cylindrical tank at a rate of 3 cubic feet per second. If the radius of the tank is 4 feet, at what rate is the depth of the water changing at any instant during the leak? 1. Identify all of the variables involved in the problem. 2. Identify which, if any, of the variables in the problem that remain constant. 3. Identify the rate(s) that are given and the rate that you wish to find. 4. Write an equation, often a geometric formula or trigonometric equation, that relates all of the variables in the problem for which a rate is given or for which a rate is to be determined. Substitute any value that represents a variable that is constant throughout the problem. It is important to keep in mind that you can have only one more variable than you have rates. You may have to make a substitution that relates one variable in terms of another. 5. Implicitly differentiate both sides of the equation with respect to time. 6. Substitute all instantaneous rates and values of the variable and solve for the remaining rate or variable. Daily Lessons and Assessments for AP* Calculus AB, A Complete Course Mark Sparks 2012 Page 297 A cone has a diameter of 10 inches and a height of 15 inches. Water is being poured into the cone so that the height of the water level is changing at a rate of 1.2 inches per second. At the instant when the radius of the expose surface area of the water is 2 inches, at what rate is the volume of the water changing? 1. Identify all of the variables involved in the problem. 2. Identify which, if any, of the variables in the problem that remain constant. 3. Identify the rate(s) that are given and the rate that you wish to find. 4. Write an equation, often a geometric formula or trigonometric equation, that relates all of the variables in the problem for which a rate is given or for which a rate is to be determined. Substitute any value that represents a variable that is constant throughout the problem. It is important to keep in mind that you can have only one more variable than you have rates. You may have to make a substitution that relates one variable in terms of another. 5. Implicitly differentiate both sides of the equation with respect to time. 6. Substitute all instantaneous rates and values of the variable and solve for the remaining rate or variable. Daily Lessons and Assessments for AP* Calculus AB, A Complete Course Mark Sparks 2012 Page 298 A ladder 25 feet long is leaning against the wall of a house. The base of the ladder is pulled away from the wall at a rate of 2 feet per second. a. How fast is the top of the ladder moving down the wall when the base of the ladder is 7 feet from the wall? b. Consider the triangle formed by the side of the house, the ladder, and the ground. Find the rate at which the area of the triangle is changing when the base of the ladder is 7 feet from the wall. c. Find the rate at which the angle formed by the ladder and the wall of the house is changing when the base of the ladder is 9 feet from the wall. Daily Lessons and Assessments for AP* Calculus AB, A Complete Course Mark Sparks 2012 Page 299 An airplane is flying at an altitude of 5 miles and passes directly over a radar antenna. When the plane is 10 miles past the antenna, the rate at which the distance between the antenna and the plane is changing is 240 miles per hour. What is the speed of the plane? The radius of a sphere is increasing at a rate of 2 inches per minute. Find the rate of change of the surface area of the sphere when the radius is 6 inches. A spherical balloon is expanding at a rate of 60π cubic inches per second. How fast is the surface area of the balloon expanding when the radius of the balloon is 4 inches. Daily Lessons and Assessments for AP* Calculus AB, A Complete Course Mark Sparks 2012 Page 300 Name_________________________________________Date____________________Class__________ Day #29 & 30 Homework 1. The radius r and area A of a circle are related by the equation A = πr2. Write an equation that relates dA to dr . dt dt 2. The radius r and surface area S of a sphere are related by the equation S = 4 πr2. Write an equation that relates dS to dr . dt dt The radius r, height h, and volume V of a right circular cylinder are related by the equation V = πr2h. Use this relationship to answer questions 3 – 5. 3. How is dV related to dr if h is constant? dt dt 4. How is dV related to dh if r is constant? dt dt 5. How is dV related to dr and dh if neither r nor h is constant? dt dt dt Daily Lessons and Assessments for AP* Calculus AB, A Complete Course Mark Sparks 2012 Page 301 Use the following information to solve problems 7 – 9. The length of a rectangle is decreasing at the rate of 2 cm/sec while the width is increasing at the rate of 2 cm/sec. When l = 12 cm and w = 5 cm, find each of the rates of change of each quantity indicated below. 7. Area 8. Perimeter 9. Length of a diagonal of the rectangle Daily Lessons and Assessments for AP* Calculus AB, A Complete Course Mark Sparks 2012 Page 302 A spherical balloon is inflated with helium at the rate of 100π ft3/min. 10. How fast is the balloon’s radius increasing when the radius is 5 feet? 11. How fast is the surface area increasing when the radius is 5 feet? 12. A conical tank with the vertex down is 10 feet across the top and 12 feet deep. If water is flowing into the tank at a rate of 10 cubic feet per minute, find the rate of change of the depth of the water when the water is 8 feet deep. Daily Lessons and Assessments for AP* Calculus AB, A Complete Course Mark Sparks 2012 Page 303 A ladder is 25 feet long and is leaning against the wall of a house. The base of the ladder is pulled away from the wall at a rate of 2 feet per second. Use this information to complete exercises 13 – 15. 13. How fast is the top of the ladder moving down the wall when its base is 15 feet from the wall? 14. Consider the triangle formed by the side of the house, the ladder and the ground. Find the rate at which the area of the triangle is changing when the ladder is 9 feet from the wall. 15. Find the rate at which the angle between the ladder and the ground is changing when the base of the ladder is 7 feet from the wall. Daily Lessons and Assessments for AP* Calculus AB, A Complete Course Mark Sparks 2012 Page 304 Water is flowing at a rate of 50 cubic meters per minute from a concrete conical reservoir. The radius of the reservoir is 45 m and the height is 6 m. Use this information to complete exercises 16 and 17. 16. How fast is the water level falling when the water is 5 meters deep? 17. How fast is the radius of the water’s surface changing when the water is 5 meters deep? 18. A man 6 feet tall walks at a rate of 5 feet per second toward a street light that is 16 feet tall. At what rate is the length of his shadow changing when he is 10 feet from the base of the light? Daily Lessons and Assessments for AP* Calculus AB, A Complete Course Mark Sparks 2012 Page 305 19. If the volume of a cube is increasing at a rate of 24 in3/min and each edge is increasing at 2 in/min, what is the length of each edge of the cube? 20. If the volume of a cube is increasing at a rate of 24 in3/min and the surface area is increasing at 12 in2/min, what is the length of each edge of the cube? 21. On a morning when the sun will pass directly overhead, the shadow of an 80—foot building on level ground is 60 feet long. At the moment in question, the angle θ the sun makes with the ground is increasing at the rate of 0.27 radians per minute. At what rate is the length of the shadow decreasing? Daily Lessons and Assessments for AP* Calculus AB, A Complete Course Mark Sparks 2012 Page 306 Daily Lessons and Assessments for AP* Calculus AB, A Complete Course Mark Sparks 2012 Page 307 Question #5 A container has the shape of an open right circular cone, as shown in the figure above. The height of the container is 10 cm and the diameter of the opening is 10 cm. Water in the container is evaporating so that its depth, h, is changing at the rate of 3 centimeters per hour. 10 (Note: The volume of a cone of height h and radius r is given by V 1 r 2 h .) 3 (a) Suppose you were asked to find the rate of change of the volume of water in the container when the depth of the water is 5 cm. Explain why the formula V 1 r 2 h could not be implicitly differentiated 3 with respect to time in order to determine this rate? What would have to be done in order to find this rate? (b) Find the rate of change of the volume of water in the container, with respect to time, when h = 5 cm. Indicate correct units of measure. (c) The surface area of the exposed water is changing at a rate of 9π cm2 per hour when the depth of the water is 4 cm. At what rate is the radius of the exposed water changing at this point in time? Indicate correct units of measure. Daily Lessons and Assessments for AP* Calculus AB, A Complete Course Mark Sparks 2012 Page 308 Daily Lessons and Assessments for AP* Calculus AB, A Complete Course Mark Sparks 2012 Page 309 AP Calculus Test #4 Answer Key & Rubrics Multiple Choice ** 1. E C 2. B C 3. D A 4. E E 5. A C 6. D E 7. E B 8. C B Free Response #1 Part A – 3 points total _____ 1 Rewrites the volume formula correctly in terms of only V and r: V 1 r 2 h 1 r 2 (2r ) 2 r 3 3 2 dr dV 2r dt dt _____ 1 Correctly differentiates implicitly with respect to time: _____ 1 Answer with correct units: 3 3 9 feet per minute 8 Free Response #1 Part B – 3 points total _____ 1 Correctly differentiates the area of a circle (A = πr2) with respect to time: _____ 1 Uses the value from part a) for dr 9 : dt 8 dA 2r dr dt dt dA 2 (2) 9 dt 8 _____ 1 Answer with correct units: 9 ft2 per minute 2 Free Response #2 Part A – 3 points total _____ 1 Substitutes the value for the length of the ladder into the equation before differentiating: x2 + y2 = 132 _____ 1 Correctly differentiates implicitly with respect to time: 2 x _____ 1 Answer with correct units: dy dx 2y 0 dt dt dy 12 feet per second dt Free Response #2 Part B – 3 points total cos x _____ 1 Uses an appropriate trigonometric equation sin _____ 1 Correctly differentiates implicitly with respect to time cos d 1 sin d 1 dx 1213 ddt 131 (12) 5 d 1 (5) _____ 1 Answer with correct units dt y 13 dy 13 dt d 1 rad/sec dt 13 dt 13 dt 13 dt 13 d 1 rad/sec dt Daily Lessons and Assessments for AP* Calculus AB, A Complete Course Mark Sparks 2012 tan sec 2 d dt y x x dydt y dxdt x2 1213 2 ddt 12(1212) 5(5) 2 d 1 rad/sec dt Page 310 Free Response #3 Part A – 2 points total 3 dy dy _____ 1 Correctly differentiates: 2 x 2 4 y dx 4 dx 0 _____ 1 Correctly solves for dy 2 3x 2 2( 3x 1) ( x31) dx 4 y 4 4( y 1) 2( y 1) x 1 2( y 3 1) Free Response #3 Part B – 2 points total _____ 1 Correctly finds the value of dy dx x 1 at the point (−2, 1) to be 1 4 2( y 3 1) _____ 1 Uses the opposite reciprocal of the slope, −4, to write the equation of the normal line to be y – 1 = −4(x + 2) Free Response #3 Part C – 2 points total dy _____ 1 Sets the denominator of dx , 2(y3 + 1) = 0, and correctly solves for y = −1. _____ 1 Substitutes y = −1 into the equation of the curve, x2 + 2x + y4 + 4y = 5, and correctly solves for x = −4 and x = 2. The two points are (−4, −1) and (2, −1) Daily Lessons and Assessments for AP* Calculus AB, A Complete Course Mark Sparks 2012 Page 311 AP CALCULUS TEST #4 Unit #4 – Applications of the Derivative (Part I) A Focus on Implicit Differentiation and Related Rates Name__________________________________________________________Date__________________ 1. 2. 3. 4. 5. 6. 7. 8. MULTIPLE CHOICE − Calculator Permitted 1. What is the slope of the line tangent to the curve defined by y2 + xy – x2 = 11x at the point (2, 3)? A. 5 2 B. 0 C. 1 8 D. 4 E. 7 3 2 2. The radius of a circle is increasing at a constant rate of 2 meters per second. What is the rate of increase in the area of the circle at the instant when the circumference of the circle is 20 meters? 2 A. 0.04 m / sec B. 40 m 2 / sec C. 4 m 2 / sec D. 20 m 2 / sec E. 100 m 2 / sec Daily Lessons and Assessments for AP* Calculus AB, A Complete Course Mark Sparks 2012 Page 312 3. If xy2 – y3 = x2 – 5, then dy dx A. 2x 2 y 3 y 2 B. y 2 2x 5 3 y 2 2 xy C. 2x 5 2 y 3 y 2 D. y 2 2x 3 y 2 2 xy E. x y2 xy 4. A spherical snowball is melting in such a way that its volume is decreasing at a rate of 2 cm 3 / min . At what rate is the radius changing when the radius is 7 cm? [The volume of a sphere is given by V 4 r 3 .] 3 A. 1 cm / min 7 B. 1 cm / min 49 C. 1 cm / min 98 D. 1 cm / min 196 E. 1 cm / min 98 5. The volume of a cube is decreasing at a rate of 24 cubic inches per minute. At the instant when an edge of the cube is 4 inches, at what rate is the edge of the cube changing? A. −0.5 in/min B. 2 in/min C. −2 in/min D. 0.5 in/min E. −2.5 in/min Daily Lessons and Assessments for AP* Calculus AB, A Complete Course Mark Sparks 2012 Page 313 6. In the triangle shown above, if θ increases at a constant rate of 3 radians per minute, at what rate is y decreasing in units per minute when x equals 3 units? A. 3 B. 15 4 C. 4 D. 9 E. 12 7. What is the slope of the tangent line to the curve y2 – 2x2 = 6 – 2xy at the point (2, 3)? A. 0 B. 4 9 C. 7 9 D. 6 7 E. 1 5 8. The radius of a sphere is decreasing at a rate of 2 centimeters per second. At the instant when the radius of the sphere is 3 centimeters, what is the rate of change, in square centimeters per second, of the surface area of the sphere? (The surface area, S, of a sphere with radius r is S = 4πr2.) A. –108π B. –72π C. –48π D. –24π E. –16π Daily Lessons and Assessments for AP* Calculus AB, A Complete Course Mark Sparks 2012 Page 314 Free Response #1 Water is running into an open conical tank at the rate of 9 cubic feet per minute. The tank is standing, inverted, and has a height of 10 feet and a base diameter of 10 feet. [Remember, the volume of a cone is given by the formula V = 1 3 r2h.] a. At what rate is the radius of the water in the tank increasing when the radius is 2 feet? b. At what rate is the exposed surface area of the water changing when the radius is 2 feet? Daily Lessons and Assessments for AP* Calculus AB, A Complete Course Mark Sparks 2012 Page 315 Free Response #2 A 13 foot ladder is leaning against a house when its base starts to slide away. By the time the base is 12 feet from the house, the base is moving at the rate of 5 feet per second. a. How fast is the top of the ladder sliding down the wall at the when the base of the ladder is 12 feet from the side of the house? b. At what rate is the angle, , between the ladder and the ground changing when the ladder is 12 feet from the side of the house? Daily Lessons and Assessments for AP* Calculus AB, A Complete Course Mark Sparks 2012 Page 316 Free Response #3 Consider the closed curve in the xy – plane given by the equation x 2 2 x y 4 4 y 5 . a. Show that dy dx x 1 . 2( y 3 1) b. Find the equation of the line normal to the curve at the point (−2, 1). c. Find the coordiates of the two points on the curve where the line tangent to the curve is vertical. Daily Lessons and Assessments for AP* Calculus AB, A Complete Course Mark Sparks 2012 Page 317 AP CALCULUS *TEST #4* Unit #4 – Applications of the Derivative (Part I) A Focus on Implicit Differentiation and Related Rates Name__________________________________________________________Date__________________ 1. 2. 3. 4. 5. 6. 7. 8. MULTIPLE CHOICE − Graphing Calculator Permitted 1. What is the slope of the line tangent to the curve defined by y2 + xy – x2 = 11 at the point (2, 3)? A. 5 2 B. 0 C. 1 D. E. 8 4 7 3 2 2. The radius of a circle is increasing at a constant rate of 0.2 meters per second. What is the rate of increase in the area of the circle at the instant when the circumference of the circle is 20 meters? A. 0.04 m 2 / sec B. 0.4 m 2 / sec C. 4 m 2 / sec D. 20 m 2 / sec E. 100 m 2 / sec Daily Lessons and Assessments for AP* Calculus AB, A Complete Course Mark Sparks 2012 Page 318 3. If xy2 – y3 = x2 – 5, then dy dx A. y 2 2x 3 y 2 2 xy B. y 2 2x 5 3 y 2 2 xy C. 2x 5 2 y 3 y 2 D. 2x 2 y 3 y 2 E. x y2 xy 4. A spherical snowball is melting in such a way that its volume is decreasing at a rate of 2 cm 3 / min . At what rate is the radius changing when the radius is 7 cm? [The volume of a sphere is given by V 4 r 3 .] 3 A. 1 cm / min 7 B. 1 cm / min 49 C. 1 cm / min 49 D. 1 cm / min 196 E. 1 cm / min 98 5. The volume of a cube is decreasing at a rate of 24 cubic inches per minute. At the instant when an edge of the cube is 2 inches, at what rate is the edge of the cube changing? A. −0.5 in/min B. 2 in/min C. −2 in/min D. 0.5 in/min E. −2.5 in/min Daily Lessons and Assessments for AP* Calculus AB, A Complete Course Mark Sparks 2012 Page 319 6. In the triangle shown above, if θ increases at a constant rate of 3 radians per minute, at what rate is y decreasing in units per minute when x equals 3 units? A. 3 B. 15 4 C. 4 D. 9 E. 12 7. What is the slope of the tangent line to the curve 3y2 – 2x2 = 6 – 2xy at the point (3, 2)? A. 0 B. 4 9 C. 7 9 D. 6 7 E. 5 3 8. The radius of a sphere is decreasing at a rate of 2 centimeters per second. At the instant when the radius of the sphere is 4.5 centimeters, what is the rate of change, in square centimeters per second, of the surface area of the sphere? (The surface area, S, of a sphere with radius r is S = 4πr2.) A. –108π B. –72π C. –48π D. –24π E. –16π Daily Lessons and Assessments for AP* Calculus AB, A Complete Course Mark Sparks 2012 Page 320 Free Response #1 Water is running into an open conical tank at the rate of 9 cubic feet per minute. The tank is standing, inverted, and has a height of 10 feet and a base diameter of 10 feet. [Remember, the volume of a cone is given by the formula V = 1 3 r2h.] a. At what rate is the radius of the water in the tank increasing when the radius is 2 feet? b. At what rate is the exposed surface area of the water changing when the radius is 2 feet? Daily Lessons and Assessments for AP* Calculus AB, A Complete Course Mark Sparks 2012 Page 321 Free Response #2 A 13 foot ladder is leaning against a house when its base starts to slide away. By the time the base is 12 feet from the house, the base is moving at the rate of 5 feet per second. a. How fast is the top of the ladder sliding down the wall at the when the base of the ladder is 12 feet from the side of the house? b. At what rate is the angle, , between the ladder and the ground changing when the ladder is 12 feet from the side of the house? Daily Lessons and Assessments for AP* Calculus AB, A Complete Course Mark Sparks 2012 Page 322 Free Response #3 Consider the closed curve in the xy – plane given by the equation x 2 2 x y 4 4 y 5 . a. Show that dy dx x 1 . 2( y 3 1) b. Find the equation of the line normal to the curve at the point (−2, 1). c. Find the coordiates of the two points on the curve where the line tangent to the curve is vertical. Daily Lessons and Assessments for AP* Calculus AB, A Complete Course Mark Sparks 2012 Page 323 Analytical and Graphical Connections between f ( x ) , f ' ( x ) , and f ' ' ( x ) Let’s begin by filling in the following charts about the relationships that exist between the graphs of a function and its first derivative. If F’(x)… Then F(x)… Is = 0 or is undefined at x = a, Is > 0, Is < 0, Changes from positive to negative, Changes from negative to positive, You have already seen this in action with polynomial functions. For the polynomial function below, determine the interval(s) where the graph is increasing, decreasing, has a relative maximum, and/ or has a relative minimum. Show your analysis and justify your answers. f ( x) x 4 4 x 3 Daily Lessons and Assessments for AP* Calculus AB, A Complete Course Mark Sparks 2012 Page 324 Polynomial functions deal with critical numbers where the first derivative is equal to zero. Since that initial investigation of these properties, we have talked about places where a function is not differentiable. In other words, places where the graph of a function has a cusp, but that does not exclude the existence of a relative maximum or minimum. Let’s investigate such a case now. Analytically find all intervals where f(x) = x 2 3 + 1 is increasing/decreasing or has a relative maximum or minimum. Sketch a graph using a graphing calculator on the grid to the right to verify your analytical results. 2 The FIRST DERIVATIVE of a function identifies intervals where a function is increasing, decreasing, has a relative maximum or has a relative minimum. In a similar fashion, the SECOND DERIVATIVE identifies intervals where a function is concave up, concave down, or has a point of inflection. Since the SECOND DERIVATIVE is the FIRST DERIVATIVE of F ' ( x) , then the same relationships that exist between F(x) and F ' ( x) must exist between F ' ( x) and F ' ' ( x) . Daily Lessons and Assessments for AP* Calculus AB, A Complete Course Mark Sparks 2012 Page 325 With these relationships in mind, complete the following table. If F’’(x)… then F(x)… And F’(x)… Is = 0 or is undefined at x = a, Is > 0 Is < 0 Changes from positive to negative Changes from negative to positive On the previous page, you found that the first derivative of the function f(x) = x 2 3 + 1 was the 2 function f ' ( x) 2 . Find f ' ' ( x) and perform a sign analysis to determine intervals of concavity and 3 3 x2 point(s) of inflection. Again, verify your analytical results by looking at the graph on the previous page. Daily Lessons and Assessments for AP* Calculus AB, A Complete Course Mark Sparks 2012 Page 326 Based on these relationships between a function and its first and second derivative, complete the following statements. 1. f(x) is increasing ↔ f ' ( x) _______________________________ 2. f(x) is decreasing ↔ f ' ( x) _______________________________ 3. f(x) has a relative maximum or minimum ↔ f ' ( x) _______________________________ 4. f(x) has a point of inflection ↔ f ' ( x) _______________________________ 5. f(x) is concave up ↔ f ' ' ( x) _______________________________ 6. f(x) is concave down ↔ f ' ' ( x) _______________________________ 7. f(x) has a point of inflection ↔ f ' ' ( x) _______________________________ 8. f ' ( x) is increasing ↔ f ' ' ( x) _______________________________ 9. f ' ( x) is decreasing ↔ f ' ' ( x) _______________________________ 10. f ' ( x) has a relative maximum or minimum ↔ f ' ' ( x) _______________________________ 11. f ' ( x) has a point of inflection ↔ f ' ' ( x) _______________________________ 12. f ' ( x) changes from negative to positive ↔ f (x)_________________________________ 13. f ' ( x) changes from positive to negative ↔ f (x)_________________________________ 14. f ' ( x) has a relative maximum or minimum ↔ f (x)_________________________________ 15. f ' ' ( x) changes from positive to negative ↔ f ' ( x) ________________________________ 16. f ' ' ( x) changes from negative to positive ↔ f ' ( x) ________________________________ On the following pages, you will find 15 graphs of polynomial functions. The 15 graphs can be put into five groups of three so that the three graphs in each group would represent f (x), f ' ( x) , and f ' ' ( x) . Once you have the graphs grouped, record the groups in the table to the right using the numbers 1 through 15. This works best if the graphs are cut out so that students can arrange them in the different groups. f (x ) f ' ( x) f ' ' ( x) 1. 2. 3. 4. 5. Daily Lessons and Assessments for AP* Calculus AB, A Complete Course Mark Sparks 2012 Page 327 Daily Lessons and Assessments for AP* Calculus AB, A Complete Course Mark Sparks 2012 Page 328 Daily Lessons and Assessments for AP* Calculus AB, A Complete Course Mark Sparks 2012 Page 329 Given the equation of g ' ( x) below, determine the interval(s) where the graph of g(x) is concave up, is concave down, and determine the x – values of the point(s) of inflection. Show your analysis and give justification for your answers. g ' ( x) 12 x x 6x 2 9 4 While the second derivative is primarily used to determine intervals of concavity and points of inflection, it can also be used to identify relative maximums and minimums of a function . THE SECOND DERIVATIVE TEST An Alternate Way to Identify Relative Maximums and Minimums using the 2nd Derivative Daily Lessons and Assessments for AP* Calculus AB, A Complete Course Mark Sparks 2012 Page 330 Use the second derivative test to locate and classify all x − values of relative extrema for each function. 1. f(x) = x4 – 4x3 + 2 2. f ( ) 2 cos , on the interval (0, 2π) Daily Lessons and Assessments for AP* Calculus AB, A Complete Course Mark Sparks 2012 Page 331 Name_________________________________________Date____________________Class__________ Day #33 & 34 Homework For exercises 1 – 2, determine the open intervals on which the given function is increasing or decreasing and the coordinates of any relative extrema. Show your analysis and explain your reasoning. 1. g ( x) ( x 2) 2 ( x 1) 2. h( x) x 2 3x 4 x2 Daily Lessons and Assessments for AP* Calculus AB, A Complete Course Mark Sparks 2012 Page 332 3. If F ' ( x) ( x 1) 2 ( x 2)( x 4) , where is the graph of F(x) increasing, decreasing, and/or reaching a relative maximum or minimum? Show your work and justify your reasoning. For exercises 4 and 5, use the Second Derivative Test to find the local extrema for the given function. Show your analysis and justify your reasoning. 4. g ( x) 3x x 3 5 5. h( x) x 3 3x 2 2 Daily Lessons and Assessments for AP* Calculus AB, A Complete Course Mark Sparks 2012 Page 333 For exercises 6 – 7, identify the intervals where the function is concave up and concave down. Also, identify the x – values of any points of inflection. Show your work and justify your reasoning. 6. g ' ( x) x 9 x 2 7. f ( x) xe 2 x Daily Lessons and Assessments for AP* Calculus AB, A Complete Course Mark Sparks 2012 Page 334 The graph of f ' ( x) , a polynomial function, is given. First, state what type of functions f (x) and f ' ' ( x) should be. Then, based on the graph of f ' ( x) , sketch possible graphs of f (x) and f ' ' ( x) . 8. 9. f (x) ________________________ f (x) _______________________ f ' ( x) f ' ( x) f ' ' ( x) _______________________ f ' ' ( x) _______________________ Daily Lessons and Assessments for AP* Calculus AB, A Complete Course Mark Sparks 2012 Page 335 10. 11. f (x) ________________________ f (x) _______________________ f ' ( x) f ' ( x) f ' ' ( x) _______________________ f ' ' ( x) _______________________ Daily Lessons and Assessments for AP* Calculus AB, A Complete Course Mark Sparks 2012 Page 336 Connecting the Graphs of f ( x ) , f ' ( x ) , and f ' ' ( x ) Given below is the graph of a function, F(x). State all of the conclusions that you can state about the graphs of F ' ( x) and F ' ' ( x) . Justify each of your conclusions. Graph of F(x) Conclusions about F ' ( x) Conclusions about F ' ' ( x) Daily Lessons and Assessments for AP* Calculus AB, A Complete Course Mark Sparks 2012 Page 337 Given below is the graph of a function, F ' ( x) . State all of the conclusions that you can state about the graphs of F(x) and F ' ' ( x) . Justify each of your conclusions. Graph of F ' ( x) Conclusions about F(x) Conclusions about F ' ' ( x) Daily Lessons and Assessments for AP* Calculus AB, A Complete Course Mark Sparks 2012 Page 338 Given below is the graph of a function, F ' ' ( x) . State all of the conclusions that you can state about the graphs of F(x) and F ' ( x) . Justify each of your conclusions. Graph of F ' ' ( x) Conclusions about F(x) Conclusions about F ' ( x) Daily Lessons and Assessments for AP* Calculus AB, A Complete Course Mark Sparks 2012 Page 339 Calculator Active Questions The function f ' ( x) cos(ln x) is the first derivative of a twice differentiable function, f(x). a. On the interval 0 < x < 10, find the x – value(s) where f(x) has a relative maximum. Justify your answer. b. On the interval 0 < x < 10, find the x – value(s) where f(x) has a relative minimum. Justify your answer. c. On the interval 0 < x < 10, find the x – value(s) where f(x) has a point of inflection. Justify your answer. On the interval 0 < x <10, how many relative minimums does the graph of g(x) have if g ' ( x) sin x ? x2 A. 0 B. 1 C. 2 D. 3 E. 4 Daily Lessons and Assessments for AP* Calculus AB, A Complete Course Mark Sparks 2012 Page 340 The derivative g ' of a function g is continuous and has exactly two zeros. Selected values of g ' are given in the table above. If the domain of g is the set of all real numbers, then g is decreasing on which of the following intervals? A. –2 < x < 2 only B. –1 < x < 1 only D. x > 2 only E. x < –2 or x > 2 C. x > –2 The second derivative of the function f is given by f ' ' ( x) x( x a)( x b) . The graph of f ' ' ( x) is shown 2 to the right. For what values of x does the graph of f ' ( x) have a relative maximum? A. j and k only B. a and b only C. a only D. 0 only E. a and 0 only If h(x) is a twice differentiable function such that h( x) 0 for all values of x, then at what value(s) does the graph of g(x) have a relative maximum if g ' ( x) 9 x 2 h( x) ? A. x = 3 and x = –3 B. x = 3 only C. x = 9 only D. x = –3 only E. g(x) does not have a relative maximum A table of function values for a twice differentiable function, f(x), is pictured to the right. Which of the following statements is/are true if f(x) has only one zero on the –3 < x < 3? I. f ' ( x) 0 on the interval –3 < x < 3. II. f(x) has a zero between x = 1 and x = 3. III. f ' ' ( x) 0 on the interval –3 < x < 3. A. I only B. I and II only D. II and III only E. I, II and III C. III only Daily Lessons and Assessments for AP* Calculus AB, A Complete Course Mark Sparks 2012 Page 341 Name_________________________________________Date____________________Class__________ Day #35 Homework 1. Pictured below is a function, f(m). Complete the chart below indicating the sign (+ or – or 0) for f(m), f ' (m) and f ' ' (m) at each of the indicated points. Point f(m) f ' (m) f ' ' (m) A B C D F 2. If, for all real numbers x, f ' ( x) < 0 and f ' ' ( x) > 0, which of the following curves could be part of the graph of f(x)? Explain your reasoning FOR EACH GRAPH. Graph A Graph B Graph C 3. The graph of a twice differentiable function is shown below. Order the values of f(2), f ' (2) and f ' ' (2) in order from least to greatest. Explain your reasoning. Daily Lessons and Assessments for AP* Calculus AB, A Complete Course Mark Sparks 2012 Page 342 The graph of f ' ( x) , the derivative of f(x) is shown in each of the following questions. Answer the questions 4 – 6 using this graph. 4. How many relative maximums does f(x) have? Label these x values with the letter C. Explain your reasoning. 5. How many relative minimums does f(x) have? Label these x values with the letter D. Explain your reasoning. 6. How many points of inflection does the graph of f(x) have? Label these x values with the letter E. Explain your reasoning. Pictured to the right is the graph of f ' ( x) . Use the graph to answer questions 7 – 13. 7. What are the value(s) of x where f(x) has a relative maximum? Explain your reasoning. 8. What are the value(s) of x where f(x) has a relative minimum? Explain your reasoning. 9. On what interval(s) is the graph of f(x) increasing? Explain your reasoning. 10. At what value(s) of x does the graph of f(x) have a point of inflection? Explain your reasoning. Daily Lessons and Assessments for AP* Calculus AB, A Complete Course Mark Sparks 2012 Page 343 11. On what interval(s) is the graph of f(x) concave up or concave down? Explain your reasoning. 12. If f(2) = 4, what is the equation of the normal line to the graph of f(x) when x = 2? 13. If f(2) = 4, what is the tangent line approximation of f(1.9)? Is this an over or under approximation of f(1.9)? Explain your reasoning. A function, F, is continuous on its domain of [–2, 4]. Additionally, F(–2) = 5, F(4) = 1 with F ' and F ' ' have the properties shown in the table below. Use this information to answer questions 14 – 17. x F ' ( x) F ' ' ( x) –2 < x < 0 Positive Positive x=0 Does not exist Does not Exist 0<x<2 Negative Positive x=2 0 0 2<x<4 Negative Negative 14. At what value(s) of x does F have relative extrema? Classify the extrema by type and give a reason for your answer. 15. At what value(s) of x does F have a point of inflection? Justify your answer. 16. On what interval(s) is the graph of F increasing, decreasing, concave up or concave down? Justify your reasoning. 17. Suppose the equation of the tangent line drawn to F at x = 2 were used to evaluate F(1.6) and F(2.4). Would the approximations be under or over approximations? Justify your answers. Daily Lessons and Assessments for AP* Calculus AB, A Complete Course Mark Sparks 2012 Page 344 Daily Lessons and Assessments for AP* Calculus AB, A Complete Course Mark Sparks 2012 Page 345 (A Modified Version) Let f be the function defined for x > 0 with f(2) = 5.623 and f ' , the first derivative of f, given by f ' ( x) e x / 4 sin x 2 . The graph of f ' is shown to the right. (a) Use the graph of f ' to determine whether the graph of f is concave up, down or neither on the interval 1.7 < x < 1.9. Explain your reasoning. (b) On the interval 0 < x < 3, at what x – value(s) does the graph of f have a relative maximum? A relative minimum? Justify your answers. (c) Write an equation for the tangent line to the graph of f at x = 2. Will the tangent line be above or below the graph of f ? Give a reason for your answer. Daily Lessons and Assessments for AP* Calculus AB, A Complete Course Mark Sparks 2012 Page 346 Daily Lessons and Assessments for AP* Calculus AB, A Complete Course Mark Sparks 2012 Page 347 Test #4 Extra Practice Problems Unless otherwise indicated, all problems are NONCALCULATOR ACTIVE 1. Let f be a function with a second derivative given by f ' ' ( x) x ( x 3)( x 6) . What are the x – coordinates of the points of inflection of the graph of f ? 2 A. 0 only B. 3 only C. 0 and 6 only D. 3 and 6 only E. 0, 3, and 6 2. The graph of f ' , the derivative of the function f, is shown to the right. Which of the following statements is true about f ? A. f is decreasing for –1 < x < 1. B. f is increasing for –2 < x < 0. C. f is increasing for 1 < x < 2. D. f has a local minimum at x = 0. E. f is not differentiable at x = –1 and x = 1. 3. The function f has the property that f ( x), f ' ( x), and f ' ' ( x) are negative for all real values of x. Which of the following could be the graph of f ? A. B. C. D. E. 4. Which of the following statements is true of f(x) = x 3 6 x 2 9 x 2 ? A. f is increasing on (–3, –1) B. f is increasing on ,3 1, C. f is increasing on ,5 D. f is increasing on 2, E. f is never increasing. Daily Lessons and Assessments for AP* Calculus AB, A Complete Course Mark Sparks 2012 Page 348 5. The function g(x) is continuous on the closed interval [–2, 0] and twice differentiable on the open interval (–2, 0). If g ' (1) 2 and g"( x) 0 on the open interval (–2, 0), which of the following could be a table of values of g? A. B. C. D. E. 6. Pictured to the right is the graph of f ' , the first derivative of f. At which of the following value(s) of x does the graph of f have a horizontal tangent but NOT a relative maximum or minimum? I. x = −3 II. x = −1 III. x = 1 A. I only B. I and III only C. II only D. II and III only E. I, II , and III 7. The function f has a first derivative given by f ' ( x) x . What is the x–coordinate of the point 1 x x3 of inflection of the graph of f ? (CALCULATOR PROBLEM) A. 1.008 B. 0.473 C. 0 D. –0.278 E. The graph has no points of inflection. Daily Lessons and Assessments for AP* Calculus AB, A Complete Course Mark Sparks 2012 Page 349 The graph given to the right is the graph of f ' , the first derivative of a differentiable function, f. Use the graph to answer the questions below. 8. On the interval [0, 8], are there any values where f(x) is not differentiable? Give a reason for your answer. 9. On what interval(s) is f ' ' > 0? < 0? Give reasons for your answers. 10. At what value(s) of x does the graph of f have a horizontal tangent? Give a reason for your answer. 11. What is the value of f ' ' (4) ? Explain your reasoning. 12. What is the value of f ' ' (8) ? Explain your reasoning. 13. If g ( x) e 2 x f ( x) and f(2) = –3, what is the equation of the normal line to the graph of g at x = 2? Daily Lessons and Assessments for AP* Calculus AB, A Complete Course Mark Sparks 2012 Page 350 14. Consider the function f ( x) 2 x 3 ax 2 bx 5 . Given the table of information below, answer the questions that follow. x < –1 –1 –1 < x < 1 1 2 1<x<2 2 2 >2 2 f' Positive 0 Negative Negative Negative 0 Positive f '' Negative Negative Negative 0 Positive Positive Positive a. Determine intervals of increasing and decreasing values of f. Justify your answers. b. Determine and classify all x – values of relative extrema of f. Justify your answers. c. Determine the intervals of concavity of f. Justify your answer. d. Determine the values of a and b in the equation of f. Show your work. Daily Lessons and Assessments for AP* Calculus AB, A Complete Course Mark Sparks 2012 Page 351 AP Calculus Quiz #6 Answer Key & Rubric Multiple Choice 1. C 2. A 3. E 4. E 5. A 6. C 7. C A B C C D A C Calculator NOT Permitted Free Response Part A – 2 points total ____ 1 The graph of f(x) has a horizontal tangent anytime that f ' ( x) = 0. ____ 1 The graph of f ' ( x) is on the x – axis at x = –2, 1, and 4. Calculator NOT Permitted Free Response Part B – 2 points total ____ 1 f(x) is increasing when the graph of f ' ( x) > 0 which occurs on 3, 2 4, 5 . ____ 1 f(x) is decreasing when the graph of f ' ( x) < 0 which occurs on 2,1 1, 4 . Calculator NOT Permitted Free Response Part C – 3 points total ____ 1 f(x) is concave up when f ' ' ( x) > 0. ____ 1 When f ' ( x) is increasing, then f ' ' ( x) > 0. ____ 1 Since f ' ( x) is increasing on the intervals (–1, 1) and (3, 5), then f(x) is concave up on these intervals. Calculator NOT Permitted Free Response Part D – 2 points total ____ 1 The graph is decreasing and concave up on the interval (0, 1) ____ 1 The graph is decreasing and concave down on the interval (1, 2) Daily Lessons and Assessments for AP* Calculus AB, A Complete Course Mark Sparks 2012 Page 352 AP CALCULUS QUIZ #6 Name______________________________________________________Date_____________________ 1. 2. 3. 4. 5. 6. 7. MULTIPLE CHOICE – Calculator NOT Permitted 1. The graph of y = 3x2 – x3 has a relative maximum at… A. (0, 0) only B. (1, 2) only C. (2, 4) only D. (4, –16) only E. (0, 0) and (2, 4) Daily Lessons and Assessments for AP* Calculus AB, A Complete Course Mark Sparks 2012 Page 353 2. The graph of a function f is pictured above. Which of the following graphs could be the graph of its derivative, f ' ? A. B. C. D. E. 3. Let g be a twice–differentiable function with g ' ( x) 0 and g ' ' ( x) 0 for all real numbers x, such that g(4) = 12 and g(5) = 18. Of the following, which is a possible value for g(6)? A. 15 B. 18 C. 21 D. 24 E. 27 4. Determine which of the following statements is/are true about the functions f(x), f ' ( x) and f ' ' ( x) . I. If f ' ( x) = 0 when x = c and f ' ' (c) > 0, then x = cis a relative minimum of f(x). II. If f ' ( x) is positive, then the graph of f(x) is increasing. III. If f(x) has a point of inflection, then f ' ( x) has a relative maximum or minimum. A. I and II B. II only C. II and III only D. III only E. I, II, and III Daily Lessons and Assessments for AP* Calculus AB, A Complete Course Mark Sparks 2012 Page 354 5. The second derivative of a function is given by F ' ' ( x) = (x – 2)2(x + 3). Which of the following conclusions can be made? I. F(x) is concave up on 3,2 and (2, ∞). II. F(x) has a point of inflection at x = –3. III. F ' ( x) has a relative maximum at x = –3. A. I and II only B. II only C. I and III only D. II and III only E. I, II and III 6. If g ( x) 2kx3 / 2 2 x ln x , for what value(s) of k would g(x) have a horizontal tangent at x = 4? A. ln 4 B. 1 C. 1 ln 4 D. 2 2 ln 4 2 3 12 3 E. No such value of k exists. 7. The total number of relative minimums of the function F(x) whose derivative, for all x, is given by F ' ( x) x( x 3)( x 1) 4 is… A. 3 B. 2 C. 1 D. 0 E. None of these Daily Lessons and Assessments for AP* Calculus AB, A Complete Course Mark Sparks 2012 Page 355 Free Response The figure above shows the graph of f ' ( x) , the derivative of f(x). The domain of f(x) is the set of all real numbers x such that –3 < x < 5. a. At what value(s) of x does the graph of f have a horizontal tangent? Justify your answer. b. On what open interval(s) is the graph of f increasing? Decreasing? Justify your answers. Daily Lessons and Assessments for AP* Calculus AB, A Complete Course Mark Sparks 2012 Page 356 c. On what open intervals is the graph of f(x) concave upward? Justify your answer. d. Suppose that f(1) = 0. In the xy-plane provided, draw a sketch that shows the general shape of the graph of the function f(x) on the open interval (0, 2). Daily Lessons and Assessments for AP* Calculus AB, A Complete Course Mark Sparks 2012 Page 357 AP CALCULUS *QUIZ #6* Name______________________________________________________Date_____________________ 1. 2. 3. 4. 5. 6. 7. MULTIPLE CHOICE − Calculator Not Permitted 1. The graph of y = 3x2 – x3 has a relative minimum at… A. (0, 0) only B. (1, 2) only C. (2, 4) only D. (4, –16) only E. (0, 0) and (2, 4) Daily Lessons and Assessments for AP* Calculus AB, A Complete Course Mark Sparks 2012 Page 358 2. Pictured above is the graph of f ' ( x) , the derivative of a function, f(x). Which of the following graphs could be the graph of f(x)? A. B. C. D. E. 3. Let g be a twice–differentiable function with g ' ( x) 0 and g ' ' ( x) 0 for all real numbers x, such that g(4) = 12 and g(5) = 18. Of the following, which is a possible value for g(6)? A. 15 B. 18 C. 21 D. 24 E. 27 4. Determine which of the following statements is/are true about the functions f(x), f ' ( x) and f ' ' ( x) . I. If f ' ( x) = 0 when x = c and f ' ' (c) > 0, then x = c is a relative maximum of f(x). II. If f ' ( x) is positive, then the graph of f(x) is increasing. III. If f(x) has a point of inflection, then f ' ( x) has a relative maximum or minimum. A. I and II B. II only C. II and III only D. III only E. I, II, and III Daily Lessons and Assessments for AP* Calculus AB, A Complete Course Mark Sparks 2012 Page 359 5. The second derivative of a function is given by F ' ' ( x) = (x – 2)2(x + 3). Which of the following conclusions can be made? I. F(x) is concave up on 3, II. F(x) has a point of inflection at x = –3. III. F ' ( x) has a relative minimum at x = –3. A. I and II only B. II only C. I and III only D. II and III only E. I, II and III 6. If g ( x) 2kx1 / 2 2 xe 2 x , for what value(s) of k would g(x) have a horizontal tangent at x = 1? 2 A. 6e 2 B. e C. 1 D. 4e 2 3 4 E. No such value of k exists. 7. The total number of relative maximums of the function F(x) whose derivative, for all x, is given by F ' ( x) x( x 3)( x 1) 4 is… A. 3 B. 2 C. 1 D. 0 E. None of these Daily Lessons and Assessments for AP* Calculus AB, A Complete Course Mark Sparks 2012 Page 360 Free Response The figure above shows the graph of f ' ( x) , the derivative of f(x). The domain of f(x) is the set of all real numbers x such that –3 < x < 5. a. At what value(s) of x does the graph of f have a horizontal tangent? Justify your answer. b. On what open interval(s) is the graph of f increasing? Decreasing? Justify your answers. Daily Lessons and Assessments for AP* Calculus AB, A Complete Course Mark Sparks 2012 Page 361 c. On what open intervals is the graph of f(x) concave upward? Justify your answer. d. Suppose that f(1) = 0. In the xy-plane provided, draw a sketch that shows the general shape of the graph of the function f(x) on the open interval (0, 2). Daily Lessons and Assessments for AP* Calculus AB, A Complete Course Mark Sparks 2012 Page 362 AP Calculus Test #5 Answer Key & Rubrics Multiple Choice Calculator Permitted * 1. D C 2. E C 3. B A 4. C A 5. D B 6. B E 7. C B Raw Score to Percentage Conversion Calculator NOT Calculator * 8. E B 9. C A 10. A B 11. A D 12. E C 13. B A 14. D E Calculator Permitted Free Response Part A – 2 points total ____ 1 Since f ' ( x) is increasing on the interval 5.3 < x < 6.7, then f ' ' ( x) > 0. ____ 1 Since f ' ' ( x) > 0 on the interval 5.3 < x < 6.7, then f is concave up. Calculator Permitted Free Response Part B – 2 points total ____ 1 f has a relative maximum anytime the graph of f ' ( x) changes from positive to negative. ____ 1 Since the graph of f ' ( x) goes from above the x – axis to below the x – axis, f has a relative maximum at x = 3.142 and x = 9.425. Calculator Permitted Free Response Part C – 2 points total ____ 1 f has a point of inflection when f ' ' ( x) changes signs. If f ' ' ( x) changes signs then the graph of f ' ( x) has a relative maximum or minimum. ____ 1 Thus, f has points of inflection at x = 1.792, 4.597, and 7.654 Calculator Permitted Free Response Part D – 3 points total ____ 1 Correct equation of the tangent line: y – 8.843 = 3.735(x – 2) ____ 1 f (2.4) 10.337 ____ 1 When x = 2, the graph of f ' ( x) is decreasing which means that f ' ' ( x) < 0 which means that f is concave down. Since f is concave down, then the tangent line will be above the graph of f which means the approximation is an over approximation. Daily Lessons and Assessments for AP* Calculus AB, A Complete Course Mark Sparks 2012 Page 363 Calculator Permitted Free Response Part A – 3 points total ____ 1 f(x) reaches at relative minimum at x = 0. ____ 1 The values of f(x) decrease on the interval (–1.5, 0) and increase on the interval (0, 1.5) ____ 1 The values of f ' ( x) change from negative to positive at x = 0. Calculator Permitted Free Response Part A – 2 points total ____ 1 If f ' ' ( x) > 0 then the graph of f ' ( x) must be increasing. _____ 1 If f ' ' ( x) > 0, then the graph of f(x) is concave up. OR ____ 1 The values of f ' ( x) are always increasing on the interval (–1.5, 1.5) _____ 1 The values of f(x) are decreasing at a decreasing rate on (–1.5, 0) and increasing at an increasing rate on (0, 1.5), both indicating that the graph is concave up. Calculator Permitted Free Response Part C – 2 points total ____ 1 Equation of tangent line: y + 4 = –5(x + 1) or y = –5x – 9 ____ 1 Uses the found equation of the tangent line to estimate f(–1.1): y ≈ –5(–1.1) – 9 ≈ –3.5 Calculator Permitted Free Response Part D – 2 points total ____ 1 The approximation of f(–1.1) is an under approximation because… ____ 1 …the graph of f is concave up on the interval (–1.5, 1.5) which means any tangent line would be below the actual graph of f(x). Daily Lessons and Assessments for AP* Calculus AB, A Complete Course Mark Sparks 2012 Page 364 AP CALCULUS AB TEST #5 Unit #4 – Applications of the Derivative (Part I) A Focus on the Relationships between f ( x ) , f ' ( x ) , and f ' ' ( x ) Name______________________________________________________Date_____________________ A GRAPHING CALCULATOR IS REQUIRED FOR SOME PROBLEMS OR PARTS OF PROBLEMS IN THIS PART OF THE EXAMINATION. (1) The exact numerical value of the correct answer does not always appear among the choices given. When this happens, select from among the choices the number that best approximates the exact numerical value. (2) Unless otherwise specified, the domain of a function f is assumed to be the set of all real numbers x for which f(x) is a real number. Multiple Choice -- Calculator Permitted Section 1. If f ' ' ( x) ( x 1)( x 2) 3 ( x 4) 2 , then the graph of f has inflection points when x = A. –2 only B. 1 only C. 1 and 4 only D. –2 and 1 only E. –2, 1, and 4 only Daily Lessons and Assessments for AP* Calculus AB, A Complete Course Mark Sparks 2012 Page 365 2. The graph of the function f is shown in the figure above. For which of the following values is f ' ( x) both positive and increasing? A. a B. b C. c D. d E. e 3. The figure below shows the graph of the first derivative of a function f. How many points of inflection does f have on the interval shown? A. Four B. Three C. Two D. One E. None 4. Let F(x) be a polynomial function such that F(4) = –1 and F’(4) = 0. If x < 4, then F’(x) < 0 and if x > 4, then F’(x) > 0, what can be said about the point (4, –1)? A. The point is a point of inflection of F(x) B. The point is a relative maximum of F(x). C. The point is a relative minimum of F(x). D. The point is a critical point but not an extrema of F(x). E. None of these conclusions can be drawn from the given information. Daily Lessons and Assessments for AP* Calculus AB, A Complete Course Mark Sparks 2012 Page 366 5. Let f be the function with derivative given by f ' ( x) sin x 1 . How many relative extrema does f have on the interval 2 < x < 4? A. One B. Two 2 C. Three D. Four E. Five 6. The first derivative of the function f is defined by f ' ( x) sin x x for 0 < x < 2. On what intervals is f increasing? 3 A. 1 < x < 1.445 only B. 1 < x < 1.691 C. 1.445 < x < 1.875 D. 0.577 < x < 1.445 and 1.875 < x < 2 E. 0 < x < 1 and 1.691 < x < 2 7. For all x in the closed interval [2, 5], the function f has a negative first derivative and a positive second derivative. Which of the following could be a table of values for f ? A. B. C. D. Daily Lessons and Assessments for AP* Calculus AB, A Complete Course Mark Sparks 2012 E. Page 367 Free Response Pictured above is the graph of f ' , the first derivative of a twice differentiable function, f(x), on the interval 0 < x < 10 such that f(2) = 8.843. The equation of f ' is defined by f ' ( x) 4 xe x / 3 sin x . Answer the questions that follow. a. On the interval 5.3 < x < 6.7, is the graph of f(x) concave up or concave down? Justify your answer. b. For what value(s) of x does the graph of f(x) have a relative maximum? Justify your answer. Daily Lessons and Assessments for AP* Calculus AB, A Complete Course Mark Sparks 2012 Page 368 c. On the interval 0 < x < 10, at what value(s) of x does the graph of f(x) have a point of inflection? Justify your answer. d. Approximate the value of f(2.4) using the equation of the tangent line drawn to the graph of f at x = 2. Is this approximation an over or under approximation of the actual value of f(2.4). Give a reason for your answer. Daily Lessons and Assessments for AP* Calculus AB, A Complete Course Mark Sparks 2012 Page 369 AP CALCULUS AB TEST #5 Unit #4 – Applications of the Derivative (Part I) A Focus on the Relationships between f ( x ) , f ' ( x ) , and f ' ' ( x ) Name______________________________________________________Date_____________________ A GRAPHING CALCULATOR IS NOT ALLOWED FOR THIS SECTION OF THE EXAM. (1) The exact numerical value of the correct answer does not always appear among the choices given. When this happens, select from among the choices the number that best approximates the exact numerical value. (2) Unless otherwise specified, the domain of a function f is assumed to be the set of all real numbers x for which f(x) is a real number. Multiple Choice − Calculator Not Permitted Section 8. If, for all real numbers x, f ' ( x) < 0 and f ' ' ( x) > 0, which of the following curves could be part of the graph of f? A. B. C. D. Daily Lessons and Assessments for AP* Calculus AB, A Complete Course Mark Sparks 2012 E. Page 370 9. Let g be the function given by g ( x) x 2 e kx , where k is a constant. For what value of k does g have a critical point at x 1 ? 3 A. 1 B. 3 −6 D. −3 3 C. 2 E. There is no such value of k. 10. Pictured to the right is the graph of f ' , the first derivative of f. For how many values of x does the graph of f have a horizontal tangent but not a relative maximum or minimum? A. One B. Two C. Three D. Four E. Five 11. The graph of a twice differentiable function, f(x), with a relative minimum at (–1, –2) is shown to the right. Which of the following inequality statements is true? A. f(–1) < f ' (1) < f ' ' (1) B. f ' (1) < f(–1) < f ' ' (1) C. f(–1) < f ' ' (1) < f ' (1) D. f ' ' (1) < f ' (1) < f(–1) E. f ' ' (1) = f ' (1) = f(–1) Daily Lessons and Assessments for AP* Calculus AB, A Complete Course Mark Sparks 2012 Page 371 12. The graph of the second derivative of a function f(x) is shown below. Which of the following is/are true? I. The graph of f(x) has a point of inflection at x = –1. II. The graph of f(x) is concave down on the interval –1 < x < 3. III. The graph f ' ( x) is decreasing at x = 2. A. I only B. II only D. I and II only E. I, II, and III C. III only 13. If g is a differentiable function such that g(x) < 0 for all real numbers x and if f ' ( x) x 4 g ( x) , which of the following is true? 2 A. f has a relative maximum at x = –2 and a relative minimum at x = 2. B. f has a relative minimum at x = –2 and a relative maximum at x = 2. C. f has relative minima at x = –2 and at x = 2. D. f has relative maxima at x = –2 and at x = 2. E. It cannot be determined if f as any relative extrema. 14. The graph of f ' ' , the second derivative of f, is shown to the right for −2 < x < 4. What are the intervals on which the graph of the function f is concave down? A. −2 < x < −1 only B. 0 < x < 2 C. 1 < x < 3 only D. −2 < x < −1 and 1 < x < 3 E. −1 < x < 1 Daily Lessons and Assessments for AP* Calculus AB, A Complete Course Mark Sparks 2012 Page 372 Free Response x −1.5 −1 −0.5 0 0.5 1 1.5 f (x) −1 −4 −6 −7 −6 −4 −1 f ' ( x) −7 −5 −3 0 3 5 7 Let f be a differentiable function such that f ' ' ( x) > 0 for [–1.5, 1.5]. The table above shows values of f(x) and f ' ( x) for selected values of x on the closed interval [–1.5, 1.5]. a. At what value of x does the graph of f(x) reach a relative minimum? Give one reason based on the values of f(x) AND one reason based on the values of f ' ( x) . b. Using the values of either f(x) or f ' ( x) , explain why the values in the table confirm the assertion f ' ' ( x) > 0. Daily Lessons and Assessments for AP* Calculus AB, A Complete Course Mark Sparks 2012 Page 373 c. Write an equation of the tangent line to the graph of f when x = –1 and use the equation of the tangent line to approximate the value of f(–1.1). d. Is the approximation in part c for f(–1.1) an under or over approximation of the actual value? Justify your answer. Daily Lessons and Assessments for AP* Calculus AB, A Complete Course Mark Sparks 2012 Page 374 AP CALCULUS AB *TEST #5* Unit #4 – Applications of the Derivative (Part I) A Focus on the Relationships between f ( x ) , f ' ( x ) , and f ' ' ( x ) Name______________________________________________________Date_____________________ A GRAPHING CALCULATOR IS REQUIRED FOR SOME PROBLEMS OR PARTS OF PROBLEMS IN THIS PART OF THE EXAMINATION. (1) The exact numerical value of the correct answer does not always appear among the choices given. When this happens, select from among the choices the number that best approximates the exact numerical value. (2) Unless otherwise specified, the domain of a function f is assumed to be the set of all real numbers x for which f(x) is a real number. Multiple Choice − Calculator Permitted Section 1. If f ' ' ( x) ( x 1)( x 2) 2 ( x 4) 3 , then the graph of f has inflection points when x = A. –2 only B. –2 and 1 only C. 1 and 4 only D. 1 only E. –2, 1, and 4 only Daily Lessons and Assessments for AP* Calculus AB, A Complete Course Mark Sparks 2012 Page 375 2. The graph of the function f is shown in the figure above. For which of the following values is f ' ( x) both positive and decreasing? A. a B. b C. c D. d E. e 3. The figure below shows the graph of the first derivative of a function f. How many points of inflection does f have on the interval shown? A. Four B. Three C. Two D. One E. None 4. Let F(x) be a polynomial function such that F(4) = –1 and F’(4) = 0. If x < 4, then F’(x) < 0 and if x > 4, then F’(x) < 0, what can be said about the point (4, –1)? A. The point is a critical point but not an extrema of F(x). B. The point is a relative maximum of F(x). C. The point is a relative minimum of F(x). D. The point is a point of inflection of F(x). E. None of these conclusions can be drawn from the given information. Daily Lessons and Assessments for AP* Calculus AB, A Complete Course Mark Sparks 2012 Page 376 5. Let f be the function with derivative given by f ' ( x) sin x 1 . How many relative maximums does f have on the interval 2 < x < 4? A. One B. Two 2 C. Three D. Four E. Five 6. The first derivative of the function f is defined by f ' ( x) sin x x for 0 < x < 2. On what intervals is f decreasing? 3 A. 1 < x < 1.445 only B. 1 < x < 1.691 C. 1.445 < x < 1.875 D. 0.577 < x < 1.445 and 1.875 < x < 2 E. 0 < x < 1 and 1.691 < x < 2 7. For all x in the closed interval [2, 5], the function f has a positive first derivative and a negative second derivative. Which of the following could be a table of values for f ? A. B. C. D. Daily Lessons and Assessments for AP* Calculus AB, A Complete Course Mark Sparks 2012 E. Page 377 Free Response Pictured above is the graph of f ' , the first derivative of a twice differentiable function, f(x), on the interval 0 < x < 10 such that f(2) = 8.843. The equation of f ' is defined by f ' ( x) 4 xe x / 3 sin x . Answer the questions that follow. a. On the interval 5.3 < x < 6.7, is the graph of f(x) concave up or concave down? Justify your answer. b. For what value(s) of x does the graph of f(x) have a relative maximum? Justify your answer. Daily Lessons and Assessments for AP* Calculus AB, A Complete Course Mark Sparks 2012 Page 378 c. On the interval 0 < x < 10, at what value(s) of x does the graph of f(x) have a point of inflection? Justify your answer. d. Approximate the value of f(2.4) using the equation of the tangent line drawn to the graph of f at x = 2. Is this approximation an over or under approximation of the actual value of f(2.4). Give a reason for your answer. Daily Lessons and Assessments for AP* Calculus AB, A Complete Course Mark Sparks 2012 Page 379 AP CALCULUS AB *TEST #5* Unit #4 – Applications of the Derivative (Part I) A Focus on the Relationships between f ( x ) , f ' ( x ) , and f ' ' ( x ) Name______________________________________________________Date_____________________ A GRAPHING CALCULATOR IS NOT ALLOWED FOR THIS SECTION OF THE EXAM. (1) The exact numerical value of the correct answer does not always appear among the choices given. When this happens, select from among the choices the number that best approximates the exact numerical value. (2) Unless otherwise specified, the domain of a function f is assumed to be the set of all real numbers x for which f(x) is a real number. Multiple Choice − Calculator Not Permitted Section 8. If, for all real numbers x, f ' ( x) > 0 and f ' ' ( x) > 0, which of the following curves could be part of the graph of f? A. B. C. D. Daily Lessons and Assessments for AP* Calculus AB, A Complete Course Mark Sparks 2012 E. Page 380 9. Let g be the function given by g ( x) x 2 e kx , where k is a constant. For what value of k does g have a critical point at x 2 ? 3 A. −3 B. 3 C. 1 D. 0 3 2 E. There is no such value of k. 10. Pictured to the right is the graph of f ' , the first derivative of f. For how many values of x does the graph of f have a horizontal tangent but not a relative maximum or minimum? A. One B. Two C. Three D. Four E. Five 11. The graph of a twice differentiable function, f(x), with a relative maximum at (–1, 2) is shown to the right. Which of the following inequality statements is true? A. f(–1) < f ' (1) < f ' ' (1) B. f ' ' (1) = f ' (1) = f(–1) C. f(–1) < f ' ' (1) < f ' (1) D. f ' ' (1) < f ' (1) < f(–1) E. f ' (1) < f(–1) < f ' ' (1) Daily Lessons and Assessments for AP* Calculus AB, A Complete Course Mark Sparks 2012 Page 381 12. The graph of the second derivative of a function f(x) is shown below. Which of the following is/are true? I. The graph of f(x) has a point of inflection at x = –1. II. The graph of f(x) is concave down on the interval –1 < x < 3. III. The graph f ' ( x) is increasing at x = 2. A. I only B. II only D. III only E. I, II, and III C. I and II only 13. If g is a differentiable function such that g(x) > 0 for all real numbers x and if f ' ( x) x 4 g ( x) , which of the following is true? 2 A. f has a relative maximum at x = –2 and a relative minimum at x = 2. B. f has a relative minimum at x = –2 and a relative maximum at x = 2. C. f has relative minima at x = –2 and at x = 2. D. f has relative maxima at x = –2 and at x = 2. E. It cannot be determined if f as any relative extrema. 14. The graph of f ' ' , the second derivative of f, is shown to the right for −2 < x < 4. What are the intervals on which the graph of the function f is concave up? A. −2 < x < −1 only B. 0 < x < 2 C. 1 < x < 3 only D. −2 < x < −1 and 1 < x < 3 E. −1 < x < 1 Daily Lessons and Assessments for AP* Calculus AB, A Complete Course Mark Sparks 2012 Page 382 Free Response x −1.5 −1 −0.5 0 0.5 1 1.5 f (x) −1 −4 −6 −7 −6 −4 −1 f ' ( x) −7 −5 −3 0 3 5 7 Let f be a differentiable function such that f ' ' ( x) > 0 for [–1.5, 1.5]. The table above shows values of f(x) and f ' ( x) for selected values of x on the closed interval [–1.5, 1.5]. a. At what value of x does the graph of f(x) reach a relative minimum? Give one reason based on the values of f(x) AND one reason based on the values of f ' ( x) . b. Using the values of either f(x) or f ' ( x) , explain why the values in the table confirm the assertion f ' ' ( x) > 0. Daily Lessons and Assessments for AP* Calculus AB, A Complete Course Mark Sparks 2012 Page 383 c. Write an equation of the tangent line to the graph of f when x = –1 and use the equation of the tangent line to approximate the value of f(–1.1). d. Is the approximation in part c for f(–1.1) an under or over approximation of the actual value? Justify your answer. Daily Lessons and Assessments for AP* Calculus AB, A Complete Course Mark Sparks 2012 Page 384 The graph of f ' ( x) , the derivative of a function f(x), is pictured below on the closed interval [–10, 10]. f ' ( x) has relative extreme values at x = –3, x = 2, and x = 6. Use the graph to answer the following questions. a. On what interval(s) of x is the graph of f(x) increasing? Justify your reasoning. b. At what value(s) of x does the graph of f(x) have relative minimum(s)? Justify your reasoning. c. On what interval(s) is the graph of f(x) concave up? Justify your reasoning. d. Determine all x values at which f(x) has a point of inflection. Justify your reasoning. Daily Lessons and Assessments for AP* Calculus AB, A Complete Course Mark Sparks 2012 Page 385 Free Response Part A − 2 points total ____ 1 f(x) is decreasing on the intervals (–10, –7), (–1, 4) and (8, 10)… ____ 1 …b/c f ' ( x) is negative. Free Response Part B − 2 points total ____ 1 f(x) has a relative maximum at x = –1 and x = 8… ____ 1 …b/c the graph of f ' ( x) changes from positive to negative at those values. Free Response Part C − 3 points total ____ 1 f(x) is concave down on the intervals (−3, 2) and (6, 10) ____ 1 f(x) is concave down when f ' ' ( x) < 0 ____ 1 When f ' ' ( x) < 0,then the graph of f ' ( x) is decreasing Free Response Part D − 2 points total ____ 1 f(x) has a point of inflection at x = –3, 2 and 6 b/c f(x) has a point of inflection when f ' ' ( x) changes signs. ____ 1 f ' ' ( x) changes signs at x = –3, 2 and 6 because f ' ( x) has a relative maximum or minimum Daily Lessons and Assessments for AP* Calculus AB, A Complete Course Mark Sparks 2012 Page 386 Unit #5 – Applications of the Derivative – Part II AP Calculus Day# Objective 39 Learn and apply the Extreme Value Theorem to find absolute extrema of a function on a closed interval. 40 41 Distinguish between the instantaneous rate of change of a function at a point and the average rate of change of a function on an interval, extending these ideas to understand and apply the Mean Value Theorem and Rolle’s Theorem. Apply major theorems numerically, graphically, and verbally. Note Handouts & Assignments Daily Lessons pages 388 – 392 Day #39 HW: #1 – 9 Daily Lessons pages 395 – 398 Day #40 HW: #1 – 12 Daily Lessons pages 402 – 406 Day #41 HW: #1 – 8 Study for Quiz #7 42 Quiz #7 43 Learn and apply the basic ideas—average velocity, average acceleration, instantaneous velocity, instantaneous acceleration—of motion and how they relate to calculus. Learn and apply the Five Commandments of Particle Motion 44 45 1998 AB #3 Parts a, b, and c 2000 AB #2 Parts a and b 2002 AB #3 Parts a and b Continue discussing particle motion extending the ideas to finding net and total distance of a moving particle over a given interval. Solve optimization—applied maximum and minimum— problems. 46 Quiz #8 47 Test #6: Unit #5 – Applications of the Derivative – Part II Daily Lessons pages 420 – 427 Day #43 HW: #1 – 11 Daily Lessons pages 431 – 435 Day #44 HW: #1 – 16 Students receive AP Free Response and Multiple Choice Practice (Daily Lessons pgs 440 – 443). This assignment will be turned in at the beginning of class on Day #46 for a grade. I will not discuss these problems. Daily Lessons pages 444 – 447 Day #45 HW: #1 – 7 AP Free Response and Multiple Choice Practice is Due – 25 points Daily Lessons and Assessments for AP* Calculus AB, A Complete Course Mark Sparks 2012 Page 387 The Extreme Value Theorem In the previous unit, we investigated heavily how to locate relative extrema of the graph of a function by using the derivative. In pre-calculus, we talked about the difference between relative and absolute extrema. In the space below, distinguish between the two. Definitions of Relative and Absolute Extrema of a Function Pictured below are the graphs of f and g. Answer the questions about these two functions. Graph of f(x) Identify the coordinates of the relative extrema of f. Graph of g(x) Identify the coordinates of the relative extrema of g. On the domain of f, what are the coordinates of the absolute extrema of f? On the domain of g, what are the coordinates of the absolute extrema of g? On the domain of the given function, did the absolute extrema occur at the function’s relative extrema? Daily Lessons and Assessments for AP* Calculus AB, A Complete Course Mark Sparks 2012 Page 388 Graph of f(x) On the interval −2 < x < 3, what are the absolute extrema of f? Graph of g(x) On the interval −4 < x < 5, what are the absolute extrema of g? On the interval −4 < x < 1, what are the absolute extrema of f? On the interval −2 < x < 6, what are the absolute extrema of g? When the domain is restricted to a particular closed interval, at what three places that the absolute extrema could exist? The Extreme Value Theorem (E. V. T.): Daily Lessons and Assessments for AP* Calculus AB, A Complete Course Mark Sparks 2012 Page 389 Consider the cubic function f ( x) x 3 6 x 2 9 x 2 to answer the following questions. a. Determine the intervals where f is increasing and decreasing. Justify your answers. b. Determine the coordinates of the relative extrema of f. Justify your answers. Pictured below is a graph of the function f on the closed interval −4 < x < 1. Identify the absolute maximum of f on the closed interval −4 < x < −1. Identify the absolute minimum of f on the closed interval −4 < x < −1. Identify the absolute maximum of f on the closed interval −4 < x < 1. Identify the absolute minimum of f on the closed interval −4 < x < 1. Daily Lessons and Assessments for AP* Calculus AB, A Complete Course Mark Sparks 2012 Page 390 Use the extreme value theorem to locate the absolute extrema of the function f ( x) x 3 6 x 2 9 x 2 on the given closed intervals. Your algebraic results should concur with your graphical conclusions from the previous page. Interval: −4 < x < −1 Interval: −4 < x < 1 For each of the following functions, state specifically why the E. V. T. is or is not applicable on the given interval. Interval: −5 < x < 0 H ( x) 3 x 2 x 3 G ( x) 2 x x 3 f ( x) ln( x 7) Daily Lessons and Assessments for AP* Calculus AB, A Complete Course Mark Sparks 2012 Page 391 Given the functions below, determine the absolute extreme values of the function on the given interval, provided the extreme value theorem is applicable. If it is not, state specifically why it is not. 1. f ( x) x 3 2 x 2 3x 2 on [–1, 3] 2. g ( x) sin 2 x cos x on x 2 3. f ( x) x 2 3 on [-3, 6] 4. h( x) ln x 2 4 on [–1, 3] 2 2 Daily Lessons and Assessments for AP* Calculus AB, A Complete Course Mark Sparks 2012 Page 392 Name_________________________________________Date____________________Class__________ Day #39 Homework 1. For which of the following functions is the Extreme Value Theorem NOT APPLICABLE on the interval [a, b]? Give a reason for your answer. Graph I Graph II Graph III For exercises 2 – 4, determine the critical numbers for each of the functions below. 2. 3. g ( x) ln( x 2 4) 4. h( x) 3 x 3 5. Given the function below, use a calculator to help determine the absolute extrema on the given interval. f ( x) sin x ln( x 1) on the interval [1, 6] Daily Lessons and Assessments for AP* Calculus AB, A Complete Course Mark Sparks 2012 Page 393 For exercises 6 – 9, determine the absolute extreme values on the given interval. You should do each of these independent from a calculator. 6. f ( x) x 3 3x 2 8. h( x) x x2 on the interval [–1, 3] on the interval [–4, 0] 7. g ( x) 3 x 2 on the interval [–3, 6] 2 9. f ( x) 3x 3 2 x on the interval [–1, 1] Daily Lessons and Assessments for AP* Calculus AB, A Complete Course Mark Sparks 2012 Page 394 The Derivate as a Rate of Change Mean Value Theorem and Rolle’s Theorem Consider the values of a differentiable function, f(x), in the table below to answer the questions that follow. Plot the points and connect them on the grid below. x 0 2 4 6 8 10 12 14 16 f(x) 1 5 8 10 11 10 8 5 1 In calculus, the derivative has many interpretations. One of the most important interpretations is that the derivative represents the Rate of Change of a Function. When speaking of rate of change, there are two rates of change that can be found that are associated with a function—average rate of change and instantaneous rate of change. Average Rate of Change of f(x) on an Interval Instantaneous Rate of Change of f(x) at a Point Find the average rate of change of f(x) on the interval [2, 12]. Is the instantaneous rate of change of f at x = 4 greater than the rate of change at x = 6? Justify. Daily Lessons and Assessments for AP* Calculus AB, A Complete Course Mark Sparks 2012 Page 395 Rolle’s Theorem Consider the function, f(x), presented on the previous page. Does Rolle’s Theorem apply on the following intervals? Explain why or why not? Interval [2, 14] Interval [2, 8] For each of the functions below, determine whether Rolle’s Theorem is applicable or not. Then, apply the theorem to find the values of c guaranteed to exist. 1. g ( x) 9 x 2 x 4 on the interval [–3, 0] 2. g ( x) sin 2 x on the interval [–4, –1] x2 Daily Lessons and Assessments for AP* Calculus AB, A Complete Course Mark Sparks 2012 Page 396 Rolle’s Theorem guarantees that if a function is continuous on the closed interval [a, b], differentiable on the open interval (a, b), and f(a) = f(b), then there is guaranteed to exist a value of c on (a, b) where the instantaneous rate of change is equal to zero. The Mean Value Theorem is similar. In fact, Rolle’s Theorem is a specific case of what is known in calculus as the Mean Value Theorem. Mean Value Theorem Consider the function h( x) 3 5 . The graph of h(x) is pictured below. Does the M.V.T. apply on the x interval [–1, 5]? Explain why or why not. Does the M.V.T. apply on the interval [1, 5]? Why or why not? Graphically, what does the M.V.T. guarantee for the function on the interval [1, 5]? Draw this on the graph to the left. Apply the M.V.T. to find the value(s) of c guaranteed for h(x) on the interval [1, 5] Daily Lessons and Assessments for AP* Calculus AB, A Complete Course Mark Sparks 2012 Page 397 2 Explain why you cannot apply the Mean Value Theorem for f ( x) x 3 2 on the interval [−1, 1]. Find the equation of the tangent line to the graph of f ( x) 2 x sin x 1on the interval (0, π) at the point which is guaranteed by the mean value theorem. The Mean Value Theorem guarantees that if a function is continuous on the closed interval [a, b] and differentiable on the open interval (a, b), then there is guaranteed to exist a value of c where the instantaneous rate of change at x = c is equal to the average rate of change of f on the interval [a, b]. Daily Lessons and Assessments for AP* Calculus AB, A Complete Course Mark Sparks 2012 Page 398 Name_________________________________________Date____________________Class__________ Day #40 Homework For the exercises 1 – 5, determine whether Rolle’s Theorem can be applied to the function on the indicated interval. If Rolle’s Theorem can be applied, find all values of c that satisfy the theorem. 1. f ( x) x 2 4 x on the interval 0 < x < 4 2. f ( x) ( x 4) 2 ( x 3) on the interval –4 < x < 3 3. f ( x) 4 x 2 on the interval –3 < x < 7 4. f ( x) sin x on the interval 0 < x < 2π 5. f ( x) cos 2 x on the interval x 2 3 3 Daily Lessons and Assessments for AP* Calculus AB, A Complete Course Mark Sparks 2012 Page 399 For exercises 6 – 9, determine whether the Mean Value Theorem can be applied to the function on the indicated interval. If the Mean Value Theorem can be applied, find all values of c that satisfy the theorem. 6. f ( x) x 3 x 2 2 x on –1 < x < 1 7. f ( x) x 3 on 3 < x < 7 8. f ( x) x 2 on 1 x 2 2 9. h( x) 2 cos x cos 2 x on 0 < x < π x Daily Lessons and Assessments for AP* Calculus AB, A Complete Course Mark Sparks 2012 Page 400 Using the graph of the function, f(x), pictured below, and given the intervals in the table below, determine if Rolle’s or Mean Value Theorem, whichever is indicated, can be applied or not. Give reasons for your answers. 10. [–5, –1] Rolle’s Theorem 11. [–2, 8] Rolle’s Theorem 12. [–1, 8] Mean Value Theorem Daily Lessons and Assessments for AP* Calculus AB, A Complete Course Mark Sparks 2012 Page 401 Applying Theorems in Calculus Intermediate Value Theorem, Extreme Value Theorem, Rolle’s Theorem, and Mean Value Theorem Before we begin, let’s remember what each of these theorems says about a function. Intermediate Value Theorem Extreme Value Theorem Rolle’s Theorem Mean Value Theorem Daily Lessons and Assessments for AP* Calculus AB, A Complete Course Mark Sparks 2012 Page 402 The rate at which water flows out of a pipe, in gallons per hour, is given by a differentiable function R of time t. The table below shows the rate as measured every 3 hours for a 24-hour period. t (hours) R(t) (gallons per hour) 0 3 6 9 12 15 18 21 24 9.6 10.4 10.8 11.2 11.4 11.3 10.7 10.2 9.6 a. Estimate the value of R ' (5) , indicating correct units of measure. Explain what this value means about R(t). b. Using correct units of measure, find the average rate of change of R(t) from t = 3 to t = 18. c. Is there some time t, 0 < t < 24, such that R ' (t ) = 0? Justify your answer. Daily Lessons and Assessments for AP* Calculus AB, A Complete Course Mark Sparks 2012 Page 403 The total order and transportation cost C(x), measured in dollars, of bottles of Pepsi Cola is approximated by the function x 1 C ( x) 10,000 , x x 3 where x is the order size in number of bottles of Pepsi Cola in hundreds. Answer the following questions. a. Is there guaranteed a value of r on the interval 0 < r < 3 such that the average rate of change of cost is equal to C ' (r ) ? Give a reason for your answer. b. Is there a value of r on the interval 3 < r < 6 such that C ' (r ) 0 . Give a reason for your answer and if such a value of r exists, then find that value of r. c. For 3 < x < 9, what is the greatest cost for order and transportation? Daily Lessons and Assessments for AP* Calculus AB, A Complete Course Mark Sparks 2012 Page 404 A car company introduces a new car for which the number of cars sold, S, is modeled by the function 9 S (t ) 300 5 , t 2 where t is the time in months. a. Find the value of S ' (2.5) . Using correct units, explain what this value represents in the context of this problem. b. Find the average rate of change of cars sold over the first 12 months. Indicate correct units of measure and explain what this value represents in the context of this problem. c. Is it possible that a value of c for 0 < c < 12 exists such that S ' (c) is equal to the average rate of change? Give a reason for your answer and if such a value of c exists, find the value. Daily Lessons and Assessments for AP* Calculus AB, A Complete Course Mark Sparks 2012 Page 405 x f(x) f ' ( x) g(x) g ' ( x) 1 6 4 2 5 2 9 2 3 1 3 10 −4 4 2 4 –1 3 6 7 The functions f and g are differentiable for all real numbers, and g is strictly increasing. The table above gives values of the functions and their first derivatives at selected values of x. The function h is given by the equation h( x) f ( g ( x)) 6 . a. Find the equation of the tangent line drawn to the graph of h when x = 3. b. Find the rate of change of h for the interval 1 < x < 3. c. Explain why there must be a value of r for 1 < r < 3 such that h(r) = –2. d. Explain why there is a value of c for 1 < c < 3 such that h' (c) 5 . Daily Lessons and Assessments for AP* Calculus AB, A Complete Course Mark Sparks 2012 Page 406 Name_________________________________________Date____________________Class__________ Day #41 Homework For the functions in exercises 1 and 2, determine if the Mean Value Theorem holds true for 0 < c < 5? Give a reason for your answer. If it does hold true, find the guaranteed value(s) of c. [CALC] 1. f ( x) 2 1 x 3 2 2. g ( x) 2 x sin 2 x 3. Administrators at a hospital believe that the number of beds in use is given by the function B(t ) 20 sin t 50 , 10 where t is measured in days. [CALC] a. Find the value of B ' (7) . Using correct units of measure, explain what this value means in the context of the problem. b. For 12 < t < 20, what is the maximum number of beds in use? Daily Lessons and Assessments for AP* Calculus AB, A Complete Course Mark Sparks 2012 Page 407 4. For t > 0, the temperature of a cup of coffee in degrees Fahrenheit t minutes after it is poured is modeled by the function F (t ) 68 93(0.91)t . Find the value of F ' (4) . Using correct units of measure, explain what this value means in the context of the problem. [CALC] For questions 5 – 8, use the table given below which represents values of a differentiable function g on the interval 0 < x < 6. Be sure to completely justify your reasoning when asked, citing appropriate theorems, when necessary. x 0 2 3 4 6 g(x) –3 1 5 2 1 5. Estimate the value of g ' (2.5) . 6. If one exists, on what interval is there guaranteed to be a value of c such that g(c) = –1? Justify your reasoning. 7. If one exists, on what interval is there guaranteed to be a value of c such that g ' (c) 0 ? Justify your reasoning. 8. If one exists, on what interval is there guaranteed to be a value of c such that g ' (c) 4 ? Justify your reasoning. Daily Lessons and Assessments for AP* Calculus AB, A Complete Course Mark Sparks 2012 Page 408 AP Calculus Quiz #7 Answer Key & Rubric Multiple Choice 1. B 2. D 3. A 4. E 5. B 6. E 7. A C B E D A C C Free Response Part A – 2 points total ____ 1 Correctly finds F ' (10) 1.309 ____ 1 F ' (10) 1.309 means that at 10 a.m., the temperature is increasing at a rate of 1.309 ºF per hour Free Response Part B – 3 points total ____ 1 Since F(t) is continuous and differentiable on the interval [0, 24] and F(0) = F(24) = 70, then Rolle’s Theorem guarantees the existence of a value, c, on the interval (0, 24) such that F ' (c) 0 12 ____ 1 Correctly finds the equation of F ' (t ) 5 sin t OR explains that the F(t) was graphed on the 6 interval 0 < t < 24 and the x – values of the relative maximum/minimum/horizontal tangent(s) were calculated on the graphing calculator. 12 ____ 1 Solves the equation F ' (t ) 5 sin t 0 to find the guaranteed value of c = 12. 6 Free Response Part C – 2 points total ____ 1 Finds F ' (21) 1.851 ____ 1 Since F ' (21) 0 , then the temperature is decreasing at 9 p.m. Free Response Part C – 2 points total ____ 1 Finds F (9) 87.071 °F, F (12) 90 °F, and F (23) 70.341 °F ____ 1 According to the Extreme Value Theorem, the lowest recorded temperature between 9 A.M. and 11 P.M. is 70.341 °F Daily Lessons and Assessments for AP* Calculus AB, A Complete Course Mark Sparks 2012 Page 409 AP CALCULUS QUIZ #7 Name______________________________________________________Date_____________________ 1. 2. 3. 4. 5. 6. 7. MULTIPLE CHOICE − Graphing Calculator Permitted 1. The table below represents values for a differentiable function, h(x), on the interval 0 < x < 10. Which of the statements below can be made? x 0 2 3 4 5 6 7 8 10 h(x) −6 −4 2 0 −4 −5 −8 −3 −6 I. On the interval 2 < x < 5, there is guaranteed to exist a value of c such that h ' (c) 0 . II. On the interval 3 < x < 6, there exists a value of c such that h(c) = −9. III. On the interval 0 < x < 3, the value of h ' ' ( x) 0 . A. I and II only B. I and III only D. I only E. II and III only Daily Lessons and Assessments for AP* Calculus AB, A Complete Course Mark Sparks 2012 C. I, II, and III Page 410 2. If c is the number that satisfies the conclusion of the Mean Value Theorem for f ( x) x3 2 x 2 on the interval 0 < x < 2, then c = … A. 0 B. 1 2 C. 1 D. 4 3 E. 2 3. Let f ( x) x . If the rate of change of f at x = c is twice its rate of change at x = 1, then c = … A. 1 4 B. 1 C. 4 D. 1 2 E. 1 2 2 4. Let f be a function that is differentiable on the open interval 0 < x < 10. If f(2) = −5, f(5) = 5, and f(9) = −5, which of the following statements must be true? I. f has at least 2 zeros. II. The graph of f has at least one horizontal tangent. III. For some c, 2 < c < 5, f(c) = 3. A. None B. I only D. I and III only E. I, II, and III Daily Lessons and Assessments for AP* Calculus AB, A Complete Course Mark Sparks 2012 C. I and II only Page 411 5. What is the maximum value of the function f ( x) x 2 sin x on the interval 1 < x < π? (A) 0.841 (B) 3.945 (C) 2.468 (D) 2.289 (E) 2.223 6. For t > 0 hours, H is a differentiable function of t that gives the temperature, in degrees Celsius, at an Arctic weather station. Which of the following is the best interpretation of H ' (24) ? A. The change in temperature during the first day B. The change in the temperature during the 24th hour. C. The average rate at which the temperature changed during the 24th hour D. The average rate at which the temperature is changing during the first day E. The rate at which the temperature is changing at the end of the 24th hour. 7. Let f be a function that is continuous on the closed interval [2, 4] with f(2) = 10 and f(4) = 20. Which of the following is guaranteed by the Intermediate Value Theorem? A. f(x) = 13 has at least one solution in the interval 2 < x < 4. B. f(3) = 15 C. f attains a maximum on the open interval 2 < x < 4. D. f ' ( x) 5 has at least one solution on the interval 2 < x < 4. E. f ' ( x) 0 for all x in the interval 2 < x < 4. Daily Lessons and Assessments for AP* Calculus AB, A Complete Course Mark Sparks 2012 Page 412 FREE RESPONSE The function, F(t), below represents the temperature outside of a house during a twenty-four hour period of time, where F(t) is measured in degrees Fahrenheit, t is measured in hours and t = 0 corresponds with 12 A.M. F (t ) 80 10 cos t , for 0 < t < 24 12 a. What is the value of F ' (10) ? Interpret this value in the context of the problem, using correct units. b. Is there guaranteed to exist a value c on the interval 0 < t < 24 such that F ' (t ) 0 ? Justify your answer. If such a value exists, find that value. Either show algebraically or explain graphically how you found the value of c. Daily Lessons and Assessments for AP* Calculus AB, A Complete Course Mark Sparks 2012 Page 413 c. Is the temperature increasing or decreasing at 9 P.M.? Give a reason for your answer. d. Between the hours of 9 A.M. and 11 P.M., what is the lowest temperature recorded? Show and explain the analysis that leads to your answer. Daily Lessons and Assessments for AP* Calculus AB, A Complete Course Mark Sparks 2012 Page 414 AP CALCULUS *QUIZ #7* Name______________________________________________________Date_____________________ 1. 2. 3. 4. 5. 6. 7. MULTIPLE CHOICE − Graphing Calculator Permitted 1. The table below represents values for a differentiable function, h(x), on the interval 0 < x < 10. Which of the statements below can be made? x 0 2 3 4 5 6 7 8 10 h(x) −6 −4 2 0 −4 −5 −8 −3 −6 I. On the interval 2 < x < 5, there is guaranteed to exist a value of c such that h ' (c) 0 . II. On the interval 3 < x < 6, there exists a value of c such that h(c) = −2. III. On the interval 0 < x < 3, the value of h ' ' ( x) 0 . A. I and II only B. I and III only D. I only E. II and III only Daily Lessons and Assessments for AP* Calculus AB, A Complete Course Mark Sparks 2012 C. I, II, and III Page 415 2. If c is the number that satisfies the conclusion of the Mean Value Theorem for f ( x) x 3 3x 2 on the interval 0 < x < 3, then c = A. 1 3 B. 2 D. 4 C. 3 E. 0 3 3. Let f ( x) x . If the rate of change of f at x = 2c is three times its rate of change at x = 1, then c = A. 2 9 B. 1 2 C. 1 9 D. 1 2 2 E. 1 18 4. Let f be a function that is differentiable on the open interval (1, 10). If f(2) = –5, f(5) = 5, and f(9) = –5, which of the following must be true? I. f has at least 2 zeros on the interval (2, 9). II. The graph has at least one horizontal tangent. III. For some c on 2 < c < 5, f(c) = 3. A. None B. I only D. I, II and III E. I and III only Daily Lessons and Assessments for AP* Calculus AB, A Complete Course Mark Sparks 2012 C. I and II only Page 416 5. What is the maximum value of the function f ( x) x sin x on the interval 1 < x < π? (A) 1.820 (B) 3.945 (C) 2.468 (D) 0.841 (E) 3.124 6. For t > 0 hours, H is a differentiable function of t that gives the temperature, in degrees at an Arctic weather station. Which of the following is the best interpretation of H (0) H ( 24) ? 0 24 A. The change in temperature during the first day B. The change in temperature during the 24th hour C. The average rate at which the temperature has changed during the 24 hour period D. The rate at which the temperature is changing during the 24th hour E. The rate at which the temperature is changing at the end of the 24th hour 7. Let g be a function that is continuous on the closed interval [2, 4] and differentiable on the open interval (2, 4) with g(2) = 10 and g(4) = 3. Which of the following is guaranteed by the Mean Value Theorem? A. g(x) = 6.5 has at least one solution in the open interval (2, 4). B. g attains an absolute maximum on the open interval (2, 4). C. g’(x) = −3.5 for some value of x on the open interval (2, 4). D. g’(x) < 0 for all values of x in the open interval (2, 4). E. g’(x) = 0 for some value of x on the open interval (2, 4). Daily Lessons and Assessments for AP* Calculus AB, A Complete Course Mark Sparks 2012 Page 417 FREE RESPONSE The function, F(t), below represents the temperature outside of a house during a twenty-four hour period of time, where F(t) is measured in degrees Fahrenheit, t is measured in hours and t = 0 corresponds with 12 A.M. F (t ) 80 10 cos t , for 0 < t < 24 12 a. What is the value of F ' (10) ? Interpret this value in the context of the problem, using correct units. b. Is there guaranteed to exist a value c on the interval 0 < t < 24 such that F ' (t ) 0 ? Justify your answer. If such a value exists, find that value. Either show algebraically or explain graphically how you found the value of c. Daily Lessons and Assessments for AP* Calculus AB, A Complete Course Mark Sparks 2012 Page 418 c. Is the temperature increasing or decreasing at 9 P.M.? Give a reason for your answer. d. Between the hours of 9 A.M. and 11 P.M., what is the lowest temperature recorded? Show and explain the analysis that leads to your answer. Daily Lessons and Assessments for AP* Calculus AB, A Complete Course Mark Sparks 2012 Page 419 Particle Motion Problems Particle motion problems deal with particles that are moving along the x – or y – axis. Thus, we are speaking of horizontal or vertical movement. The position, velocity, or acceleration of a particle’s motion are DEFINED by functions, but the particle DOES NOT move along the graph of the function. It moves along an axis. Most of the time, we speak of movement along the x – axis. In units 6 and 7, particle motion is revisited. At that time, we will deal more with vertical motion. For the time, we will focus on horizontal motion of particles. In this lesson, we develop the ideas of velocity and acceleration in terms of position. We will speak of two types of velocities and accelerations. Let’s define average and instantaneous velocity in the box below. Average and Instantaneous Velocity A particle’s position is given by the function p(t ) e t sin t , where p(t) is measured in centimeters and t is measured in seconds. Answer the following questions. What is the average velocity on the interval t = 1 to t = 3 seconds? Indicate appropriate units of measure. What is the instantaneous velocity of the particle at time t = 1.5. Indicate appropriate units of measure. Daily Lessons and Assessments for AP* Calculus AB, A Complete Course Mark Sparks 2012 Page 420 Before we proceed, a connection needs to be made. When given a function, f(x), how did we find the slope of the secant line on the interval from x = a to x = b? In terms of position of a particle, to what does the slope of the secant line correspond? To what does the instantaneous velocity correspond? Average and Instantaneous Acceleration A particle’s position is given by the function p(t ) e t sin t , where p(t) is measured in centimeters and t is measured in seconds. Answer the following questions. What is the average acceleration on the interval t = 1 to t = 3 seconds? Indicate appropriate units of measure. What is the instantaneous acceleration of the particle at time t = 1.5. Daily Lessons and Assessments for AP* Calculus AB, A Complete Course Mark Sparks 2012 Page 421 In summary, let’s correlate the concepts of position, velocity, and acceleration to what we already know about a function and its first and second derivative. corresponds with corresponds with corresponds with Let’s summarize our relationships between position, velocity and acceleration below. Velocity Position (Motion of the Particle) Is = 0 or is undefined Is > 0 Is < 0 Changes from positive to negative Changes from negative to positive Acceleration Velocity Is = 0 or is undefined Is > 0 Is < 0 Changes from positive to negative Changes from negative to positive Daily Lessons and Assessments for AP* Calculus AB, A Complete Course Mark Sparks 2012 Page 422 The graph below represents the position, s(t), of a particle which is moving along the x axis. At which point(s) is the velocity equal to zero? Justify your answer. At which point(s) does the acceleration equal zero? Justify your answer. On what interval(s) is the particle’s velocity positive? Justify your answer. On what interval(s) is the particle’s velocity negative? Justify your answer. On what interval(s) is the particle’s acceleration positive? Justify your answer. On what interval(s) is the particle’s acceleration negative? Justify your answer. Daily Lessons and Assessments for AP* Calculus AB, A Complete Course Mark Sparks 2012 Page 423 Five Commandments of Particle Motion 1. 2. 3. 4. 5. Suppose the velocity of a particle is given by the function v(t ) (t 2)(t 4) 2 for t > 0, where t is measured in minutes and v(t) is measured in inches per minute. Answer the questions that follow. a. Find the values of v(3) and v' (3) . Based on these values, describe the speed of the particle at t = 3. b. On what interval(s) is the particle moving to the left? Right? Show your analysis and justify your answer. Daily Lessons and Assessments for AP* Calculus AB, A Complete Course Mark Sparks 2012 Page 424 1998 AP Calculus AB #3 (Modified) The graph of the velocity v(t), in feet per second, of a car traveling on a straight road, for 0 < t < 50 is shown below. A table of values for v(t), at 5 second intervals of time, is also shown to the right of the graph. a. During what interval(s) of time is the acceleration of the car positive? Give a reason for your answer. b. Find the average acceleration of the car over the interval 0 < t < 50. Indicate units of measure. c. Find one approximation for the acceleration of the car at t = 40. Show the computations you used to arrive at your answer. Indicate units of measure. d. Is the speed of the car increasing or decreasing at t = 40? Give a reason for your answer. Daily Lessons and Assessments for AP* Calculus AB, A Complete Course Mark Sparks 2012 Page 425 2000 AP Calculus AB #2 (Partial) Two runners, A and B, run on a straight racetrack for 0 < t < 10 seconds. The graph below, which consists of two line segments, shows the velocity, in meters per second, of Runner A. The velocity, in meters per second, of Runner B is given by the function v defined by v(t ) 24t . 2t 3 a. Find the velocity of Runner A and the velocity of Runner B at t = 2 seconds. Indicate units of measure. b. Find the acceleration of Runner A and the acceleration of Runner B at time t = 2 seconds. Indicate units of measure. Daily Lessons and Assessments for AP* Calculus AB, A Complete Course Mark Sparks 2012 Page 426 2002 AP Calculus AB #3 (Partial) An object moves along the x – axis with initial position x(0) = 2. The velocity of the object at time t > 0 is given by the function v(t ) sin t . 3 a. What is the acceleration of the object at time t = 4? b. Consider the following two statements. Statement I: For 3 < t < 4.5, the velocity of the object is decreasing. Statement II: For 3 < t < 4.5, the speed of the object is decreasing. Are either or both of these statements correct? For each statement, provide a reason why it is correct or not correct. Daily Lessons and Assessments for AP* Calculus AB, A Complete Course Mark Sparks 2012 Page 427 Name_________________________________________Date____________________Class__________ Day #43 Homework A particle moves along the x axis such that its position, for t > 0, is given by the function p(t) = e2t – 5t. Use this information to complete exercises 1 – 4. 1. What are the values of p' (2) and p' ' (2) ? Explain what each value represents. 2. Based on the values found in part (a), what can be concluded about the speed of the particle at t = 2? Give a reason for your answer. 3. On what interval(s) of t is the particle moving to the left? To the right? Justify your answers. 4. Does the particle ever change directions? Justify your answer. 5. The graph of v(t), the velocity of a moving particle, is given below. What conclusions can be made about the movement of the particle along the x – axis and the acceleration, a(t), of the particle for t > 0? Give reasons for your answers. Daily Lessons and Assessments for AP* Calculus AB, A Complete Course Mark Sparks 2012 Page 428 3 2 6. If the position of a particle is defined by the function x(t) = t 9t 24t for t > 0, is the speed of the particle increasing or decreasing when t = 2.5? Justify your answer. The position of a particle is given by the function p(t ) (2t 3)e 2 t for t > 0. Answer questions 7 – 9. 7. What is the average velocity from t = 1 to t = 3? 8. Find an equation for v(t), the velocity of the particle. 9. For what value(s) of t will the v(t) = 0? Daily Lessons and Assessments for AP* Calculus AB, A Complete Course Mark Sparks 2012 Page 429 2003 AP Calculus AB #2 (Partial) A particle moves along the x – axis so that its velocity at time t is given by 2 v(t ) (t 1) sin t . 2 10. Find the acceleration of the particle at t = 2. Is the speed of the particle increasing at t = 2? Explain why or why not. 11. Find all times in the open interval 0 < t < 3 when the particle changes direction. Justify your answer. Daily Lessons and Assessments for AP* Calculus AB, A Complete Course Mark Sparks 2012 Page 430 More on Particle Motion Finding Net and Total Distance The graph below represents the velocity, v(t) which is measured in meters per second, of a particle moving along the x – axis. At what value(s) of t does the particle have no acceleration on the interval (0, 10)? Justify your answer. Express the acceleration, a(t), as a piecewise-defined function on the interval (0, 10). For what value(s) of t is the particle moving to the right? To the left? Justify your answer. Find the average acceleration of the particle on the interval [1, 8]. Show your work. Daily Lessons and Assessments for AP* Calculus AB, A Complete Course Mark Sparks 2012 Page 431 Definition of Net Distance: Definition of Total Distance: If a particle is moving in the same direction the entire amount of time, what can be said about the net distance and the total distance? To Find the Net Distance a Particle Travels on an Interval To Find the Total Distance a Particle Travels on an Interval The position of a particle is given by the function p(t ) 2t 3 6t 2 8t where p(t) is measured in centimeters. Find the net and total distance the particle travels from t = 1.5 seconds to t = 4 seconds. Daily Lessons and Assessments for AP* Calculus AB, A Complete Course Mark Sparks 2012 Page 432 The position of a particle is given by the function p(t ) e 2t 8t where p(t) is measured in feet. Find the net and total distance the particle travels from t = 0.5 minutes to t = 1.5 minutes. The position of a particle is given by the function p(t ) t 2 sin t where p(t) is measured in feet. Find the net and total distance the particle travels from t = minutes to t = 5 minutes. 6 4 Daily Lessons and Assessments for AP* Calculus AB, A Complete Course Mark Sparks 2012 Page 433 NO CALCULATOR PERMITTED A particle moves along the x – axis so that its position at any time t > 0 is given by the function p(t ) t 3 4t 2 3t 1, where p is measured in feet and t is measured in seconds. a. Find the average velocity on the interval t = 1 and t = 2 seconds. Give your answer using correct units. b. On what interval(s) of time is the particle moving to the left? Justify your answer. c. Using appropriate units, find the value of p' (3) and p' ' (3) . Based on these values, describe the motion of the particle at t = 3 seconds. Give a reason for your answer. d. What is the maximum velocity on the interval from t = 1 to t = 3 seconds. Show the analysis that leads to your conclusion. e. Find the total distance that the particle moves on the interval [1, 5]. Show and explain your analysis. Daily Lessons and Assessments for AP* Calculus AB, A Complete Course Mark Sparks 2012 Page 434 CALCULATOR PERMITTED A test plane flies in a straight line with positive velocity v(t), in miles per minute at time t minutes, where v is a differentiable function of t. Selected values of v(t) for 0 < t < 40 are shown in the table below t (min) v(t) (miles per min) 0 5 10 15 20 25 30 35 40 7.0 9.2 9.5 7.0 4.5 2.4 2.4 4.3 7.3 a. Find the average acceleration on the interval 5 < t < 20. Express your answer using correct units of measure. b. Based on the values in the table, on what interval(s) is the acceleration of the plane guaranteed to equal zero on the open interval 0 < t < 40? Justify your answer. c. Does the data represent velocity values of the plane moving away from its point of origin or returning to its point of origin? Give a reason for your answer. d. The function f, defined by f (t ) 6 cos t 3sin 7t , is used to model the velocity of the plane, in 10 40 miles per minute, for 0 < t < 40. According to this model, what is the acceleration of the plane at t = 23? What does this value indicate about the velocity at t = 23? Justify your answer, indicating units of measure. Daily Lessons and Assessments for AP* Calculus AB, A Complete Course Mark Sparks 2012 Page 435 Name_________________________________________Date____________________Class__________ Day #44 Homework The function whose graph is pictured below, represents the velocity, v(t), of a particle for t = 0 to t = 9 seconds moving along the x – axis. Use the graph to complete exercises 1 – 4. 1. On what interval(s) is the particle moving to the right? Left? Justify your answer. 2. On what interval(s) is the particle slowing down? Speeding up? Justify your answer. 3. At what value(s) of t is the particle momentarily stopped and changing directions? Justify your answer. 4. On what interval of the time is the acceleration 0? Justify your answer. Daily Lessons and Assessments for AP* Calculus AB, A Complete Course Mark Sparks 2012 Page 436 The graph below represents the position, p(t), of a particle that is moving along the x – axis on the interval 0 < t < 6. Use the graph to complete exercises 5 – 9. p(t) is measured in centimeters and t is measured in seconds. 5. For what interval(s) of time is the particle moving to the right? Justify your answer. 6. For what interval(s) of time is the particle moving to the left? Justify your answer. 7. Express the velocity, v(t), as a piecewise-defined function on the interval 0 < t < 6. 8. At what value(s) of t is the velocity undefined on the interval 1 < t < 6? Graphically justify your reasoning. 9. Find the average velocity of the particle on the interval 1 < t < 6. Daily Lessons and Assessments for AP* Calculus AB, A Complete Course Mark Sparks 2012 Page 437 A particle moves along the x – axis so that at any time 0 < t < 5, the velocity, in meters per second, is given by the function v(t ) (t 2) 2 cos 2t . Use a graphing calculator to complete exercises 10 – 12. 10. On the interval 0 < t < 5, at how many times does the particle change directions? Give a reason for your answer. 11. Using appropriate units, what is the value of v' (2) . Describe the motion of the particle at this time. Justify your answer. 12. Using appropriate units, what is the average acceleration between t = 1 and t = 3.5 seconds? Daily Lessons and Assessments for AP* Calculus AB, A Complete Course Mark Sparks 2012 Page 438 Jeff leaves his house riding his bicycle toward school. His velocity v(t), measured in feet per minute, on the interval 0 < t < 15, for t minutes, is shown in the graph to the right. Use the graph to complete exercises 13 – 16. 13. Find the value of v ' (4) . Explain, using appropriate units, what this value represents. 14. On the interval 0 < t < 5, is there any interval of time at which a(t) = 0? Explain how you know. 15. On the interval 0 < t < 5, does Rolle’s Theorem guaranteed that there will be a value of t such that a(t) = 0? Justify your answer. 16. At some point, Jeff realizes that he forgot something at home and has to turn around. After how many minutes does he turn around? Give a reason for your answer. Daily Lessons and Assessments for AP* Calculus AB, A Complete Course Mark Sparks 2012 Page 439 AP Free Response and Multiple Choice Practice NO CALCULATOR A particle moves along the x – axis with velocity at time t > 0 given by v(t ) 1 e1 t . a. Find the acceleration of the particle at t = 3. b. Is the speed of the particle increasing at t = 3? Give a reason for your answer. c. Find all values of t at which the particle changes direction. Justify your answer. d. The function p(t ) e1 t t models the position of the particle for t > 0. Find the total distance that particle traveled on the time interval 0 < t < 3. Daily Lessons and Assessments for AP* Calculus AB, A Complete Course Mark Sparks 2012 Page 440 NO CALCULATOR A car is traveling on a straight road. For 0 < t < 24 seconds, the car’s velocity, v(t), in meters per second, is modeled by the piecewise-linear function defined by the graph below. a. For what interval(s) of time does the car have zero acceleration? Show the work and explain the analysis that leads to your answer. b. For each value of v ' (4) and v ' (20) , find the value or explain why it does not exist. Indicate units of measure. c. Let a(t) be the car’s acceleration at time t in meters per second per second. For 0 < t < 24, write a piecewise-defined function for a(t). d. Find the average rate of change of v over the interval 8 < t < 20. Does the Mean Value Theorem guarantee a value of c, for 8 < c < 20, such that v ' (c) is equal to this average rate of change? Why or why not? Daily Lessons and Assessments for AP* Calculus AB, A Complete Course Mark Sparks 2012 Page 441 You must show your work to earn credit for the following. You will need to use a calculator for these. 2 If f ( x) sin x , then there exists a number c on the interval x 3 that satisfies the conclusion of 2 2 the Mean Value Theorem. Which of the following values could be c? (A) 2 3 (B) 3 4 (C) 5 6 (D) π (E) 3 2 A particle moves along a line so that at time t, where 0 < t < π, its position is given by 2 s(t ) 4 cos t t 10 . What is the velocity of the particle when its acceleration is zero? 2 (A) –5.19 (B) 0.74 (C) 1.32 (D) 2.55 Daily Lessons and Assessments for AP* Calculus AB, A Complete Course Mark Sparks 2012 (E) 8.13 Page 442 The graph of the function y x 3 6 x 2 7 x 2 cos x changes concavity at x = (A) –1.58 (B) –1.63 (C) –1.67 (D) –1.89 (E) –2.33 (D) 0 (E) 2 If y = 2x – 8, what is the minimum value of the product of xy? (A) –16 (B) –8 (C) –4 Daily Lessons and Assessments for AP* Calculus AB, A Complete Course Mark Sparks 2012 Page 443 Solving Optimization Problems General Approach to Solving Optimization Problems 1. 2. 3. 4. 5. Example 1 A manufacturer wants to design an open box having a square base and a surface area of 108 square inches. What dimensions will produce a box with maximum volume? What is the maximum volume? Daily Lessons and Assessments for AP* Calculus AB, A Complete Course Mark Sparks 2012 Page 444 Example 2 A box is to be built from a rectangular piece of cardboard that is 25 cm wide and 40 cm long by cutting out a square from each corner and then bending up the sides. Find the size of the corner square with will produce a container that will hold the most amount of soup. Example 3 A rectangular page is to contain 24 square inches of print. The margins at the top and bottom of the page are to be 1 ½ inches, and the margins on the left and right are to be 1 inch. What should the dimensions of the page be so that the least amount of paper is used? Example 4 A rectangle is bounded by the x and y axes and the graph of y = 3 – ½x. What length and width should the rectangle have so that its area is a maximum? Daily Lessons and Assessments for AP* Calculus AB, A Complete Course Mark Sparks 2012 Page 445 Example 5 A solid is formed by adjoining two hemispheres to the ends of a right circular cylinder. The total volume of the solid is 12 cubic centimeters. Find the radius of the cylinder that produces the minimum surface area. Example 6 The profit P (in thousands of dollars) for a company spending an amount of s (in thousands of dollars) on advertising is P 10 s 6s 400 . Find the amount of money the company should spend on advertising in order to yield a maximum profit. 1 3 2 Daily Lessons and Assessments for AP* Calculus AB, A Complete Course Mark Sparks 2012 Page 446 Example 7 Determine the point on the line y = 2x + 3 so that the distance between the line and the point (1, 2) is a minimum. Example 8 A rectangle ABCD with sides parallel to the coordinate axes is inscribed in the region enclosed by the graph of y = –4x2 + 4 as shown in the figure below. Find the x and y coordinates of the point C so that the area of the rectangle is a maximum. Daily Lessons and Assessments for AP* Calculus AB, A Complete Course Mark Sparks 2012 Page 447 Name_________________________________________Date____________________Class__________ Day #45 Homework 1. Find the point on the graph of f ( x) x 8 so that the point (2, 0) is closest to the graph. 2. A rancher has 200 total feet of fencing with which to enclose two adjacent rectangular corrals. What dimensions should each corral be so that the enclosed area will be a maximum? 3. The area of a rectangle is 64 square feet. What dimensions of the rectangle would give the smallest perimeter? Daily Lessons and Assessments for AP* Calculus AB, A Complete Course Mark Sparks 2012 Page 448 4. A rectangle is bound by the x – axis and the graph of a semicircle defined by y 25 x 2 . What length and width should the rectangle have so that its area is a maximum? 5. A Norman window is constructed by adjoining a semicircle to the top of an ordinary rectangular window (see figure below). Find the dimensions of a Norman window of maximum area if the total perimeter is 16 feet. Daily Lessons and Assessments for AP* Calculus AB, A Complete Course Mark Sparks 2012 Page 449 6. Find the maximum volume of a box that can be made by cutting squares from the corners of an 8 inch by 15 inch rectangular sheet of cardboard and folding up the sides. 7. The volume of a cylindrical tin can with a top and bottom is to be 16π cubic inches. If a minimum amount of tin is to be used to construct the can, what much the height, in inches, of the can be? Daily Lessons and Assessments for AP* Calculus AB, A Complete Course Mark Sparks 2012 Page 450 AP Calculus Quiz #9 Answer Key & Rubric Multiple Choice * 1. B D 2. D A 3. C C 4. C B 5. D E 6. E D 7. D C Free Response Part A – 2 points total ____ 1: Average Acceleration = v(15) v(0) 3.1 0 15 0 15 ____ 1: 0.207 feet per second2 Free Response Part B – 1 point total ____ 1: a(43) = v(45) v(40) 3.1 2.7 0.08 feet per second2 45 40 5 Free Response Part C – 3 points total ____ 1: When acceleration is positive, the velocity is increasing. ____ 1: v(t) is increasing on (0, 15). ____ 1: v(t) is increasing on (45, 60). Free Response Part D – 2 points total ____ 1: Since the ride moves smoothly, then v(t) is differentiable, then Rolle’s theorem applies ____ 1: Rolle’s theorem guarantees a(t) = 0 on the interval (5, 25) because v(5) = v(25) = 1.6. ____ 1: Rolle’s theorem guarantees a(t) = 0 on the interval (35, 55) because v(35) = v(55) = −1.6. Daily Lessons and Assessments for AP* Calculus AB, A Complete Course Mark Sparks 2012 Page 451 AP CALCULUS QUIZ #9 Name______________________________________________________Date_____________________ 1. 2. 3. 4. 5. 6. 7. MULTIPLE CHOICE − Calculator Permitted 1. If f ' ( x) x 2 sin 2 x , what is the first value of x where the graph of f(x) changes concavity on the interval (0, 5). A. 0 B. 1.144 C. 1.571 D. 3.412 E. 2.290 2. The maximum acceleration attained on the interval 0 < t < 3 by the particle whose velocity is given by v(t) = t3 – 3t2 + 12t + 4 is … A. 9 B. 12 C. 14 D. 21 E. 40 Daily Lessons and Assessments for AP* Calculus AB, A Complete Course Mark Sparks 2012 Page 452 3. The graph below shows the distance s(t) from a reference point of a particle moving on a number line, as a function of time, t. Which of the following points marked is closest to the point where the acceleration first becomes negative? A. A B. B C. C D. D E. E 4. A particle moves along the x – axis so that at any time t > 0 its velocity is given by the function v(t ) t 2 ln(t 2) . What is the acceleration of the particle at time t = 6? A. 1.500 B. 20.453 C. 29.453 D. 74.860 E. 133.417 5. The graph of f ' ' ( x) , the second derivative, is shown at the right. Which of the following is/are true? I. The graph of f(x) has an inflection point at x = –1. The graph of f ' ' ( x) II. The graph of f(x) is concave down on the interval (–1, 3). III. The graph of f ' ( x) is increasing at x = 1. A. I only B. II only C. III only D. I and II only E. I, II, and III Daily Lessons and Assessments for AP* Calculus AB, A Complete Course Mark Sparks 2012 Page 453 6. A rectangular field, bounded on one side by a building, is to be fenced on the other three sides using a total of 3000 feet of fencing. What is the length of the field parallel to the building if the field is to be the largest possible. A. 1400 feet B. 750 feet C. 1000 feet D. 2000 feet E. 1500 feet 7. For –1.5 < x < 1.5, let f be a function with a first derivative given by f ' ( x) e x 2 x 1 2 . Which of the following are all the intervals on which the graph of f is concave down? 4 2 A. (–0.418, 0.418) only B. (–1, 1) only C. (–1.354, –0.409) and (0.409, 1.354) D. (–1.5, –1) and (0, 1) E. (–1.5, –1.354), (–0.409, 0), and (1.354, 1.5) FREE RESPONSE The table below gives the velocity, in feet per second, of a rider on a Ferris Wheel at an amusement park. The time, t, is measured in seconds after the ride starts. The rider moves smoothly and the table gives the values for one complete revolution of the wheel. t seconds v(t) ft/second 0 5 10 15 20 25 30 35 40 45 50 55 60 0 1.6 2.7 3.1 2.4 1.6 0 1.6 2.7 3.1 2.4 1.6 0 a. What is the average acceleration during the first 15 seconds of the ride? Indicate units of measure. Daily Lessons and Assessments for AP* Calculus AB, A Complete Course Mark Sparks 2012 Page 454 b. Approximate the acceleration at t = 43 seconds? Justify your work and indicate units of measure c. During what interval(s) of time, t, is the acceleration certain to be positive? Justify your reasoning. d. On what interval(s) of time for 0 < t < 60, does Roelle’s Theorem guaranteed that the acceleration is equal 0? Justify your reasoning. Daily Lessons and Assessments for AP* Calculus AB, A Complete Course Mark Sparks 2012 Page 455 AP CALCULUS *QUIZ #9* Name______________________________________________________Date_____________________ 1. 2. 3. 4. 5. 6. 7. MULTIPLE CHOICE – Calculator Permitted 1. If f ' ( x) x 2 sin 2 x , what is the first value of x where the graph of f(x) changes concavity on the interval (0, 5)? A. 0 B. 3.412 C. 1.571 D. 1.144 E. 2.290 2. The maximum acceleration attained on the interval 0 < t < 3 by the particle whose velocity is given by v(t) = t3 – 3t2 + 12t + 4 is … A. 21 B. 40 C. 14 D. 9 E. 12 Daily Lessons and Assessments for AP* Calculus AB, A Complete Course Mark Sparks 2012 Page 456 3. The graph below shows the distance s(t) from a reference point of a particle moving on a number line, as a function of time, t. Which of the following points marked is closest to the point where the acceleration first becomes negative? A. A B. B C. C D. D E. E 4. A particle moves along the x – axis so that at any time t > 0 its velocity is given by the function v(t ) t 2 ln(t 2) . What is the acceleration of the particle at time t = 6? A. 1.500 B. 29.453 C. 20.453 D. 74.860 E. 133.417 5. The graph of f ' ' ( x) , the second derivative, is shown at the right. Which of the following is/are true? I. The graph of f(x) has an inflection point at x = –1. The graph of f ' ' ( x) II. The graph of f(x) is concave down on the interval (–1, 3). III. The graph of f ' ( x) is decreasing at x = 1. A. I only B. II only C. I and II only D. III only E. I, II, and III Daily Lessons and Assessments for AP* Calculus AB, A Complete Course Mark Sparks 2012 Page 457 6. A rectangular field, bounded on one side by a building, is to be fenced on the other three sides using a total of 2000 feet of fencing. What is the length of the field parallel to the building if the field is to be the largest possible. A. 1400 feet B. 750 feet C. 2000 feet D. 1000 feet E. 1500 feet 7. For –1.5 < x < 1.5, let f be a function with a first derivative given by f ' ( x) e x 2 x 1 2 . Which of the following are all the intervals on which the graph of f is decreasing? 4 2 A. (–0.418, 0.418) only B. (–1, 1) only C. (–1.354, –0.409) and (0.409, 1.354) D. (–1.5, –1) and (0, 1) E. (–1.5, –1.354), (–0.409, 0), and (1.354, 1.5) FREE RESPONSE The table below gives the velocity, in feet per second, of a rider on a Ferris Wheel at an amusement park. The time, t, is measured in seconds after the ride starts. The rider moves smoothly and the table gives the values for one complete revolution of the wheel. t seconds v(t) ft/second 0 5 10 15 20 25 30 35 40 45 50 55 60 0 1.6 2.7 3.1 2.4 1.6 0 1.6 2.7 3.1 2.4 1.6 0 a. What is the average acceleration during the first 15 seconds of the ride? Indicate units of measure. Daily Lessons and Assessments for AP* Calculus AB, A Complete Course Mark Sparks 2012 Page 458 b. Approximate the acceleration at t = 43 seconds? Justify your work and indicate units of measure c. During what interval(s) of time, t, is the acceleration certain to be positive? Justify your reasoning. d. On what interval(s) of time for 0 < t < 60, does Roelle’s Theorem guaranteed that the acceleration is equal 0? Justify your reasoning. Daily Lessons and Assessments for AP* Calculus AB, A Complete Course Mark Sparks 2012 Page 459 AP Calculus Test #6 Answer Key & Rubrics Multiple Choice Calculator Permitted * 1. C A 2. E C 3. B A 4. C E 5. E C 6. B A 7. E B Raw Score to Percentage Conversion Calculator NOT Permitted 8. C 9. D 10. D 11. D 12. C 13. C 14. A B C A E B B C Calculator Permitted Free Response Part A – 1 point total ____ 1: Average velocity = 6.684 feet per second Calculator Permitted Free Response Part B – 2 points total ____ 1: x' (2) = 5.389 feet per second ____ 1: The particle is moving to the right since the velocity, x' (2) , is positive. Calculator Permitted Free Response Part C – 2 points total ____ 1: Finds that v(t) = x' (t ) = 0 when t = ln(2) or 0.693 ____ 1: Finds the value of x' ' (ln 2) 2 feet per second2 Calculator Permitted Free Response Part D – 3 points total ____ 1: x' (1.5) = 2.482 or is > 0 ____ 1: x' ' (1.5) = 4.482 or is > 0 ____ 1: Since the velocity and acceleration both have the same sign the speed of the particle is increasing. ____ 1: Correct units of measure on answers to parts A, B, and C Daily Lessons and Assessments for AP* Calculus AB, A Complete Course Mark Sparks 2012 Page 460 Calculator NOT Permitted Free Response Part A – 2 points total ____ 1: f is a continuous function since it is differentiable. ____ 1: Since f changes signs on –2 < r < 1, the intermediate value theorem guarantees a value of r such that f(r) = 0. Calculator NOT Permitted Free Response Part B – 2 points total ____ 1: g ' ' (0.5) g ' (0) g ' (1) 2 (5) 3 0 1 1 ____ 1: Since g ' ' (0.5) 0 , then the graph of g is concave down when x = 0.5. Calculator NOT Permitted Free Response Part C – 3 points total ____ 1: Finds h(1) f (1)2 g (1) (2) 2 2 6 ____ 1: Finds h ' (1) 2 f (1) f ' (1) g ' (1) 2(2)(2) 1 7 ____ 1: Uses h(–1) and h ' (1) to find the equation of the tangent line: y – 6 = –7(x + 1) Calculator NOT Permitted Free Response Part D – 2 points total ____ 1: Correctly finds h(1) f (1)2 g (1) (3) 2 3 9 3 6 ____ 1: Since h is differentiable and h(1) = h(–1) = 6, then Rolle’s Theorem guarantees a value of c on –1 < c < 1, such that h ' (c) 0 . Daily Lessons and Assessments for AP* Calculus AB, A Complete Course Mark Sparks 2012 Page 461 AP CALCULUS AB TEST #6 Unit #5 – Applications of the Derivative – Part II Name______________________________________________________Date_____________________ A GRAPHING CALCULATOR IS REQUIRED FOR SOME PROBLEMS OR PARTS OF PROBLEMS IN THIS PART OF THE EXAMINATION. (1) The exact numerical value of the correct answer does not always appear among the choices given. When this happens, select from among the choices the number that best approximates the exact numerical value. (2) Unless otherwise specified, the domain of a function f is assumed to be the set of all real numbers x for which f(x) is a real number. Multiple Choice 1. The derivative of the function f is given by f ' ( x) x 2 cos x 2 . How many points of inflection does the graph of f have on the open interval (–2, 2)? A. One B. Two C. Three D. Four E. Five Daily Lessons and Assessments for AP* Calculus AB, A Complete Course Mark Sparks 2012 Page 462 2. A right circular cylinder is such that the sum of the height and the circumference around the base is 30 centimeters. What is the radius that produces a maximum volume? [The volume of a cylinder is given by the formula V = πr2h] A. 3 cm B. 10 cm C. 20 cm D. 302 cm E. 10 cm 3. A particle moves along a straight line with velocity given by v(t ) 7 (1.01) t at time t > 0. What is the acceleration of the particle at time t = 3? 2 A. −0.914 B. 0.055 C. 5.486 D. 6.086 E. 18.087 4. A particle’s position is given by the function p(t ) 3t 4.1sin(t ) for t > 0. Find the total distance that the particle travels on the interval 0 < t < 3.5 A. 9.062 B. 7.939 C. 10.877 D. 9.969 E. 6.263 Daily Lessons and Assessments for AP* Calculus AB, A Complete Course Mark Sparks 2012 Page 463 5. The function f is continuous and non-linear for –3 < x < 7 and f(–3) = 5 and f(7) = –5. If there is no value of c on –3 < x < 7 for which f ' (c) 1 , which of the following statements must be true? A. For some c on –3 < c < 7, f ' (c) 1 . B. For some c on –3 < c < 7, f ' (c) 1 . C. For some c on –3 < c < 7, f ' (c) 0 . D. For –3 < c < 7, f ' (c) exists. E. For some c on –3 < c < 7, f ' (c) does not exist. 6. A new robotic dog called the iPup went on sale at 9 AM and sold out within 8 hours. The number of customers in line to purchase the iPup at time t is modeled by a differentiable function A, for 0 < t < 8. Values of A(t) are shown in the table below. For 0 < t < 8, what is the greatest number of intervals during which Rolle’s Theorem guarantees that A ' (t ) 0 ? t (hours) A(t) (number of people) A. 1 0 1 2 3 4 5 6 7 150 185 135 120 75 75 120 60 B. 2 C. 3 D. 4 E. 5 7. A particle moving along a straight line so that at any time t > 0 its velocity is given by v(t ) sin t . cos 2t For which of the following values of t is the particle speeding up? I. t 6 II. t 3 III. t 2 3 A. I only B. II only C. III only D. I and II only E. I and III only Daily Lessons and Assessments for AP* Calculus AB, A Complete Course Mark Sparks 2012 Page 464 Free Response The position of a particle moving along the x-axis at time t > 0 seconds is given by the function x(t ) e t 2t feet. a. Find the average velocity of the particle over the interval [1, 3]. Be sure to indicate correct units of measure. b. In what direction is the particle moving and what is the particle’s velocity at t = 2 seconds? Justify your answers and be sure to indicate correct units of measure. c. What is the acceleration of the particle when the velocity is zero? Show the analysis that leads to your answer and be sure to indicate correct units of measure. d. Is the speed increasing or decreasing at t = 1.5 seconds? Justify your answer. Daily Lessons and Assessments for AP* Calculus AB, A Complete Course Mark Sparks 2012 Page 465 AP CALCULUS AB TEST #6 Unit #5 – Applications of the Derivative – Part II Name______________________________________________________Date_____________________ A GRAPHING CALCULATOR IS NOT ALLOWED FOR THIS SECTION OF THE EXAM. (1) The exact numerical value of the correct answer does not always appear among the choices given. When this happens, select from among the choices the number that best approximates the exact numerical value. (2) Unless otherwise specified, the domain of a function f is assumed to be the set of all real numbers x for which f(x) is a real number. Multiple Choice 8. A particle starts at time t = 0 and moves along a number line so that its position, at time t > 0, is given by x(t) = (t – 2)3(t – 6). On which of the following intervals is the particle always moving to the right? A. 0 < t < 5 B. 2 < t < 5 C. t > 5 D. t > 0 E. never Daily Lessons and Assessments for AP* Calculus AB, A Complete Course Mark Sparks 2012 Page 466 9. The sign graphs for the velocity, v(t), and the acceleration, a(t), of a particle moving along the x – axis for t > 0 are pictured to the right. On which interval(s) of time is the speed of the particle decreasing? I. 0 < t < 2 II. 1 < t < 3 IV. 2 < t < 3 III. 0 < t < 1 V. 1 < t < 2 A. I and II only B. II only C. V only D. III and IV only E. I only 10. For what value of c is the instantaneous rate of change for the function f ( x) 2 x equal to the average rate of change on the interval 1 < x < 4? A. c = 4 B. c = 9 C. c = 3 2 D. c = 9 4 E. No value of c is guaranteed 11. The graph of f ' ( x) , the derivative of a function f(x), is shown to the right. Which of the following statements is/are true about the function f(x)? I. f is increasing on –1 < x < 5. II. f has a point of inflection at x = 2. III. f is concave up on x < 0 and 2 < x < 4. A. I and II only B. II only C. II and III only D. I, II, and III E. I and III only Daily Lessons and Assessments for AP* Calculus AB, A Complete Course Mark Sparks 2012 Page 467 12. The graph of the velocity, v(t), in ft/sec, of a car traveling on a straight road for 0 < t < 50 is shown to the right. On which of the following intervals is the acceleration of the car negative? I. 0 < t < 10 II. 35 < t < 45 III. 15 < t < 35 A. I only B. I and III only C. II only D. II and III only E. I, II and III 13. A differentiable function f on the interval –2 < x < 7 is such that f ' (c) 0 where –2 < c < 7. If f(c) < f(7), then which of the following statements must be true? A. f(7) is the absolute maximum value of f on the interval –2 < x < 7. B. f(c) is the absolute minimum value of f on the interval –2 < x < 7. C. Either f(–2) or f(c) is the absolute minimum on the interval –2 < x < 7. D. Either f(c) or f(7) is the absolute minimum on the interval –2 < x < 7. E. The value of f ' ( x) 0 on the interval c < x < 7. 14. A particle moves along a straight line. The graph of the particle’s position, x(t), at time t is shown above for 0 < t < 6. The graph has horizontal tangents at t = 1 and t = 5 and a point of inflection at t = 2. For what values of t is the velocity of the particle increasing? A. 0 < t < 2 B. 1 < t < 5 D. 3 < t < 5 only E. 1 < t < 2 and 5 < t < 6 Daily Lessons and Assessments for AP* Calculus AB, A Complete Course Mark Sparks 2012 C. 2 < t < 6 Page 468 Free Response The table below represents values of differentiable functions, f and g, and their first derivatives. Use the table of values to answer the questions that follow. x f f' g g' −2 5 –3 4 2 −1 2 –2 2 1 0 –1 4 3 –2 1 –3 1 –3 –5 a. Is there a value of r on –2 < r < 1 such that f(r) = 0? Give a reason for your answer. b. Approximate the value of g ' ' (0.5) . What does this value say about the graph of g when x = 0.5? c. If h( x) f ( x)2 g ( x) , what is the equation of the line tangent to the graph of h at x = –1? Daily Lessons and Assessments for AP* Calculus AB, A Complete Course Mark Sparks 2012 Page 469 d. Is there guaranteed to exist a value of c on –1 < c < 1, such that h ' (c) 0 ? Give a reason for your answer. Daily Lessons and Assessments for AP* Calculus AB, A Complete Course Mark Sparks 2012 Page 470 AP CALCULUS AB TEST #6 *Unit #5 – Applications of the Derivative – Part II* Name______________________________________________________Date_____________________ A GRAPHING CALCULATOR IS REQUIRED FOR SOME PROBLEMS OR PARTS OF PROBLEMS IN THIS PART OF THE EXAMINATION. (1) The exact numerical value of the correct answer does not always appear among the choices given. When this happens, select from among the choices the number that best approximates the exact numerical value. (2) Unless otherwise specified, the domain of a function f is assumed to be the set of all real numbers x for which f(x) is a real number. Multiple Choice 1. The derivative of the function f is given by f ' ( x) 2 2 x sin x 2 . How many points of inflection does the graph of f have on the open interval (–2, 2)? A. Two B. Four C. Six D. Eight E. Ten Daily Lessons and Assessments for AP* Calculus AB, A Complete Course Mark Sparks 2012 Page 471 2. A right circular cylinder is such that the sum of the height and the circumference around the base is 30 centimeters. What is the radius that produces a maximum volume? [The volume of a cylinder is given by the formula V = πr2h] A. 3 cm B. 10 cm C. 10 cm D. 302 cm E. 20 cm 3. A particle moves along a straight line with velocity given by v(t ) 7t (1.01) t at time t > 0. What is the acceleration of the particle at time t = 3? 2 A. 7.055 B. 0.055 C. 5.486 D. 6.020 E. 20.086 4. A particle’s position is given by the function p(t ) 3t 4.1sin(t ) for t > 0. Find the total distance that the particle travels on the interval 3 < t < 5 A. 5.222 B. 10.877 C. 2.188 D. 7.939 E. 2.887 Daily Lessons and Assessments for AP* Calculus AB, A Complete Course Mark Sparks 2012 Page 472 5. The function f is continuous and non-linear for –3 < x < 7 and f(–3) = 5 and f(7) = 5. If there is no value of c on –3 < x < 7 for which f ' (c) 0 , which of the following statements must be true? A. For some c on –3 < c < 7, f ' (c) 1 . B. For some c on –3 < c < 7, f ' (c) 1 . C. For some c on –3 < c < 7, f ' (c) does not exist. D. For –3 < c < 7, f ' (c) exists. E. For some c on –3 < c < 7, f (c) 0 . 6. A new robotic dog called the iPup went on sale at 9 AM and sold out within 8 hours. The number of customers in line to purchase the iPup at time t is modeled by a differentiable function A, for 0 < t < 8. Values of A(t) are shown in the table below. For 0 < t < 8, what is the greatest number of intervals of times on which Rolle’s Theorem guarantees that A ' (t ) 0 ? t (hours) A(t) (number of people) A. 1 0 1 2 3 4 5 6 7 150 185 135 120 85 75 120 60 B. 2 C. 3 D. 4 E. 5 7. A particle moving along a straight line so that at any time t > 0 its velocity is given by v(t ) sin t . cos 2t For which of the following values of t is the particle slowing down? I. t 6 II. t 3 III. t 2 3 A. I only B. II only C. III only D. I and II only E. I and III only Daily Lessons and Assessments for AP* Calculus AB, A Complete Course Mark Sparks 2012 Page 473 Free Response The position of a particle moving along the x-axis at time t > 0 seconds is given by the function x(t ) e t 2t feet. a. Find the average velocity of the particle over the interval [1, 3]. Be sure to indicate correct units of measure. b. In what direction is the particle moving and what is the particle’s velocity at t = 2 seconds? Justify your answers and be sure to indicate correct units of measure. c. What is the acceleration of the particle when the velocity is zero? Show the analysis that leads to your answer and be sure to indicate correct units of measure. d. Is the speed increasing or decreasing at t = 1.5 seconds? Justify your answer. Daily Lessons and Assessments for AP* Calculus AB, A Complete Course Mark Sparks 2012 Page 474 AP CALCULUS AB TEST #6 *Unit #5 – Applications of the Derivative – Part II* Name______________________________________________________Date_____________________ A GRAPHING CALCULATOR IS NOT ALLOWED FOR THIS SECTION OF THE EXAM. (1) The exact numerical value of the correct answer does not always appear among the choices given. When this happens, select from among the choices the number that best approximates the exact numerical value. (2) Unless otherwise specified, the domain of a function f is assumed to be the set of all real numbers x for which f(x) is a real number. Multiple Choice 8. A particle starts at time t = 0 and moves along a number line so that its position, at time t > 0, is given by x(t) = (t – 2)3(t – 6). On which of the following intervals is the particle always moving to the left? A. 0 < t < 5 B. 2 < t < 5 C. t > 5 D. t > 0 E. never Daily Lessons and Assessments for AP* Calculus AB, A Complete Course Mark Sparks 2012 Page 475 9. The sign graphs for the velocity, v(t), and the acceleration, a(t), of a particle moving along the x – axis for t > 0 are pictured to the right. On which interval(s) of time is the speed of the particle increasing? I. 0 < t < 2 II. 1 < t < 3 IV. 2 < t < 3 III. 0 < t < 1 V. 1 < t < 2 A. I and II only B. II only C. V only D. III and IV only E. I only 10. For what value of c is the instantaneous rate of change for the function f ( x) 2 x equal to the average rate of change on the interval 4 < x < 9? A. c = 25 4 B. c = 9 C. c = 3 2 D. c = 9 4 E. No value of c is guaranteed 11. The graph of f ' ( x) , the derivative of a function f(x), is shown to the right. Which of the following statements is/are true about the function f(x)? I. f is decreasing on 0 < x < 2 and x > 4. II. f has a point of inflection at x = 1 and x = 3. III. f is concave up on x < 0 and 2 < x < 4. A. I and II only B. II only C. II and III only D. I, II, and III E. III only Daily Lessons and Assessments for AP* Calculus AB, A Complete Course Mark Sparks 2012 Page 476 12. The graph of the velocity, v(t), in ft/sec, of a car traveling on a straight road for 0 < t < 50 is shown to the right. On which of the following intervals is the acceleration of the car positive? I. 0 < t < 10 II. 35 < t < 45 III. 15 < t < 35 A. I only B. I and III only C. II only D. II and III only E. I, II and III 13. A differentiable function f on the interval –2 < x < 7 is such that f ' (c) 0 where –2 < c < 7. If f(c) > f(7) > f(–2), then which of the following statements must be true? A. f(7) is the absolute minimum value of f on the interval –2 < x < 7. B. f(c) is the absolute maximum value of f on the interval –2 < x < 7. C. Either f(–2) or f(c) is the absolute maximum on the interval –2 < x < 7. D. Either f(c) or f(7) is the absolute maximum on the interval –2 < x < 7. E. The value of f ' ( x) 0 on the interval c < x < 7. 14. A particle moves along a straight line. The graph of the particle’s position, x(t), at time t is shown above for 0 < t < 6. The graph has horizontal tangents at t = 1 and t = 5 and a point of inflection at t = 2. For what values of t is the velocity of the particle decreasing? A. 0 < t < 2 B. 1 < t < 5 D. 3 < t < 5 only E. 1 < t < 2 and 5 < t < 6 Daily Lessons and Assessments for AP* Calculus AB, A Complete Course Mark Sparks 2012 C. 2 < t < 6 Page 477 Free Response The table below represents values of differentiable functions, f and g, and their first derivatives. Use the table of values to answer the questions that follow. x f f' g g' −2 5 –3 4 2 −1 2 –2 2 1 0 –1 4 3 –2 1 –3 1 –3 –5 a. Is there a value of r on –2 < r < 1 such that f(r) = 0? Give a reason for your answer. b. Approximate the value of g ' ' (0.5) . What does this value say about the graph of g when x = 0.5? c. If h( x) f ( x)2 g ( x) , what is the equation of the line tangent to the graph of h at x = –1? Daily Lessons and Assessments for AP* Calculus AB, A Complete Course Mark Sparks 2012 Page 478 d. Is there guaranteed to exist a value of c on –1 < c < 1, such that h ' (c) 0 ? Give a reason for your answer. Daily Lessons and Assessments for AP* Calculus AB, A Complete Course Mark Sparks 2012 Page 479 AP Calculus Unit #6 − Basic Integration and Applications Day Objectives # 47 Find anti-derivatives/indefinite integrals of polynomial-type and trigonometric functions. 48 49 Approximate the area between a curve and the x-axis using Riemann and Trapezoidal Sums from graphs and tables of values. Learn and apply the Fundamental Theorem of Calculus, Part 1. 50 Understand and apply the properties of definite integrals and apply the First Fundamental Theorem numerically and graphically. 51 52 Quiz #10 Problems of rectilinear motion will be reviewed and integration through numerical, graphical and analytical methods will be incorporated into these problems to find the total distance an object travels over a period of time. 53 Understand the definite integral to be the total accumulation of change of a function over an interval of time. 54 Apply the Fundamental Theorem of Calculus to find the average value of a function on an interval. 55 56 Quiz #11 Unit #6 Test – Basic Integration and Basic Applications Note Handouts & Assignments Daily Lessons pages 487 – 489 Day #47 HW: #1 – 14 Daily Lessons pages 492 – 500 Day #48 HW: #1 – 9 Daily Lessons pages 503 – 506 Day #49 HW: #1 – 12 2001 AB #5 2002 AB #6 Parts a & b Daily Lessons pages 511 – 513 Day #50 HW: #1 – 12 2003 AB #4 1998 AB #3 1999 AB #3 Parts a & b Study for Quiz #10 Daily Lessons pages 528 – 530 Day #52 HW: #1 – 16 2001 AB #3 Parts a, b, c, and 2002 (Form B) AB #3 Parts a, b, c, and d Daily Lessons pages 536 – 539 Day #53 HW: #1 – 6 and 2005 AB #2 2005 AB #3 2002 AB #2(Form B) 2000 AB #4 Parts a, b, c, and d Daily Lessons pages 545 – 547 Day #54 HW: #1 – 7 2003 AB #3 2004 AB #1 Parts a, b, and c 2001 AB #2 Parts a, b, c, and d Study for Quiz #11 Study for Unit #6 Test Daily Lessons and Assessments for AP* Calculus AB, A Complete Course Mark Sparks 2012 Page 486 Finding Anti-derivatives of Polynomial-Type Functions If you had to explain to someone how to find the derivative of a polynomial-type function, what would you say? To find the anti-derivative, you would do the opposite of each one of those operations and in the reverse order. Therefore, to find the anti-derivative of a polynomial-type function… The anti-derivative of a function, f(x), is denoted by the notation f ( x)dx . So when finding the anti- derivative of a function, you are finding the function of which f(x) is the first derivative. This will enable us, if given f ' or f " to be able to find f. However, if f ' ( x)dx f ( x) , what problem do you foresee? Find each of the following anti-derivatives. 3x 2 2x 3dx ( x 2)(2x 3)dx x 3 2 x 4 dx x Daily Lessons and Assessments for AP* Calculus AB, A Complete Course Mark Sparks 2012 2 dx x Page 487 We learned that d sin x cos x and d cos x sin x . Similarly, write what the anti-derivatives of dx dx sine and cosine are. cos xdx _________________ sin xdx _________________ Find each of the following anti-derivatives. 2 sin x cos xdx t sin t dt 4x 3 cos xdx x sin xdx 2 Use the given information about f ' and f " to find f(x). 1. f " ( x) 2 f (2) = 10 f ' (2) 5 2. f " ( x) x 3 / 2 Daily Lessons and Assessments for AP* Calculus AB, A Complete Course Mark Sparks 2012 f ' (4) 2 f (0) = 0 Page 488 An evergreen nursery usually sells a certain shrub after 6 years of growth and shaping. The growth rate during those 6 years is approximated by the differential equation dh 1.5t 5 , dt where t is the time in years and h is the height in centimeters. The seedlings are 12 centimeters tall when planted, at t = 0. a. Find the value of the differential equation above when t = 3. Using correct units of measure, explain what this value represents in the context of this problem. b. Find an equation for h(t), the height of the shrubs at any year t. Then, determine how tall the shrubs are when they are sold. A particle moves along the x – axis at a velocity of v(t ) 1 a. What is the acceleration of the particle when t = 9? b. What is the position of the particle when t = 9? t , for t > 0. At time t = 1, its position is 4. Daily Lessons and Assessments for AP* Calculus AB, A Complete Course Mark Sparks 2012 Page 489 Name_________________________________________Date____________________Class__________ Day #47 Homework For problems 1 – 12, find the indefinite integrals below. 2. 2 x 3x 2 dx x 2 2 x 2 3x dx 4. x 3 / 2 2 x 1dx x 1 dx 2 x 6. y 3 y dy 8. 1. 3 x 3dx 3. 5. 7. 3x 2 2 x 3 dx x3 1 w w dw Daily Lessons and Assessments for AP* Calculus AB, A Complete Course Mark Sparks 2012 Page 490 9. x3 3 dx x 10. 11. 2 cos d 12. x sin x 2dx ( x 3)( x 3) 2 dx For problems 13 and 14, find the indicated function based on the given information. 13. If f ' ( x) 2 x sin x and f(0) = 4, find f(x). 14. If f ' ' ( x) x 2 , f ' (0) 6, and f (0) 3 , find f(x). Daily Lessons and Assessments for AP* Calculus AB, A Complete Course Mark Sparks 2012 Page 491 Riemann Sums A Graphical Approach to Approximating the Definite Integral In calculus, the result of f ( x)dx is a function that represents the anti-derivative of the function f(x). This is also sometimes referred to as an INDEFINITE INTEGRAL. b The result of f ( x)dx is a value that represents the area of the region bounded by the curve of f(x) and the x – axis on the interval a < x < b. a b Calculating Riemann sums is a way to estimate the area under a curve, the value of f ( x)dx , for a graphed function on a particular interval. In this a activity, you will learn to calculate four types of Riemann sums: Left Hand, Right Hand, Midpoint, and Trapezoidal Sums. Approximation #1 – Left Hand Riemann Sum with intervals of length 2 units Let’s consider for a moment the function f(x) = 13 x 2 x 5 . This function is graphed below. On the interval [1, 7] subdivide the area bounded by the graph of the function and the x-axis into rectangles of length 2 units. Place the upper left hand vertex of the rectangle on the curve each time. 2 Then, calculate the area of each rectangle and sum the areas to approximate the area of the region under the curve bounded by f(x) = 13 x 2 x 5 , x = 1, x = 7, and the x-axis. 2 Daily Lessons and Assessments for AP* Calculus AB, A Complete Course Mark Sparks 2012 Page 492 Approximation #2 – Left Hand Riemann Sum with intervals of length 1 unit Approximation #3 – Left Hand Riemann Sum with intervals of length ½ unit On the interval [1, 7] subdivide the area bounded by the graph of the function and the x-axis into rectangles of length1 unit. On the interval [1, 7] subdivide the area bounded by the graph of the function and the x-axis into rectangles of length ½ unit. Daily Lessons and Assessments for AP* Calculus AB, A Complete Course Mark Sparks 2012 Page 493 Now, we are going to change things up a little bit. On the next three approximations, we are going to do a right hand sum. That is, draw your rectangles so that the upper right hand vertex of the rectangle is on the curve of the function. Approximation #4 – Right Hand Riemann Sum with intervals of Approximation #5 – Right Hand Riemann Sum with intervals of length 2 units length 1 unit On the interval [1, 7] subdivide the area bounded by the graph of the function and the x-axis into rectangles of length 2 units. On the interval [1, 7] subdivide the area bounded by the graph of the function and the x-axis into rectangles of length 1 unit. Daily Lessons and Assessments for AP* Calculus AB, A Complete Course Mark Sparks 2012 Page 494 Approximation #6 – Right Hand Riemann Sum with intervals of length ½ unit Before we continue, what inference can you make about the approximations as the lengths of the rectangles decreases? On the interval [1, 7] subdivide the area bounded by the graph of the function and the x-axis into rectangles of length ½ unit . Graphically, why do you suppose this is so? Daily Lessons and Assessments for AP* Calculus AB, A Complete Course Mark Sparks 2012 Page 495 On the next two approximations, we are going to do a midpoint sum. That is, draw your rectangles so that the upper right hand vertex of the rectangle is on the curve of the function. Approximation #7 – Midpoint Riemann Sum with intervals of length 2 Approximation #8 – Midpoint Riemann Sum with intervals of length 1 units units On the interval [1, 7] subdivide the area bounded by the graph of the function and the x-axis into rectangles of length 2 units. On the interval [1, 7] subdivide the area bounded by the graph of the function and the x-axis into rectangles of length 1 unit. Daily Lessons and Assessments for AP* Calculus AB, A Complete Course Mark Sparks 2012 Page 496 On the next two approximations, we are going to draw trapezoids between the curve and the x – axis to approximate the area. Remember that the area of a trapezoid is found using the formula A 1 h(b1 b2 ) , where b1 and b2 are the parallel sides and h is the distance between those parallel 2 sides. Approximation #9 – Trapezoidal Riemann Sum with intervals of length 2 units On the interval [1, 7] subdivide the area bounded by the graph of the function and the x-axis into trapezoids of height 2 units. Daily Lessons and Assessments for AP* Calculus AB, A Complete Course Mark Sparks 2012 Page 497 Approximation #10 – Trapezoidal Riemann Sum with intervals of length 1 unit On the interval [1, 7] subdivide the area bounded by the graph of the function and the x-axis into trapezoids of height 1 unit. Daily Lessons and Assessments for AP* Calculus AB, A Complete Course Mark Sparks 2012 Page 498 Type of Sum Numerical Approximation Do you think this is an over or under approximation of the area? Left Hand Riemann Sum Intervals of 2 units Left Hand Riemann Sum Intervals of 1 unit Left Hand Riemann Sum Intervals of ½ unit At this point, place a star next to the approximation that you feel is the most accurate to the actual area. In our next lesson, we will learn how to find the EXACT area of a region between the graph of a function and the x – axis. In the space below, we will come back to this to find the exact area once we complete the next lesson in order to see which approximation to the left is the most accurate. Right Hand Riemann Sum Intervals of 2 units Right Hand Riemann Sum Intervals of 1 unit Right Hand Riemann Sum Intervals of ½ unit Midpoint Riemann Sum Intervals of 2 units Midpoint Riemann Sum Intervals of 1 unit Trapezoidal Riemann Sum Intervals of 2 units Trapezoidal Riemann Sum Intervals of 1 unit Daily Lessons and Assessments for AP* Calculus AB, A Complete Course Mark Sparks 2012 Page 499 Given the table of values below, approximate each definite integral by finding the indicated Riemann Sum. 25 a. Approximate f ( x)dx using a midpoint sum x 0 4 7 12 15 20 25 f(x) 15 6 –5 –10 –2 8 20 x 0 4 7 12 15 20 25 f(x) 15 6 –5 –10 –2 8 20 x 0 4 7 12 15 20 25 f(x) 15 6 –5 –10 –2 8 20 x 0 4 7 12 15 20 25 f(x) 15 6 –5 –10 –2 8 20 0 and three subintervals. 15 b. Approximate f ( x)dx using a left hand sum 0 and four subintervals. 20 c. Approximate f ( x)dx using a right hand sum 4 and four subintervals. 25 d. Approximate f ( x)dx using a trapezoidal sum 0 and three subintervals Daily Lessons and Assessments for AP* Calculus AB, A Complete Course Mark Sparks 2012 Page 500 Name_________________________________________Date____________________Class__________ Day #48 Homework Given below is a table of function values of h(x). Approximate each of the following definite integrals using the indicated Riemann or Trapezoidal sum, using the indicated subintervals of equal length. x –3 –1 1 3 5 7 9 h(x) 5 2 –3 –7 –2 6 11 1 1. 3. h(x)dx 3 9 using two subintervals and a Left 2. h(x)dx 3 using three subintervals and a Right Hand Riemann sum. Hand Riemann sum. 9 3 h(x)dx 3 using three subintervals and a Midpoint Riemann sum. 4. h(x)dx 3 using three subintervals and a Trapezoidal sum. 9 5. h(x)dx 3 using six subintervals and a Trapezoidal sum. Daily Lessons and Assessments for AP* Calculus AB, A Complete Course Mark Sparks 2012 Page 501 6. Approximate 2x sin xdx using four subintervals of equal length and a Right Hand Riemann sum. 0 e 2 x 2 dx using four subintervals of equal length and a Trapezoidal sum. 10 7. Approximate 2 8. Given the table to the right, approximate x −2 0 1 3 5 8 9 P(x) 5 8 2 −4 −1 2 5 x −2 0 1 3 5 8 9 P(x) 5 8 2 −4 −1 2 5 9 P(x)dx 2 using three subintervals and a Midpoint Riemann sum. 9. Given the table to the right, approximate 9 P(x)dx 2 using six subintervals and a Trapezoidal sum. Daily Lessons and Assessments for AP* Calculus AB, A Complete Course Mark Sparks 2012 Page 502 Applying the Fundamental Theorem of Calculus Connecting the Graphical, Analytical, and Numerical Approaches The Fundamental Theorem of Calculus, Part I Consider the function f(x) = –2x + 3 whose graph is pictured below. Calculate each of the following definite integrals according to the Fundamental Theorem of Calculus. Then, shade the area of the region that the integral represents. 1 Find 2 x 3dx . 2 5 Find 2 x 3dx . 2 Daily Lessons and Assessments for AP* Calculus AB, A Complete Course Mark Sparks 2012 Page 503 4 Find 2 x 3dx . 1 3 Find 2 x 3dx . 1 5 Find 2 x 3dx . 0 Daily Lessons and Assessments for AP* Calculus AB, A Complete Course Mark Sparks 2012 Page 504 Based on the results of the five previous examples, what inferences can you make about the value of the definite integral and the amount of area bounded by the graph of the integrand and the x – axis? Find each of the following definite integrals applying the fundamental theorem of calculus. Show your work. Then, use your graphing calculator to verify your results. 2x x dx 2 2 1 4 2x 3 1 x 2 dx 3 1 x dx 2 2 x 2x cos xdx 0 Daily Lessons and Assessments for AP* Calculus AB, A Complete Course Mark Sparks 2012 Page 505 Pictured below is a table of values show the values of a function, f(x), and its first and second derivative for selected values of x. Use the information in the table to answer the questions that follow. 1 1. What is the value of 3 f ' ( x)dx . 3 2. What is the value of f ' ( x) f ' ' ( x)dx ? 1 5 3. What is the value of 3 f ' ' ( x)dx ? 1 3 4. What is the value of 1 f ' ( x) 2 f ' ' ( x)dx ? 2 3 5. What is the equation of the tangent line to the graph of f(x) at x = 3? 6. Use the equation of the tangent line in #5 to approximate the value of f(3.1). Is this an over or under approximation of f(3.1)? Give a reason for your answer. Daily Lessons and Assessments for AP* Calculus AB, A Complete Course Mark Sparks 2012 Page 506 Name_________________________________________Date____________________Class__________ Day #49 Homework For exercises 1 – 6, find the value of the definite integral. Show your algebraic work. t 2 t dt 1 1. 1 3. 4 u 2 1 0 u du 5. (1 sin x)dx 2 2. 32 1dx 1 x 1 4. x 12 dx 2 x 3 6. 3x 2 5 x 4 dx 1 Daily Lessons and Assessments for AP* Calculus AB, A Complete Course Mark Sparks 2012 Page 507 Pictured to the right is the graph of a function f. In exercises 7 – 12, find the values of each of the following definite integrals. If a value does not exist, explain why. 2 7. f ( x)dx 8. 4 f ' ( x)dx 4 f ( x)dx 0 0 10. 1 3 1 11. f ' ( x)dx 1 9. f ( x)dx 1 3 12. f ' ( x)dx Daily Lessons and Assessments for AP* Calculus AB, A Complete Course Mark Sparks 2012 1 Page 508 Daily Lessons and Assessments for AP* Calculus AB, A Complete Course Mark Sparks 2012 Page 509 Daily Lessons and Assessments for AP* Calculus AB, A Complete Course Mark Sparks 2012 Page 510 Properties of Definite Integrals Given the integral statements, write what you think each is equivalent to. Be prepared to explain your reasoning with the rest of the class. a 1. f (x)dx a b 2. Given that a < c < b, f ( x)dx = a b 3. If f (x)dx K , a f ( x)dx = then a b b f ( x)dx = 4. Given that b < a, then a b 5. If k is a constant, then k f ( x)dx = a b 6. f (x) g(x)dx = a 7. Given that f(x) is an even function, 8. Given that f(x) is an odd function, a f ( x)dx = a a f ( x)dx = a Daily Lessons and Assessments for AP* Calculus AB, A Complete Course Mark Sparks 2012 Page 511 3 7 0 3 If f ( x)dx 6 and f ( x)dx 8 , determine the value of each of the following integrals using the properties of definite integrals. Explain how you arrived at your answer for each. 0 7 3 0 3 3 f ( x)dx f ( x)dx f ( x)dx 3 f ( x)dx 3 7 7 3 (2 3 f ( x))dx 3 f ( x)dx , if f(x) is an even function 3 3 f ( x)dx , if f(x) is an odd function 3 Daily Lessons and Assessments for AP* Calculus AB, A Complete Course Mark Sparks 2012 Page 512 Pictured to the right is the graph of a function f(x). 3 What is the value of f ( x)dx ? 0 4 What is the value of f ( x)dx ? 0 If F(0) = 5, what is the value of F(3), where F is the anti-derivative of f(x)? 3 What is the value of f ( x)dx ? 3 If F(–2) = –2, what is the value of F(2), where F is the anti-derivative of f(x)? Daily Lessons and Assessments for AP* Calculus AB, A Complete Course Mark Sparks 2012 Page 513 Name_________________________________________Date____________________Class__________ Day #50 Homework 6 6 2 2 Given f ( x)dx 10 and g ( x)dx 2 , find the values of each of the following definite integrals, if possible, by rewriting the given integral using the properties of integrals. 6 6 1. [ f ( x) g ( x)]dx 2. [2 f ( x) 3g ( x)]dx 2 Given 4 2 2 f ( x)dx 6 and 4 2 2 3. 6 g ( x)dx 6 6 g ( x) 4. dx 2 2 f ( x ) g ( x)dx 4 , find the values of each of the following definite integrals. Rewrite the given integral using the properties of integrals. Then, find the value. 4 5. [ f ( x) 4]dx 6. 2 4 2 3g ( x) xdx 4 1 f ( x) 3x 2 dx 2 2 7. Daily Lessons and Assessments for AP* Calculus AB, A Complete Course Mark Sparks 2012 Page 514 Pictured below is the graph of f ' ( x) , the first derivative of a function f(x). Use the graph to answer the following questions 8 –10. Graph of f ' ( x) 7 8. What is the value of f ' ( x)dx 0 9. If f(0) = –3, what is the value of f(3)? 10. If f(3) = –1, what is the value of f(7)? The graph of f ' ( x) , the derivative of a function, f(x), is pictured below on the interval [–2, 6]. Write and find the value of a definite integral to find each of the indicated values of f(x) below. Also, f(–2) = 5. 11. Find the value of f(0). 12. Find the value of f(6). Daily Lessons and Assessments for AP* Calculus AB, A Complete Course Mark Sparks 2012 Page 515 Problem #4 Let f be a function defined on the closed interval –3 < x < 4 with f(0) = 3. The graph of f ' , the derivative of f, consists of one line segment and a semicircle, as shown above. a. On what intervals, if any, is f increasing. Justify your reasoning. b. Find the x – coordinate of each point of inflection of the graph of f on the open interval –3 < x < 4. Justify your answer. c. Find an equation for the line tangent to the graph of f at the point (0, 3). d. Find f(–3) and f(4). Show the work that leads to your answers. Daily Lessons and Assessments for AP* Calculus AB, A Complete Course Mark Sparks 2012 Page 516 Daily Lessons and Assessments for AP* Calculus AB, A Complete Course Mark Sparks 2012 Page 517 Daily Lessons and Assessments for AP* Calculus AB, A Complete Course Mark Sparks 2012 Page 518 AP Calculus Quiz #10 Answer Key & Rubric Multiple Choice: ** 1. B A 2. A C 3. D A 4. E C 5. B A 6. C B 7. A E Free Response Part A – 2 points total ____ 1 f(x) is increasing and concave down on the interval (3, 4) … ____ 1 …because f ' ( x) > 0 and f ' ' ( x) < 0 on that interval. Free Response Part B – 2 points total k ____ 1 If f ' ( x)dx f (k ) f (1) 0 , then f(k) and f(1) must be the same value. 1 ____ 1 f(1) = 2 and the only other value for which f(k) = 2 is when k = 3. Free Response Part C – 2 points total 4 ____ 1 The value of f ( x)dx is positive… 1 ____ 1 …because all of the function values are positive making the region contained entirely above the x – axis. Free Response Part D – 3 points total ____ 1 Graph of f(x) pictured to the right is given. ____ 1 Trapezoidal Approximation ½(1)(2 + 1) + ½(1)(1 + 2) + ½(1)(2 +4) 6 ____ 1 Most students will say that it looks to be a pretty accurate approximation b/c the amount of area in the first two subintervals that is above the graph of f(x) is about the same size as the amount of area in the third subinterval that is under the graph of f(x) Daily Lessons and Assessments for AP* Calculus AB, A Complete Course Mark Sparks 2012 Page 519 AP CALCULUS QUIZ #10 Name____________________________________________________Date_____________________ 1. 2. 3. 4. 5. 6. 7. MULTIPLE CHOICE − Calculator NOT Permitted Pictured to the right is the graph of the function, f(x). Use the graph to answer questions 1 and 2. 7 1. Find f ' ( x)dx 0 A. 5.5 B. 4 C. 9 D. –4 E. 6 5 2. Find f ( x)dx . 0 A. 5.5 B. 11 D. 12.5 E. 9 C. 13 Daily Lessons and Assessments for AP* Calculus AB, A Complete Course Mark Sparks 2012 Page 520 3. x4 1 x2 dx = A. 1 x 3 x c B. 1 x 3 x c C. D. 3 1 x3 3 c 3 x3 3 1 x3 1 c 3 x E. 1 x 3 1 c 3 x 2 4. If (2 x 2)dx 3 , which of the following values is a possible value of k? k A. –2 B. 0 C. 1 D. 2 E. –1 5. A function, f(x) is such that f ' ( x) > 0 and f ' ' ( x) < 0 on the interval (2, 6). Which of the following statements can be made about the Riemann sum approximation on the interval? A. The left hand approximation will be an over approximation. B. The right hand approximation will be an over approximation. C. The trapezoidal approximation will be an over approximation. D. The right hand approximation will be an under approximation. E. None of these statements can be made. Daily Lessons and Assessments for AP* Calculus AB, A Complete Course Mark Sparks 2012 Page 521 Pictured to the right is the graph of f ' ( x) , the derivative of a function f(x). Use the graph for questions 6 and 7. 6. If f(0) = –3, what is the value of f(3)? A. 10.5 + π B. 10.5 + 2π C. 4.5 + π D. 7.5 + 2π E. 4.5 + 2π 7. Which of the following statements is/are true? I. f(x) is increasing on the interval (0, 5). II. f(x) has a point of inflection at x = 3. III. 4 f ' ( x)dx 10.5 0 A. I and II only B. II only D. I, II, and III E. I and III only C. II and III only FREE RESPONSE The table below describes the graph of a function f(x) on the interval 1< x < 4. x f(x) f ' ( x) f ' ' ( x) x=1 2 ----- 1<x<2 Positive Negative Positive x=2 1 0 Positive 2<x<3 Positive Positive Positive x=3 2 0 0 3<x<4 Positive Positive Negative x=4 4 ----- a. On what interval(s) is the graph of f(x) both increasing and concave down? Justify your reasoning. Daily Lessons and Assessments for AP* Calculus AB, A Complete Course Mark Sparks 2012 Page 522 k b. Other than k = 1, for what value of k is f ' ( x)dx 0 ? Justify your answer. 1 4 c. Based on the information in the table, determine if the value of f ( x)dx is positive or negative. 1 Give a reason for your answer. d. Sketch a graph of f(x) on the interval [1, 4]. Then, find the trapezoidal approximation the value of 4 1 f ( x)dx using 3 subintervals of equal length. Does this approximation appear to be greater or less than the actual value of the definite integral? Justify your answer. Daily Lessons and Assessments for AP* Calculus AB, A Complete Course Mark Sparks 2012 Page 523 AP CALCULUS *QUIZ #10* Name____________________________________________________Date_____________________ 1. 2. 3. 4. 5. 6. 7. MULTIPLE CHOICE − Calculator NOT Permitted Pictured to the right is the graph of the function, f(x). Use the graph to answer questions 1 and 2. 7 1. Find f ' ( x)dx 3 A. 2 B. 6 C. 9 D. –4 E. 4 6 2. Find f ( x)dx . 0 A. 5.5 B. 11 D. 12.5 E. 9 C. 6.5 Daily Lessons and Assessments for AP* Calculus AB, A Complete Course Mark Sparks 2012 Page 524 3. x4 1 x2 dx = A. 1 x 3 1 c B. 1 x 3 x c C. D. 3 x 1 x3 3 c 3 x3 3 1 x3 x c 3 E. 1 x 3 1 c 3 x 2 4. If (2 x 2)dx 3 , which of the following values is a possible value of k? k A. –2 B. 0 C. –1 D. 2 E. 1 5. A function, f(x) is such that f ' ( x) < 0 and f ' ' ( x) < 0 on the interval (2, 6). Which of the following statements can be made about the Riemann sum approximation on the interval? A. The left hand approximation will be an over approximation. B. The trapezoidal approximation will be an over approximation. C. The right hand approximation will be an over approximation. D. The left hand approximation will be an under approximation. E. None of these statements can be made. Daily Lessons and Assessments for AP* Calculus AB, A Complete Course Mark Sparks 2012 Page 525 Pictured to the right is the graph of f ' ( x) , the derivative of a function f(x). Use the graph for questions 6 and 7. 6. If f(1) = 4, what is the value of f(5)? A. 8 + 4π B. 16 + 2π C. −8 + 2π D. 16 + 4π E. 8 + 2π 7. Which of the following statements is/are true? I. f(x) is concave up on the interval (3, 5). II. f(x) is decreasing on the interval (−4, 0) III. 3 f ' ( x)dx 6 1 A. I and II only B. II only D. I, II, and III E. II and III only C. I and III only FREE RESPONSE The table below describes the graph of a function f(x) on the interval 1< x < 4. x f(x) f ' ( x) f ' ' ( x) x=1 2 ----- 1<x<2 Positive Negative Positive x=2 1 0 Positive 2<x<3 Positive Positive Positive x=3 2 0 0 3<x<4 Positive Positive Negative x=4 4 ----- a. On what interval(s) is the graph of f(x) both increasing and concave down? Justify your reasoning. Daily Lessons and Assessments for AP* Calculus AB, A Complete Course Mark Sparks 2012 Page 526 k b. Other than k = 1, for what value of k is f ' ( x)dx 0 ? Justify your answer. 1 4 c. Based on the information in the table, determine if the value of f ( x)dx is positive or negative. 1 Give a reason for your answer. d. Sketch a graph of f(x) on the interval [1, 4]. Then, find the trapezoidal approximation the value of 4 1 f ( x)dx using 3 subintervals of equal length. Does this approximation appear to be greater or less than the actual value of the definite integral? Justify your answer. Daily Lessons and Assessments for AP* Calculus AB, A Complete Course Mark Sparks 2012 Page 527 The Fundamental Theorem of Calculus, Particle Motion, and Average Value b Three Things to Always Keep In Mind: (1) v(t )dt p(b) p(a) , where v(t) represents the velocity a and p(t) represents the position. b (2) v(t )dt The Net Distance the particle travels on the a interval from t = a to t = b. If v(t) > 0 on the interval (a, b), then it also represents the Total Distance. b (3) v(t ) dt The Total Distance the particle travels on a the interval (a, b), whether or not v(t) > 0. To be safe, always do this integral when asked to find total distance when given velocity. 1. The velocity of a particle that is moving along the x – axis is given by the function v(t) = 3t2 + 6. (This is a non-calculator active question.) a. If the position of the particle at t = 4 is 72, what is the position when t = 2? b. What is the total distance the particle travels on the interval t = 0 to t = 7? Daily Lessons and Assessments for AP* Calculus AB, A Complete Course Mark Sparks 2012 Page 528 2. The velocity of a particle that is moving along the x – axis is given by the function v(t ) 0.5et t 23 . (This is a calculator active question.) a. If the position of the particle at t = 1.5 is 2.551, what is the position when t = 3.5? b. What is the total distance that the object travels on the interval t = 1 to t = 5? The graph of the velocity, measured in feet per second, of a particle moving along the x – axis is pictured below. The position, p(t), of the particle at t = 8 is 12. Use the graph of v(t) to answer the questions that follow. a. What is the position of the particle at t = 3? b. What is the acceleration when t = 5? c. What is the net distance the particle travels from t = 0 to t = 10? d. What is the total distance the particle travels from t = 0 to t = 10? Daily Lessons and Assessments for AP* Calculus AB, A Complete Course Mark Sparks 2012 Page 529 The table above shows values of the velocity, V(t) in meters per second, of a particle moving along the x – axis at selected values of time, t seconds. 18 a. What does the value of V (t )dt represent? 0 18 b. Using a left Riemann sum of 6 subintervals of equal length, estimate the value of V (t )dt . Indicate 0 units of measure. 18 c. Using a right Riemann sum of 6 subintervals of equal length, estimate the value of V (t )dt . Indicate 0 units of measure. 18 d. Using a midpoint Riemann sum of 3 subintervals of equal length, estimate the value of V (t )dt . 0 Indicate units of measure. 18 e. Using a trapezoidal sum of 6 subintervals of equal length, estimate the value of V (t )dt . 0 Indicate units of measure. f. Find the average acceleration of the particle from t = 3 to t = 9. For what value of t, in the table, is this average acceleration approximately equal to v’(t)? Explain your reasoning. Daily Lessons and Assessments for AP* Calculus AB, A Complete Course Mark Sparks 2012 Page 530 Name_________________________________________Date____________________Class__________ Day #52 Homework The graph to the right represents the velocity, v(t) in meters per second, of a particle that is moving along the x – axis on the time interval 0 < t < 10. The initial position of the particle at time t = 0 is 12. 1. On what interval(s) of time is the particle moving to the left and to the right? Justify your answer. 2. What is the total distance that the particle has traveled on the time interval 0 < t < 7. Leave your answer in terms of π. Indicate units of measure. 3. What is the net distance that the particle travels on the interval 5 < t < 10? Round your answer to the nearest thousandth. Indicate units of measure. 4. What is the acceleration of the particle at time t = 2? Indicate units of measure. 5. What is the position of the particle at time t = 5? Indicate units of measure. Daily Lessons and Assessments for AP* Calculus AB, A Complete Course Mark Sparks 2012 Page 531 Pictured to the right is the graph of a function which represents a particle’s velocity on the interval [0, 4]. Answer the following questions. 6. For what values is the particle moving to the right? Justify your answer. 7. For what values is the particle moving to the left? Justify your answer. 8. For what values is the speed of the particle increasing? Justify your answer. 9. For what values is the speed of the particle decreasing? Justify your answer. 10. What is the net distance that the particle travels on the interval [0, 4]? 11. What is the total distance that the particle travels on the interval [0, 4]? Daily Lessons and Assessments for AP* Calculus AB, A Complete Course Mark Sparks 2012 Page 532 A car travels on a straight track. During the time interval 0 < t < 60 seconds, the car’s velocity, v, measured in feet per second, and acceleration, a, measured in feet per second per second, are continuous functions. The table below shows selected values of these functions. t (sec) v(t) (ft/sec) a(t) (ft/sec2) 0 15 25 30 35 50 60 −20 −30 −20 −14 −10 0 10 1 5 2 1 2 4 2 12. Using appropriate units, explain the meaning of 60 0 v(t ) dt in terms of the car’s motion. Approximate this integral using a midpoint approximation with three subintervals as determined by the table. 13. Using appropriate units, explain the meaning of 50 15 a(t )dt in terms of the car’s motion. Find the exact value of the integral. 14. Is there a value of t such that a’(t) = 0? If so, on what interval does such a value exist? Justify your reasoning. 15. Using appropriate units, approximate the value of v’(31). What does this value say about the motion of the car at t = 31. 60 16. Using appropriate units, find the value and explain the meaning of 1 a(t )dt . 35 25 Daily Lessons and Assessments for AP* Calculus AB, A Complete Course Mark Sparks 2012 Page 533 Problem #3 Daily Lessons and Assessments for AP* Calculus AB, A Complete Course Mark Sparks 2012 Page 534 Problem #3 Daily Lessons and Assessments for AP* Calculus AB, A Complete Course Mark Sparks 2012 Page 535 Interpretations and Applications of the Derivative and the Definite Integral d [AMOUNT] The rate at which that amount is changing dx For example, if water is being drained from a swimming pool and R(t) represents the amount of water, measured in cubic feet, that is in a swimming pool at any given time, measured in hours, then R ' (t ) would represent the rate at which the amount of water is changing. d [ R(t )] R ' (t ) dx What would the units of R ' (t ) be?__________________________ b RATE AMOUNTOF CHANGE a b In the context of the example situation above, explain what this value represents: R ' (t )dt R(b) R(a) a . The table given below represents the velocity of a particle at given values of t, where t is measure in minutes. 0 5 10 15 20 25 30 t minutes 0 1.6 2.7 3.1 2.4 1.6 0 v(t) ft/minute 30 a. Approximate the value of v(t )dt using a midpoint Riemann Sum. Using correct units of measure, 0 explain what this value represents. 25 b. What is the value of a (t )dt , and using correct units, explain what this value represents. 5 Daily Lessons and Assessments for AP* Calculus AB, A Complete Course Mark Sparks 2012 Page 536 The temperature of water in a tub at time t is modeled by a strictly increasing, twice differentiable function, W, where W(t) is measured in degrees Fahrenheit and t is measured in minutes. Using the data in the table, estimate the value of W ' (12) . Using correct units, interpret the meaning of this value in the context of this problem. Use the data in the table to evaluate 20 W ' (t )dt. Using correct units, interpret the meaning of this integral 0 in the context of this problem. For 20 < t < 25, the function W that models the water temperature has a first derivative given by the function W ' (t ) 0.4 t cos(0.06t ) . Based on this model, what is the temperature of the water at time t = 25? Daily Lessons and Assessments for AP* Calculus AB, A Complete Course Mark Sparks 2012 Page 537 A pan of biscuits is removed from an oven at which point in time, t = 0 minutes, the temperature of the biscuits is 100°C. The rate at which the temperature of the biscuits is changing is modeled by the function B ' (t ) 13.84e 0.173t . Find the value of B ' (3) . Using correct units, explain the meaning of this value in the context of the problem. Sketch the graph of B ' (t ) on the axes below. Explain in the context of the problem why the graph makes sense. At time t = 10, what is the temperature of the biscuits? Show your work. Daily Lessons and Assessments for AP* Calculus AB, A Complete Course Mark Sparks 2012 Page 538 A cylindrical can of radius 10 millimeters is used to measure rainfall in Stormville. During the first 5 days of a 60-day period, 3 millimeters of rainfall had been collected. The height of water in the can is modeled by the function, S, where S(t) is measured in millimeters and t is measured in days for 5 < t < 60. The rate at which the height of the water is rising is given by the function S ' (t ) 2 sin(0.03t ) 1.5 . 15 Find the value of S ' (t)dt . Using correct units, explain the meaning of this value in the context of this 10 problem. At the end of the 60-day period, what is the volume of water that had accumulated in the can? Show your work. The rate at which people enter an auditorium for a concert is modeled by the function R given by R(t ) 1380t 2 675t 3 for 0 < t < 2 hours; R(t) is measured in people per hour. V.I.P. tickets were sold to 100 people who are already in the auditorium when the when the doors open at t = 0 for general admission ticket holders to enter. The doors close and the concert begins at t = 2. If all of the V.I.P. ticket holders stayed for the start of the concert, how many people are in the auditorium when the concert begins? Daily Lessons and Assessments for AP* Calculus AB, A Complete Course Mark Sparks 2012 Page 539 Name_________________________________________Date____________________Class__________ Day #53 Homework At time t = 0, there are 120 pounds of sand in a conical tank. Sand is being added to the tank at the rate of 2 S (t ) 2e sin t 2 pounds per hour. Sand from the tank is used at a rate of R(t ) 5 sin 2 t 3 t per hour. The tank can hold a maximum of 200 pounds of sand. 4 1. Find the value of S (t )dt . Using correct units, what does this value represent? 0 3 2. Find the value of R(t )dt . Using correct units, what does this value represent? 1 4 3. Find the value of 1 S (t )dt . Using correct units, what does this value represent? 4 0 4. Write a function, A(t), containing an integral expression that represents the amount of sand in the tank at any given time, t. 5. How many pounds of sand are in the tank at time t = 7? 6. After time t = 7, sand is not used any more. Sand is, however, added until the tank is full. If k represents the value of t at which the tank is at maximum capacity, write, but do not solve, an equation using an integral expression to find how many hours it will take before the tank is completely full of sand. Daily Lessons and Assessments for AP* Calculus AB, A Complete Course Mark Sparks 2012 Page 540 Problem #2 Daily Lessons and Assessments for AP* Calculus AB, A Complete Course Mark Sparks 2012 Page 541 Problem #3 Daily Lessons and Assessments for AP* Calculus AB, A Complete Course Mark Sparks 2012 Page 542 Problem #2 Daily Lessons and Assessments for AP* Calculus AB, A Complete Course Mark Sparks 2012 Page 543 Problem #4 Daily Lessons and Assessments for AP* Calculus AB, A Complete Course Mark Sparks 2012 Page 544 Average Value of a Function How have we found Average Velocity? How have we found Average Acceleration? If p(t), v(t), and a(t) represent position, velocity and acceleration defined for any time t, write an equivalent expression for each of the following integrals based on the fundamental theorem of calculus. b 1 a(t )dt ba To what is this equivalent? a b 1 v(t )dt ba To what is this equivalent? a The average value of a function, f(x), on an interval [a, b] is defined to be: Find the average value of the function f ( x) x 3 sin 2 x on the interval 1 < x < 3. [Calculator] Find the average value of the function f(x) = 2 – 4x on the interval 2 < x < 6. [Noncalculator] Daily Lessons and Assessments for AP* Calculus AB, A Complete Course Mark Sparks 2012 Page 545 A ski resort uses a snow machine to control the snow level on a ski slope. Over a 24-hour period the 14 volume of snow added to the slope per hour is modeled by the equation S (t ) 24 t sin 2 t . The rate at 3 which the snow melts is modeled by the equation M (t ) 10 8 cos t . Both S(t) and M(t) have units of cubic yards per hour and t is measured in hours for 0 < t < 24. At time t = 0, the slope holds 50 cubic yards of snow. a. Compute the total volume of snow added to the mountain over the first 6-hour period. 6 b. Find the value of M (t )dt and 0 6 1 M (t )dt . Using correct units of measure, explain what each 6 0 represents in the context of this problem. c. Is the volume of snow increasing or decreasing at time t = 4? Justify your answer. d. How much snow is on the slope after 5 hours? Show your work. e. Suppose the snow machine is turned off at time t = 10. Write, but do not solve, an equation that could be solved to find the time t = K when the snow would all be melted. Daily Lessons and Assessments for AP* Calculus AB, A Complete Course Mark Sparks 2012 Page 546 2003 AB #6 Part b Daily Lessons and Assessments for AP* Calculus AB, A Complete Course Mark Sparks 2012 Page 547 Name_________________________________________Date____________________Class__________ Day #54 Homework 1. Using a right Riemann sum over the given intervals, estimate 35 5 F (t )dt A. 730 B. 661 C. 564 D. 474 E. 325 2 2. For the first six seconds of driving, a car accelerates at a rate of a(t ) 10 sin1 t meters per 10 2 second . Which one of the following expressions represents the velocity of the car when it first begins to decelerate? A. 0.775 0 2.389 a(t )dt B. a(t )dt C. a(t )dt D. a(t )dt E. a(t )dt 0 1.715 0 4.627 0 3.830 0 3. The rate at which gas is flowing through a large pipeline is given in thousands of gallons per month in the chart below. Use a midpoint Riemann sum with two equal subintervals to approximate the number of gallons that pass through the pipeline in a year. A. 594,000 B. 672,000 C. 732,000 D. 744,000 E. 1,068,000 Daily Lessons and Assessments for AP* Calculus AB, A Complete Course Mark Sparks 2012 Page 548 4. Let f be a continuous function on the closed interval [1, 11]. If the values of f are given below at 11 three points, use a trapezoidal approximation to find 1 f ( x)dx using two subintervals. A. 165 B. 172 C. 190.5 D. 40 E. 80 5. If f ( x)dx 2a 3b , then f ( x) 3dx = b b a a A. 2a – 3b + 3 B. 3b – 3a C. –a D. 5a – 6b E. a – 6b Use the table below to answer questions 6 and 7. Suppose the function f(x) is a continuous function and f is the derivative of F(x). 3 6. What is f ( x)dx ? 1 A. 5 B. 8 C. 4 D. 19 E. Cannot be determined 7. If the area under the curve of f(x) on the interval 0 < x < 2 is equal to the area under the curve f(x) on the interval 2 < x < 3, then what is the value of A? A. 4 B. 2 C. 5.5 D. 6 E. Cannot be determined Daily Lessons and Assessments for AP* Calculus AB, A Complete Course Mark Sparks 2012 Page 549 Problem #3 Daily Lessons and Assessments for AP* Calculus AB, A Complete Course Mark Sparks 2012 Page 550 Problem #1 Daily Lessons and Assessments for AP* Calculus AB, A Complete Course Mark Sparks 2012 Page 551 Problem #2 Daily Lessons and Assessments for AP* Calculus AB, A Complete Course Mark Sparks 2012 Page 552 AP Calculus Quiz #11 Answer Key & Rubric Multiple Choice * 1. B E 2. E D 3. A C 4. C A 5. B C 6. A D 7. E B Free Response Part A – 2 points total ____ 1 The value of v’(5) = –3 feet per second per second (ft/sec2). ____ 1 This value represents the acceleration of the particle at t = 5 seconds. Free Response Part B – 3 points total ____ 1 The particle is moving to the right on the interval (0, 5). ____ 1 The particle is moving to the right on the interval (7, 9). ____ 1 The particle is moving to the right on these two intervals because v(t) > 0. Free Response Part C – 2 points total 9 ____ 1 The value of v(t )dt = 13.5 feet. 0 9 ____ 1 v(t )dt represents the net distance that the particle travels from 0 to 9 seconds. 0 Free Response Part D – 2 points total 9 ____ 1 The value of v(t ) dt = 19.5 feet. 0 9 ____ 1 v(t ) dt represents the total distance that the particle travels from 0 to 9 seconds. 0 Daily Lessons and Assessments for AP* Calculus AB, A Complete Course Mark Sparks 2012 Page 553 AP CALCULUS QUIZ #11 Name____________________________________________________Date_____________________ 1. 2. 3. 4. 5. 6. 7. MULTIPLE CHOICE – Graphing Calculator Permitted 1. A table of values of a continuous function is shown above. If four equal subdivisions of the interval 2 [0, 2] are used, what is the trapezoidal approximation of f ( x)dx ? 0 A. 24 B. 12 C. 10 D. 16 E. 20 Daily Lessons and Assessments for AP* Calculus AB, A Complete Course Mark Sparks 2012 Page 554 2. Pictured to the right is the graph of the derivative, f ' ( x) , of a 4 function. What is the value of f ' ( x)dx ? 0 A. 4 + B. 4 – 2 C. 4 + 2 D. 3 – 2 E. 4 – 3. What is the average value of the function f(x) = x 2 x 3 1 on the interval [0, 2]? A. 26 9 B. 52 9 C. 26 3 D. 52 3 E. 24 b b a a 4. If f ( x)dx a 2b , then what is the value of f ( x) 5dx ? A. a + 2b + 5 B. 5b – 5a C. 7b – 4a D. 7b – 5a E. 7b – 6a Daily Lessons and Assessments for AP* Calculus AB, A Complete Course Mark Sparks 2012 Page 555 5. Pictured to the right is the graph of the derivative of a function, g ' ( x) and it is known that g(0) = –3. What is the value of g(–4)? A. –6 – B. –5 – C. 2 + 2 D. 5 + E. –2 – 2 6. A spherical tank contains 81.637 gallons of water at time t = 0. For the next six minutes, water flows out of the tank at the rate of 9 sin t 1 gallons per minute. How many gallons of water are in the tank at the end of the six minutes? A. 36.606 B. 45.031 C. 68.858 D. 77.355 E. 126.668 7. The velocity of a particle, v(t), is given by the function v(t ) t 2 cos(t 2) . If the position of the particle at t = 2 is 10, what is the position when t = 5. A. 16.778 B. 16.206 C. 48.789 D. −16.778 E. 36.778 Daily Lessons and Assessments for AP* Calculus AB, A Complete Course Mark Sparks 2012 Page 556 FREE RESPONSE A particle is moving along the x – axis such that its velocity, v(t), measured in feet per second, at any given time, t, is graphed in the graph to the right. Use the graph to answer the following questions. a. What is the value of v’(5) and explain, using correct units, what this value represents. b. On what interval(s) of time is the particle moving to the right? Justify your answer. Daily Lessons and Assessments for AP* Calculus AB, A Complete Course Mark Sparks 2012 Page 557 9 c. What is the value of v(t )dt and explain the meaning of this value. Be sure to include correct 0 units in your answer. 9 d. What is the value of v(t ) dt and explain the meaning of this value. Be sure to include correct 0 units in your answer. Daily Lessons and Assessments for AP* Calculus AB, A Complete Course Mark Sparks 2012 Page 558 AP CALCULUS *QUIZ #11* Name____________________________________________________Date_____________________ 1. 2. 3. 4. 5. 6. 7. MULTIPLE CHOICE – Graphing Calculator Permitted 1. A table of values of a continuous function is shown above. If four equal subdivisions of the interval 2 [0, 2] are used, what is the trapezoidal approximation of f ( x)dx ? 0 A. 24 B. 20 C. 10 D. 16 E. 12 Daily Lessons and Assessments for AP* Calculus AB, A Complete Course Mark Sparks 2012 Page 559 2. Pictured to the right is the graph of the derivative, f ' ( x) , of a 4 function. What is the value of f ' ( x)dx ? 0 A. 4 + B. 4 – 2 C. 4 + 2 D. 4 – E. 3 – 2 3. What is the average value of the function f(x) = x 2 x 3 1 on the interval [0, 2]? A. 26 3 B. 52 9 C. 26 9 D. 52 3 E. 24 b b a a 4. If f ( x)dx a 2b , then what is the value of f ( x) 5dx ? A. 7b – 4a B. 5b – 5a C. a + 2b + 5 D. 7b – 5a E. 7b – 6a Daily Lessons and Assessments for AP* Calculus AB, A Complete Course Mark Sparks 2012 Page 560 5. Pictured to the right is the graph of the derivative of a function, g ' ( x) and it is known that g(0) = –3. What is the value of g(–4)? A. –6 – B. 2 + 2 C. –5 – D. 5 + E. –2 – 2 6. A spherical tank contains 81.637 gallons of water at time t = 0. For the next six minutes, water flows out of the tank at the rate of 9 sin t 1 gallons per minute. How many gallons of water are in the tank at the end of the six minutes? A. 77.355 B. 45.031 C. 68.858 D. 36.606 E. 126.668 7. The velocity of a particle, v(t), is given by the function v(t ) t cos(t 2) . If the position of the particle at t = 2 is 10, what is the position when t = 5. A. 16.778 B. 16.206 C. 48.789 D. −16.778 E. 36.778 Daily Lessons and Assessments for AP* Calculus AB, A Complete Course Mark Sparks 2012 Page 561 FREE RESPONSE A particle is moving along the x – axis such that its velocity, v(t), measured in feet per second, at any given time, t, is graphed in the graph to the right. Use the graph to answer the following questions. a. What is the value of v’(5) and explain, using correct units, what this value represents. b. On what interval(s) of time is the particle moving to the right? Justify your answer. Daily Lessons and Assessments for AP* Calculus AB, A Complete Course Mark Sparks 2012 Page 562 9 c. What is the value of v(t )dt and explain the meaning of this value. Be sure to include correct 0 units in your answer. 9 d. What is the value of v(t ) dt and explain the meaning of this value. Be sure to include correct 0 units in your answer. Daily Lessons and Assessments for AP* Calculus AB, A Complete Course Mark Sparks 2012 Page 563 AP Calculus Test #7 Answer Key & Rubrics Raw Score to Percentage Conversion Multiple Choice Calculator Permitted * 1. C E 2. E A 3. A C 4. D C 5. D B 6. D B 7. A C Calculator NOT Permitted * 8. D A 9. C E 10. E B 11. D E 12. C E 13. D B 14. B A Calculator NOT Permitted Free Response Part A – 3 point total 30 ____ 1 1 v(t )dt 1 (6)(7.5 12.5 13.5 14 13) 12.1 meters per second 30 0 30 ____ 1 Uses correct units of meters per second ____ 1 The value of 1 30 30 0 v(t )dt represents the average velocity during the first 30 seconds. Calculator NOT Permitted Free Response Part B – 1 point total ____ 1 Average Acceleration = v(18) v(6) 14.1 10.1 4 1 m/sec2 18 6 12 12 3 Calculator NOT Permitted Free Response Part C – 3 points total ____ 1 v’(6) v(9) v(3) 12.5 7.5 5 m/sec2 93 6 6 ____ 1 v’(6) represents the acceleration of the particle at t = 6 seconds. ____ 1 Since v(6) and v’(6) have the same sign, then the speed of the particle is increasing at t = 6. Calculator NOT Permitted Free Response Part D – 2 points total ____ 1 The particle has a negative acceleration on the interval (20, 30) because… ____ 1 …velocity is decreasing on this interval. Daily Lessons and Assessments for AP* Calculus AB, A Complete Course Mark Sparks 2012 Page 564 Calculator Permitted Free Response Part A – 2 points total 17 ____ 1 Correct setup: 15 9 E (t )dt 11 23 17 E (t )dt ____ 1 Correct answer: $104,048 Calculator Permitted Free Response Part B – 4 points total 17 ____ 1 H (17) 9 E (t ) L(t )dt 3725 people ____ 1 H(17) means that at 5 p.m., there are 3725 people in the park. H ' (t ) E (t ) L(t ) ____ 1 H ' (17) E (17) L(17) H ' (17) 380.281 people per hour ____ 1 Since H ' (17) < 0, the number of people in the park is decreasing at a rate of 380 people per hour. Calculator Permitted Free Response Part C – 3 points total ____ 1 H ' (t ) E(t ) L(t ) 0 when E(t) = L(t) which will occur when t = 15.794815 ____ 1 Correctly finds the value of H(9), H(15.794815), and H(23) H(9) = 0 15.794 H (15.794815) 9 H (23) 23 9 E (t ) L(t )dt 3950.680 E (t ) L(t )dt 1.014 ____ 1 According to the Extreme Value Theorem, the maximum number of people in the park at any given time is approximately 3950 or 3951 people. Daily Lessons and Assessments for AP* Calculus AB, A Complete Course Mark Sparks 2012 Page 565 AP CALCULUS AB TEST #7 Unit #6 – Basic Integration and Applications Name______________________________________________________Date_____________________ A GRAPHING CALCULATOR IS REQUIRED FOR SOME PROBLEMS OR PARTS OF PROBLEMS IN THIS PART OF THE EXAMINATION. (1) The exact numerical value of the correct answer does not always appear among the choices given. When this happens, select from among the choices the number that best approximates the exact numerical value. (2) Unless otherwise specified, the domain of a function f is assumed to be the set of all real numbers x for which f(x) is a real number. Multiple Choice k 1. If k is a constant, find its value if 2kx x 2 dx 18. 0 A. –9 B. –3 C. 3 D. 9 E. 18 Daily Lessons and Assessments for AP* Calculus AB, A Complete Course Mark Sparks 2012 Page 566 2. After being poured into a cup, coffee cools so that its temperature, T(t), is represented by the function t T (t ) 70 110e 2 , where t is measured in minutes and T(t) is measured in degrees Fahrenheit. What is the average temperature of the coffee during the first four minutes after being poured? A. 470.226 °F B. 1356.996 °F C. 7.443 °F D. 5427.984 °F E. 117.557 °F 3. Which of the following integrals is/are true if f(x) is a differentiable function on the open interval c b a c (a, b), c is on the open interval (a, b) , f ( x)dx 6 and f ( x)dx 2 . b I. f ( x)dx 4 a a II. c a b 1 3 III. 2 f ( x)dx 9 f ( x)dx f ( x)dx c b A. I and II only B. II only C. I only D. II and III only E. III only (t 3) 2 2 gallons per minute, where 4. At time t = 0 water begins leaking from a tank at the rate of L(t) = 5e t is measured in minutes. How much water has leaked out of the tank after 5 minutes? A. 0.621 gallons B. 0.676 gallons C. 1.353 gallons D. 12.231 gallons E. 15.769 gallons Daily Lessons and Assessments for AP* Calculus AB, A Complete Course Mark Sparks 2012 Page 567 5. The graph to the right represents the rate at which people arrive at an amusement park ride throughout the day, where t is measured in hours from the time the ride begins operation. If there were 275 people in line when the ride began operation, How many people have waited in line for the ride after 4 hours? A. 3800 B. 675 C. 400 D. 4075 E. 600 6. Using a right Riemann sum over the given intervals, 35 estimate F (t )dt . 5 A. 730 B. 661 C. 564 D. 474 E. 325 7. At 10 a.m. the temperature at a ski resort begins to increase causing the snow to begin to melt at a rate defined by the equation M (t ) 10 8 cos t . If there are 178 cubic yards of snow at that point, how 3 much snow remains at 5 p.m. if no additional snow has been added and the temperature has continually increased throughout the day? A. 90.646 cubic yards B. 265.354 cubic yards C. 96.177 cubic yards D. 69.646 cubic yards E. 73.128 cubic yards Daily Lessons and Assessments for AP* Calculus AB, A Complete Course Mark Sparks 2012 Page 568 Free Response The rate at which people enter an amusement park on a given day is modeled by the function E defined by 15600 . E (t ) 2 t 24t 160 The rate at which people leave the same amusement part on the same day is modeled by the function L defined by 9890 . L(t ) t 2 38t 370 Both E(t) and L(t) are measured in people per hour and time t is measured in number of hours after midnight. These functions are valid for 9 < t < 23, the hours during which the park is open. At t = 9, there are no people in the park. a. The price of admission to the park is $15 until 5:00 p.m. After 5:00 p.m., the price of admission to the park is $11. How many dollars are collected from admissions to the park on the given day? Round your answer to the nearest whole number. b. Let H (t ) E (t ) L(t ) dt for 9 < t < 23. Find the values of H(17) and H ' (17) and explain, using t 9 correct units, the meaning of both values in the context of the park. Daily Lessons and Assessments for AP* Calculus AB, A Complete Course Mark Sparks 2012 Page 569 c. During the hours that the park is open, 9 < t < 23, what is the maximum number of people in the park at any given moment? Show your work and justify your answer. Daily Lessons and Assessments for AP* Calculus AB, A Complete Course Mark Sparks 2012 Page 570 AP CALCULUS AB TEST #7 Unit #6 – Basic Integration and Applications Name______________________________________________________Date_____________________ A GRAPHING CALCULATOR IS NOT ALLOWED FOR THIS SECTION OF THE EXAM. (1) The exact numerical value of the correct answer does not always appear among the choices given. When this happens, select from among the choices the number that best approximates the exact numerical value. (2) Unless otherwise specified, the domain of a function f is assumed to be the set of all real numbers x for which f(x) is a real number. Multiple Choice 3 5 5 0 3 0 8. If f ( x)dx 6 and f ( x)dx 4 , then (3 2 f ( x))dx A. 10 B. 20 C. 23 D. 35 E. 50 3 9. If g(x) = x2 – 3x + 4 and f(x) = g ' ( x) , then f ( x)dx = 1 A. 14 3 B. –2 C. 2 D. 4 E. 14 3 Daily Lessons and Assessments for AP* Calculus AB, A Complete Course Mark Sparks 2012 Page 571 10. Given to the right is the graph of the derivative of a function, f ' ( x) . If f(0) = 9, what is the value of f(2)? y = f ' ( x) A. 11 + 2π B. –7 + 4π C. –7 + π D. 3 + 4π E. 11 + π 11. A left Riemann sum, a right Riemann sum, and a trapezoidal sum are 1 used to approximate the value of f ( x)dx , each using the same 0 number of subintervals. The graph of the function f is shown in the figure to the right. Which of the sums give(s) an underestimate of the 1 value of f ( x)dx ? 0 I. Left Sum II. Right Sum III. Trapezoidal Sum A. I only B. II only C. III only D. I and III only E. II and III only Daily Lessons and Assessments for AP* Calculus AB, A Complete Course Mark Sparks 2012 Page 572 12. A tank contains 50 liters of oil at time t = 4 hours. Oil is being pumped into the tank at a rate of R(t), where R(t) is measured in liters per hour, and t is measured in hours. Selected values of R(t) are given in the table above. Using a right Riemann sum with three subintervals and a data from the table, what is the approximation of the number of liters of oil that are in the tank at time t = 15 hours? A. 64.9 B. 68.2 C. 114.9 D. 116.6 E. 118.2 13. The function f is continuous on the closed interval [0, 6] and has the values given in the table above. 6 The trapezoidal approximation for f ( x)dx , found with 3 subintervals of equal length is 52. What 0 is the value of k? A. 2 B. 6 C. 7 D. 10 E. 14 Daily Lessons and Assessments for AP* Calculus AB, A Complete Course Mark Sparks 2012 Page 573 Use the table of values below to answer question 14. f(x) is a twice-differentiable function for which values of f, f ' , and f ' ' are given. x –3 –2 –1 0 1 f(x) 7 3 1 3 7 f ’(x) –5 –3 0 3 5 f ’’(x) 2 –1 –3 –2 0 1 14. Find the value of 2 f ' ( x) 3 f " ( x)dx . 3 A. 13 B. 30 C. 0 D. 9 E. 4 Free Response A particle is moving along a straight path. The velocity of the particle for 0 < t < 30 is shown in the table below for selected values of t and velocity is at a maximum at t = 20 sec. Answer the questions that follow. t 0 3 6 9 12 15 18 21 24 27 30 v(t) m/sec 0 7.5 10.1 12.5 13 13.5 14.1 14 13.9 13 12.2 30 v(t )dt . 30 0 a. Using the midpoints of five subintervals of equal length, approximate the value of 1 30 v(t )dt . 30 0 Using correct units, explain the meaning of the value of 1 Daily Lessons and Assessments for AP* Calculus AB, A Complete Course Mark Sparks 2012 Page 574 b. Find the average acceleration of the particle over the interval 6 < t < 18. Express your answer using correct units. c. Find an approximation of v ' (6) . Using correct units, explain what this value represents and state, providing justification, if the speed of the particle is increasing or decreasing at t = 6? d. During what interval(s) of time is the acceleration negative? Justify your answer. Daily Lessons and Assessments for AP* Calculus AB, A Complete Course Mark Sparks 2012 Page 575 AP CALCULUS AB TEST #7 *Unit #6 – Basic Integration and Applications* Name______________________________________________________Date_____________________ A GRAPHING CALCULATOR IS REQUIRED FOR SOME PROBLEMS OR PARTS OF PROBLEMS IN THIS PART OF THE EXAMINATION. (1) The exact numerical value of the correct answer does not always appear among the choices given. When this happens, select from among the choices the number that best approximates the exact numerical value. (2) Unless otherwise specified, the domain of a function f is assumed to be the set of all real numbers x for which f(x) is a real number. Multiple Choice k 1. If k is a constant, find its value if 2kx x 2 dx 18. 0 A. –9 B. 18 C. –3 D. 9 E. 3 Daily Lessons and Assessments for AP* Calculus AB, A Complete Course Mark Sparks 2012 Page 576 2. After being poured into a cup, coffee cools so that its temperature, T(t), is represented by the function t T (t ) 70 110e 2 , where t is measured in minutes and T(t) is measured in degrees Fahrenheit. What is the average temperature of the coffee during the first four minutes after being poured? A. 117.557 °F B. 1356.996 °F C. 7.443 °F D. 470.226 °F E. 5427.984 °F 3. Which of the following integrals is/are true if f(x) is a differentiable function on the open interval c b a c (a, b), c is on the open interval (a, b) , f ( x)dx 6 and f ( x)dx 2 . b I. f ( x)dx 4 a a II. c a b 1 3 III. 2 f ( x)dx 9 f ( x)dx f ( x)dx c b A. I only B. II only C. I and II only D. III only E. II and III only (t 3) 2 2 gallons per minute, where 4. At time t = 0 water begins leaking from a tank at the rate of L(t) = 5e t is measured in minutes. How much water has leaked out of the tank after 5 minutes? A. 0.621 gallons B. 0.676 gallons C. 12.231 gallons D. 1.353 gallons E. 15.769 gallons Daily Lessons and Assessments for AP* Calculus AB, A Complete Course Mark Sparks 2012 Page 577 5. The graph to the right represents the rate at which people arrive at an amusement park ride throughout the day, where t is measured in hours from the time the ride begins operation. If there were 275 people in line when the ride began operation, How many people have waited in line for the ride after 4 hours? A. 600 B. 4075 C. 400 D. 675 E. 3800 6. Using a left Riemann sum over the given intervals, 35 estimate F (t )dt . 5 A. 730 B. 661 C. 564 D. 474 E. 325 7. At 10 a.m. the temperature at a ski resort begins to increase causing the snow to begin to melt at a rate defined by the equation M (t ) 10 8 cos t . If there are 178 cubic yards of snow at that point, how 3 much snow remains at 4 p.m. if no additional snow has been added and the temperature has continually increased throughout the day? A. 90.646 cubic yards B. 265.354 cubic yards C. 96.177 cubic yards D. 69.646 cubic yards E. 259.823 cubic yards Daily Lessons and Assessments for AP* Calculus AB, A Complete Course Mark Sparks 2012 Page 578 Free Response The rate at which people enter an amusement park on a given day is modeled by the function E defined by 15600 . E (t ) 2 t 24t 160 The rate at which people leave the same amusement part on the same day is modeled by the function L defined by 9890 . L(t ) t 2 38t 370 Both E(t) and L(t) are measured in people per hour and time t is measured in number of hours after midnight. These functions are valid for 9 < t < 23, the hours during which the park is open. At t = 9, there are no people in the park. a. The price of admission to the park is $15 until 5:00 p.m. After 5:00 p.m., the price of admission to the park is $11. How many dollars are collected from admissions to the park on the given day? Round your answer to the nearest whole number. b. Let H (t ) E (t ) L(t ) dt for 9 < t < 23. Find the values of H(17) and H ' (17) and explain, using t 9 correct units, the meaning of both values in the context of the park. Daily Lessons and Assessments for AP* Calculus AB, A Complete Course Mark Sparks 2012 Page 579 c. During the hours that the park is open, 9 < t < 23, what is the maximum number of people in the park at any given moment? Show your work and justify your answer. Daily Lessons and Assessments for AP* Calculus AB, A Complete Course Mark Sparks 2012 Page 580 AP CALCULUS AB TEST #7 *Unit #6 – Basic Integration and Applications* Name______________________________________________________Date_____________________ A GRAPHING CALCULATOR IS NOT ALLOWED FOR THIS SECTION OF THE EXAM. (1) The exact numerical value of the correct answer does not always appear among the choices given. When this happens, select from among the choices the number that best approximates the exact numerical value. (2) Unless otherwise specified, the domain of a function f is assumed to be the set of all real numbers x for which f(x) is a real number. Multiple Choice 3 5 5 0 3 0 8. If f ( x)dx 6 and f ( x)dx 4 , then (3 2 f ( x))dx A. 35 B. 20 C. 10 D. 23 E. 50 3 9. If g(x) = x2 – 3x + 4 and f(x) = g ' ( x) , then f ( x)dx = 1 A. 14 3 B. –2 C. 14 3 D. 4 E. 2 Daily Lessons and Assessments for AP* Calculus AB, A Complete Course Mark Sparks 2012 Page 581 10. Given to the right is the graph of the derivative of a function, f ' ( x) . If f(0) = 9, what is the value of f(2)? y = f ' ( x) A. –7 + 4π B. 11 + π C. –7 + π D. 3 + 4π E. 11 + 2π 11. A left Riemann sum, a right Riemann sum, and a trapezoidal sum are 1 used to approximate the value of f ( x)dx , each using the same 0 number of subintervals. The graph of the function f is shown in the figure to the right. Which of the sums give(s) an underestimate of the 1 value of f ( x)dx ? 0 I. Left Sum II. Right Sum III. Trapezoidal Sum A. I only B. II only C. III only D. II and III only E. I and III only Daily Lessons and Assessments for AP* Calculus AB, A Complete Course Mark Sparks 2012 Page 582 12. A tank contains 50 liters of oil at time t = 4 hours. Oil is being pumped into the tank at a rate of R(t), where R(t) is measured in liters per hour, and t is measured in hours. Selected values of R(t) are given in the table above. Using a left Riemann sum with three subintervals and a data from the table, what is the approximation of the number of liters of oil that are in the tank at time t = 15 hours? A. 64.9 B. 68.2 C. 114.9 D. 116.6 E. 118.2 13. The function f is continuous on the closed interval [0, 6] and has the values given in the table above. 6 The trapezoidal approximation for f ( x)dx , found with 3 subintervals of equal length is 52. What 0 is the value of k? A. 6 B. 10 C. 7 D. 2 E. 14 Daily Lessons and Assessments for AP* Calculus AB, A Complete Course Mark Sparks 2012 Page 583 Use the table of values below to answer question 14. f(x) is a twice-differentiable function for which values of f, f ' , and f ' ' are given. x –3 –2 –1 0 1 f(x) 7 3 1 3 7 f ’(x) –5 –3 0 3 5 f ’’(x) 2 –1 –3 –2 0 0 14. Find the value of 2 f ' ( x) 3 f " ( x)dx . 1 A. 13 B. 30 C. 0 D. 9 E. 4 Free Response A particle is moving along a straight path. The velocity of the particle for 0 < t < 30 is shown in the table below for selected values of t and velocity is at a maximum at t = 20 sec. Answer the questions that follow. t 0 3 6 9 12 15 18 21 24 27 30 v(t) m/sec 0 7.5 10.1 12.5 13 13.5 14.1 14 13.9 13 12.2 a. Using the midpoints of five subintervals of equal length, approximate the value of 1 30 30 0 v(t )dt . 30 v(t )dt . 30 0 Using correct units, explain the meaning of the value of 1 Daily Lessons and Assessments for AP* Calculus AB, A Complete Course Mark Sparks 2012 Page 584 b. Find the average acceleration of the particle over the interval 6 < t < 18. Express your answer using correct units. c. Find an approximation of v ' (6) . Using correct units, explain what this value represents and state, providing justification, if the speed of the particle is increasing or decreasing at t = 6? d. During what interval(s) of time is the acceleration negative? Justify your answer. Daily Lessons and Assessments for AP* Calculus AB, A Complete Course Mark Sparks 2012 Page 585 Unit #7 – Advanced Integration and Applications AP Calculus Day Objective 57 Understand and use the Second Fundamental Theorem of Calculus and functions defined by integrals. Assignment Daily Lessons pages 591 – 595 Day #57 Homework 58 59 Finding integrals by substitution, including changing of variables Daily Lessons pages 599 – 602 and limits of integration. Day #58 Homework Find general and particular solutions for differential equations. Daily Lessons pages 605 – 608 2000 AB #6 Parts a and b 60 Identify and create slope fields that represent the general solutions of differential equations. 2008 AP #5 61 2010 (Form B) AP #5 Quiz #12 Finding the area between two curves. 62 Find volumes of solids of revolution using the disk method. Day #59 Homework 2001 AB #6 Parts a and b 2002 (Form B) AB #5 Parts a and b Daily Lessons pages 611 – 618 Day #60 Homework 2004 AB #6 Parts a, b, and c 2004 (Form B) AB #5 Parts a, b, and c Daily Lessons pages 635 – 638 Finish note packet begun in class. Daily Lessons pages 639 – 641 Day #62 Homework 63 More work with volumes of solids of revolution and find volumes of solids of known cross sections. 64 Quiz #13 2006 AP #3 2011 AB #3 65 Daily Lessons pages 645 – 647 Day #63 Homework Daily Lessons pages 652 – 655 2004 AP #5 2011 AB #4 Finish any of the 4 Free Responses not completed in class Study for Unit #7 Test Unit #7 Test – Advanced Integration and Applications Daily Lessons and Assessments for AP* Calculus AB, A Complete Course Mark Sparks 2012 Page 590 The Second Fundamental Theorem of Calculus Functions Defined by Integrals Given the functions, f(t), below, use F ( x) x 1 f (t )dt to find F(x) and F’(x) in terms of x. 1. f(t) = 4t – t2 2. f(t) = cos t Given the functions, f(t), below, use F ( x) x2 1 3. f(t) = t3 f (t )dt to find F(x) and F’(x) in terms of x. 4. f(t) = 6 t Daily Lessons and Assessments for AP* Calculus AB, A Complete Course Mark Sparks 2012 Page 591 Second Fundamental Theorem of Calculus Complete the table below for each function. Function F ' ( x) from page 584 Find F ' ( x) by applying the Second Fundamental Theorem of Calculus x F ( x) 4t t 2 dt 1 x F ( x) 4t t 2 dt 1 F ( x) cos t dt x 1 F ( x) cos t dt x 1 x2 3 t dt 1 F ( x) x2 3 t dt 1 F ( x) F ( x) x2 1 F ( x) x2 1 6 t dt 6 t dt Daily Lessons and Assessments for AP* Calculus AB, A Complete Course Mark Sparks 2012 Page 592 Find the derivative of each of the following functions. F ( x) 2x 2 2 t 2 dt G ( x) 3 x 2 e cos t dt H ( x) cos x 0 t 2 dt Pictured to the right is the graph of g(t) and the function f(x) is defined to be f ( x) 2x 4 g (t )dt . 1. Find the value of f(0). 2. Find the value of f(2). 3. Find the value of f ' (1) . 4. Find the value of f ' (2) . 5. Find the value of f ' ' (2) . Daily Lessons and Assessments for AP* Calculus AB, A Complete Course Mark Sparks 2012 Page 593 Given to the right is the graph of f(t) which consists of three line segments and one semicircle. Additionally, let the function g(x) be defined to be g ( x) x 1 f (t )dt . 1. Find g(–6). 2. Find g(6). 3. Find g ' (6) . 4. Find g ' (2) . 5. Find g ' ' (2) . Give a reason for your answer. 6. Find g ' ' (4) . Give a reason for your Answer. Daily Lessons and Assessments for AP* Calculus AB, A Complete Course Mark Sparks 2012 Page 594 The continuous function f is defined on the interval –4 < x < 3. The graph consists of two quarter circles x and one line segment, as show in the figure above. Let g ( x) 1 x 2 f (t )dt . 2 0 Find the value of g(3). Find the value of g(−4). Find the value of g ' (3) . Find the value of g ' ' (2) . Daily Lessons and Assessments for AP* Calculus AB, A Complete Course Mark Sparks 2012 Page 595 Name_________________________________________Date____________________Class__________ Day #57 Homework Find the derivative of each of the following functions defined by integrals. 1. g ( x) 3x x4 (2t 3)dt 2. h( x) t 2t dt 2x 4. H ( x) 2 3. f ( x) 1 2 5. P( x) x 2 2x 2 3t 2dt 2 3 t dt cos x 5 2t 2 dt e t t dt ln x 6. f ( x) 2 Daily Lessons and Assessments for AP* Calculus AB, A Complete Course Mark Sparks 2012 Page 596 Pictured to the right is the graph of f(t)and F ( x) 2x 6f (t )dt . Use the graph and F(x) to answer the questions 7 – 11. 7. Find the value of F(0). 8. Find the value of F 1 . 2 9. Find the value of F ' (2) . 10. Find the value of F ' (2.5) . Pictured to the right is the graph of f and G ( x) 11. Find the value of F ' ' (0) x 2f (t)dt . Use the graph to answer questions 12 – 15. 12. Find the value of G(3). 13. Find the value of G(–4). 14. Find the value of G ' (2) . 15. Find the value of G ' ' (5) . Daily Lessons and Assessments for AP* Calculus AB, A Complete Course Mark Sparks 2012 Page 597 x If g ( x) t 3e t dt , find each of the following values in questions 16 – 17. 0 16. Find the value of g ' (1). 17. Find the value of g ' ' (1). 2 If h( x) 2 1 t 4 dt , find each of the following values in questions 18 – 19. x 18. Find h' ( x). 19. Find h' ' (1). Daily Lessons and Assessments for AP* Calculus AB, A Complete Course Mark Sparks 2012 Page 598 Integration of Composite Functions For each of the functions given below, find both f ' ( x) and f ' ( x)dx . f ' ( x) f(x) f ' ( x)dx f ( x) sin 3x f ( x) e cos x f ( x) ln x 2 3 f ( x) x2 3 Anti-differentiation by Pattern Recognition d f ( g ( x)) ____________________________________ dx f ' ( g ( x)) g ' ( x) dx _____________________________ Daily Lessons and Assessments for AP* Calculus AB, A Complete Course Mark Sparks 2012 Page 599 Find each of the following indefinite integrals by pattern recognition. 3 cos 3x dx 2x x 5 dx 2 sin(2x 3) dx x2x22x dx cos(3x 2)dx 3x x 2 dx 5e dx 2x 3 dx 3x 3 2 2 2 Daily Lessons and Assessments for AP* Calculus AB, A Complete Course Mark Sparks 2012 3x 2 Page 600 Ant-differentiation by U–Substitution In each of the eight examples above, the g ' ( x) , or “license to integrate,” existed in the integrand of f ' ( g ( x)) g ' ( x) dx or g ' ( x) was attainable by multiplying by a constant. The g ' ( x) does not always exist and there are times when it is not attainable by multiplication of a constant. Consider the example below. x(2x 1) dx 3 Identify the “inner function,” g(x): _________________________ What is g ' ( x) ? ___________________ Is g ' ( x) part of the integrand? _____________________ Is g ' ( x) attainable by multiplying the integrand by a constant? ________________________ In this case, we must find the anti-derivative by a method known as U-Substitution. Here is how it works. 1. Let u = the inner function, g(x). 4. Rewrite the entire integrand as a polynomial or polynomial type of function in terms of u. Then, anti-differentiate. 2. Find du and solve the equation for dx. 3. Find an expression for x in terms of u. Daily Lessons and Assessments for AP* Calculus AB, A Complete Course Mark Sparks 2012 Page 601 2xx14 dx 1. Let u = the inner function, g(x). 4. Rewrite the entire integrand as a polynomial or polynomial type of function in terms of u. Then, anti-differentiate. 2. Find du and solve the equation for dx. 3. Find an expression for x in terms of u. 4 Find the value of x 2x 1 dx . Then, check the result using the graphing calculator. 0 Daily Lessons and Assessments for AP* Calculus AB, A Complete Course Mark Sparks 2012 Page 602 Name_________________________________________Date____________________Class__________ Day #58 Homework In problems 1 – 6, find the indefinite integral. x 3 x 4 3 dx 3 2. x 1 2x 2 dx 3. x 3 sin x 4 dx 4. 1 x dx 5. 5x 1 x 2 dx 6. u u 2du 1. 3 x3 4 2 2 3 Daily Lessons and Assessments for AP* Calculus AB, A Complete Course Mark Sparks 2012 Page 603 For problems 7 and 8, find the indefinite integral by using substitution. 7. x 2x 1 dx 8. 5 9. Find the value of x 1 2 x dx 2x 1 dx . Show your work and then check using a graphing calculator. x 1 Daily Lessons and Assessments for AP* Calculus AB, A Complete Course Mark Sparks 2012 Page 604 Solving Differential Equations Examples of Variable Separable Differential Equations Given below are differential equations with given initial condition values. Find the particular solution for each differential equation. 1. dy 6 x 2 6 x 2 and f(–1) = 2 3. 3 dx 2 2. dy 1 12 x and f(0) = 2 dx 2 x dy x 2 2 x and f(0) = 2 dx 2y 4. dy x 2 and f(1) = –3 dx y Daily Lessons and Assessments for AP* Calculus AB, A Complete Course Mark Sparks 2012 Page 605 5. dy x 4 ( y 2) and f(0) = 0 dx 6. dy y 1 and f(2) = 0 dx x2 Daily Lessons and Assessments for AP* Calculus AB, A Complete Course Mark Sparks 2012 Page 606 Daily Lessons and Assessments for AP* Calculus AB, A Complete Course Mark Sparks 2012 Page 607 The acceleration of a particle moving along the x – axis at time t is given by a(t) = 6t – 2. If the velocity is 25 when t = 3 and the position is 10 when t = 1, then the position x(t) = A. 9t2 + 1 B. 3t2 – 2t + 4 C. t3 – t2 + 4t + 6 D. t3 – t2 + 9t – 20 E. 36t3 – 4t2 – 77t + 55 A particle moves along the x-axis so that, at any time t > 0, its acceleration is given by a(t) = 6t + 6. At time t = 0, the velocity of the particle is –9 and its position is –27. a. Find v(t), the velocity of the particle at any time t. b. Find the net distance traveled by the particle over the interval [0, 2]. c. Find the total distance traveled by the particle over the interval [0, 2]. Daily Lessons and Assessments for AP* Calculus AB, A Complete Course Mark Sparks 2012 Page 608 Name_________________________________________Date____________________Class__________ Day #59 Homework Daily Lessons and Assessments for AP* Calculus AB, A Complete Course Mark Sparks 2012 Page 609 Daily Lessons and Assessments for AP* Calculus AB, A Complete Course Mark Sparks 2012 Page 610 Slope Fields Graphical Representations of Solutions to Differential Equations A slope field is a pictorial representation of all of the possible solutions to a given differential equation. Remember that a differential equation is the first derivative of a function, f ' ( x) or dy . Thus, the dx solution to a differential equation is the function, f(x) or y. There is an infinite number of solutions to the differential equation dy x 3 . Show your work and dx explain why. For the AP Exam, you are expected to be able to do the following four things with slope fields: 1.________________________________________________________________________________ ________________________________________________________________________________ 2.________________________________________________________________________________ ________________________________________________________________________________ 3.________________________________________________________________________________ ________________________________________________________________________________ 4.________________________________________________________________________________ ________________________________________________________________________________ Daily Lessons and Assessments for AP* Calculus AB, A Complete Course Mark Sparks 2012 Page 611 #1 Sketch a slope field for a given differential equation. Given the differential equation below, compute the slope for each point Indicated on the grid to the right. Then, make a small mark that approximates the slope through the point. dy x 1 dx Given the differential equation below, compute the slope for each point Indicated on the grid to the right. Then, make a small mark that approximates the slope through the point. dy x y dx Daily Lessons and Assessments for AP* Calculus AB, A Complete Course Mark Sparks 2012 Page 612 #2 Given a slope field, sketch a solution curve through a given point. To the right is pictured the slope field that you developed for the differential equation on the previous page. dy x 1 dx Sketch the solution curve through the point (1, -1). To do this, you find the point and then follow the slopes as you connect the lines. Daily Lessons and Assessments for AP* Calculus AB, A Complete Course Mark Sparks 2012 Page 613 #3 Match a slope field to a differential equation. Since the slope field represents all of the particular solutions to a differential equation, and the solution represents the ANTIDERIVATIVE of a differential equation, then the slope field should take the shape of the antiderivative of dy/dx. Match the slope fields to the differential equations on the next page. A. B. C. D. E. F. G. H. I. J. Daily Lessons and Assessments for AP* Calculus AB, A Complete Course Mark Sparks 2012 Page 614 Separate the variables and find the general solution to each differential equation below to determine what the slope field should look like for each. Then, match to the graphs of slope fields on the previous page. 1. dy sin x dx 2. dy 2x 4 dx 3. dy ex dx 4. dy 2 dx 5. dy x 3 3x dx 6. dy 2 cos x dx 7. dy 4 2x dx 8. dy x dx 9. dy x2 dx 10. dy 1 dx x Daily Lessons and Assessments for AP* Calculus AB, A Complete Course Mark Sparks 2012 Page 615 #4 Match a slope field to a solution to a differential equation. When given a slope field and a solution to a differential equation, then the slope field should look like the solution, or y. Match the slope fields below to the solutions on the next page. A. B. C. D. E. F. G. H. I. J. Daily Lessons and Assessments for AP* Calculus AB, A Complete Course Mark Sparks 2012 Page 616 1. y x 2. y x 2 3. y e x 4. y 5. y x 3 6. y sin x 7. y cos x 8. y x 9. y 1 10. y tan x 1 x2 Daily Lessons and Assessments for AP* Calculus AB, A Complete Course Mark Sparks 2012 Page 617 Shown below is a slope field for which of the following differential equations? Explain your reasoning for each of the choices below. Consider the differential equation dy x to answer the following questions. dx y a. On the axes below, sketch a slope field for the equation. b. Sketch a solution curve that passes through the point (0, –1) on your slope field. c. Find the particular solution y = f(x) to the differential equation with the initial condition f(0) = –1. Daily Lessons and Assessments for AP* Calculus AB, A Complete Course Mark Sparks 2012 Page 618 Name_________________________________________Date____________________Class__________ Day #60 Homework For the indicated points on each grid, draw the slope field for the given differential equation. 1. dy x y dx 2. dy y dx x 3. dy x 1 dx 4. dy 1 dx x 1 Daily Lessons and Assessments for AP* Calculus AB, A Complete Course Mark Sparks 2012 Page 619 Daily Lessons and Assessments for AP* Calculus AB, A Complete Course Mark Sparks 2012 Page 620 Daily Lessons and Assessments for AP* Calculus AB, A Complete Course Mark Sparks 2012 Page 621 Daily Lessons and Assessments for AP* Calculus AB, A Complete Course Mark Sparks 2012 Page 622 Daily Lessons and Assessments for AP* Calculus AB, A Complete Course Mark Sparks 2012 Page 623 AP Calculus Quiz #12 Answer Key & Rubric Multiple Choice: * 1. D C 2. A E 3. C A 4. E A 5. A B 6. A E 7. B A Free Response Part A – 2 points total 4 ____ 1 g (4) f (t )dt 5 2 1 ____ 1 g (2) 2 1 f (t )dt 2 1 f (t )dt 6 Free Response Part B – 2 points total x ____ 1 If g ( x) f (t )dt , then g ' ( x) f ( x) 1 . So, g ' (1) f (1) 4 1 ____ 1 If g ' ( x) f ( x) , then g ' ' ( x) f ' ( x) . g ' ' (1) f ' (1) is undefined because the graph of f has a cusp when x = 1. Free Response Part C – 3 points total ____ 1 Identifies the only relative maximum of g on (–2, 4) to be x = 3. 3 ____ 1 Finds the value of g (3) f (t )dt 1 (1)(1 4) 1 (1)(1) 3 . 1 2 2 ____ 1 On a closed interval, the absolute maximum occurs at an endpoint of the interval or at any relative maximum on the interval. g(–2) = –6, g(3) = 3, and g(4) = 5 . Thus, since g(3) is the 2 greatest, the point (3, 3) is the absolute maximum of g on the interval [–2, 4]. Free Response Part D – 2 points total ____ 1 The graph of g has a point of inflection when g ' ' f ' changes signs. f ' changes signs when the graph of f has a relative maximum or minimum. ____ 1 The graph of f changes from increasing to decreasing at x = 1. Thus, g has a point of inflection at x = 1. Daily Lessons and Assessments for AP* Calculus AB, A Complete Course Mark Sparks 2012 Page 624 AP CALCULUS QUIZ #12 Name____________________________________________________Date_____________________ Calculator Permitted 1. 2. 3. 4. 5. 6. 7. MULTIPLE CHOICE − Graphing Calculator NOT Permitted The graph of a function, f, which consists of a two line segments and a semi-circle is pictured below. Let G ( x) x 2 2x 2f (t)dt . Use this information to answer questions 1 and 2. 1. What is the value of G(2)? A. 4 + 2π B. –1 + 2π C. 5 + 2π D. 3 + 2π E. 4 Daily Lessons and Assessments for AP* Calculus AB, A Complete Course Mark Sparks 2012 Page 625 2. Find the value of G ' (2) . A. 4 B. 12 C. 7 D. 27 E. 8 3. If g(x) = x2 3t 1 3 t 1 dt , then what is the value of g ' (2) ? A. –3 B. 8 3 C. 48 65 2 D. 3 E. 12 65 4. If g ( x) x 0 t 3e t dt , find g ' ' (1) . A. e B. 2e C. e – 1 D. 3e E. 4e Daily Lessons and Assessments for AP* Calculus AB, A Complete Course Mark Sparks 2012 Page 626 5. 2x 2 x 2 dx 3 C 3 B. 1 x 3 2 C 3 C. 2 x 3 2 C 3 D. 2x 3 2 C E. 3x 3 2 C 1 A. 4 x 3 2 1 1 1 1 2 2 2 2 2 x 3 2x 3dx 6. If u = 2x – 3, then C. 1 3 u du 2 E. 1 (2u 3)3 u du 2 4 1 A. 1 u 3 3u 3 du 4 7. If 2 dy x dx y D. 1 3 u du 4 2 4 B. 1 u 3 3u 3 du 2 and f(0) = –4, find the particular solution to the differential equation. A. f(x) = 1 x 3 4 3 B. f(x) = 2 x 3 16 3 2 x 3 16 3 C. f(x) = D. f(x) = 1 x 3 3 E. f(x) = 2 x 3 8 3 Daily Lessons and Assessments for AP* Calculus AB, A Complete Course Mark Sparks 2012 Page 627 FREE RESPONSE The graph of a function f, consisting of three line segments, is pictured above. Let g ( x) x 1 f (t)dt . a. Compute the values of g(4) and g(–2). b. Find g ' (1) and g ' ' (1) . Show or explain your work. Daily Lessons and Assessments for AP* Calculus AB, A Complete Course Mark Sparks 2012 Page 628 c. Find the coordinates of the absolute maximum of g on the closed interval [–2, 4]. Justify your answer. d. The second derivative of g is not defined at x = 1 and x = 2. Which of these two values is/are coordinates of points of inflection of the graph of g? Justify your answer. Daily Lessons and Assessments for AP* Calculus AB, A Complete Course Mark Sparks 2012 Page 629 AP CALCULUS *QUIZ #12* Name____________________________________________________Date_______________________ Calculator Permitted 1. 2. 3. 4. 5. 6. 7. MULTIPLE CHOICE − Graphing Calculator NOT Permitted The graph of a function, f, which consists of a two line segments and a semi-circle is pictured below. Let G ( x) x 2 2x 2f (t)dt . Use this information to answer questions 1 and 2. 1. What is the value of G(1)? A. 2 + π B. –1 + 2π C. 1 + 2π D. 3 + 2π E. 1 + π Daily Lessons and Assessments for AP* Calculus AB, A Complete Course Mark Sparks 2012 Page 630 2. Find the value of G ' (1) . A. 4 B. 8 C. 7 D. 27 E. 2 x2 3t 1 3 3. If g(x) = t 1 dt , then what is the value of g ' (1) ? A. –3 B. 8 3 C. 24 65 3 D. 2 E. 12 65 4. If g ( x) x 3 t 0 t e dt , find g ' ' (1) . A. 4e B. 2e C. e – 1 D. 3e E. e Daily Lessons and Assessments for AP* Calculus AB, A Complete Course Mark Sparks 2012 Page 631 5. x 2 dx 2x 2 3 C 3 B. 4 x 3 2 C 3 C. 2 x 3 2 C 3 D. 2x 3 2 C E. 3x 3 2 C A. 1 x 3 2 1 2 1 1 1 1 2 2 2 2 x 2x 3dx 3 6. If u = 2x – 3, then C. u du E. u 3u du A. 1 (2u 3)3 u du 2 1 2 1 4 7. If 2 dy x dx y 3 4 1 3 D. u du 2 4 B. 1 u 3 3u 3 du 2 1 4 3 3 and f(0) = –4, find the particular solution to the differential equation. A. f(x) = 2 x 3 16 3 B. f(x) = 1 x 3 4 3 2 x 3 16 3 C. f(x) = D. f(x) = 1 x 3 3 E. f(x) = 2 x 3 8 3 Daily Lessons and Assessments for AP* Calculus AB, A Complete Course Mark Sparks 2012 Page 632 FREE RESPONSE The graph of a function f, consisting of three line segments, is pictured above. Let g ( x) x 1 f (t)dt . a. Compute the values of g(4) and g(–2). b. Find g ' (1) and g ' ' (1) . Show or explain your work. Daily Lessons and Assessments for AP* Calculus AB, A Complete Course Mark Sparks 2012 Page 633 c. Find the coordinates of the absolute maximum of g on the closed interval [–2, 4]. Justify your answer. d. The second derivative of g is not defined at x = 1 and x = 2. Which of these two values is/are coordinates of points of inflection of the graph of g? Justify your answer. Daily Lessons and Assessments for AP* Calculus AB, A Complete Course Mark Sparks 2012 Page 634 Finding the Area between Two Curves An Application of Integration Graph the function f ( x) 12 x 2 2 x 4 and find the value of 0 5 f ( x)dx . Using one color, shade the region for which this value represents the area. Graph the function g ( x) 1 x 4 on the same grid above and then find the value of 2 0 5g ( x)dx . Using a different color, shade the region for which this value represents the area. What do you suppose you would do to find the area of the region that is located in between the graphs of f(x) and g(x)? Find this value. Daily Lessons and Assessments for AP* Calculus AB, A Complete Course Mark Sparks 2012 Page 635 Now, find the value of the definite integral below if f ( x) 12 x 2 2 x 4 and g ( x) 1 x 4 . Show 2 your work. 0 f ( x) g ( x)dx 5 What do you notice about this value? This brings about the general way that we will find the area between two curves. Find the area of the shaded region, R, that is bounded by y = sin(x) and y = x3 – 4x. Daily Lessons and Assessments for AP* Calculus AB, A Complete Course Mark Sparks 2012 Page 636 3 2 4 3 Pictured to the right is the graph of f ( x) x x x 3 cos x 2 and a line, l, which is tangent to f(x) at the point (0, 3). Find the area of Region R. Find the equation of line l if it is tangent to the graph of f(x) at (0, 3). At what ordered pair, other than (0, 3), does the graph of line l intersect the graph of f(x)? Find the area of Region S. Daily Lessons and Assessments for AP* Calculus AB, A Complete Course Mark Sparks 2012 Page 637 Pictured to the right are regions R and S, which are formed by the graphs of f ( x) 1 sin(x) and g ( x) 4 x 4 Identify the points of intersection of f(x) and g(x). Find the area of Region R. Find the area of Region S. Find the area of the unshaded region bounded by the graphs of f, g, and the x – axis. Daily Lessons and Assessments for AP* Calculus AB, A Complete Course Mark Sparks 2012 Page 638 Volumes of Solids of Revolution A solid of revolution is formed when a flat, two-dimensional shape is rotated around an axis. Consider the flat region below to the left. When that region is rotated about the x – axis, the solid pictured below to the right is formed. This objective of this lesson is to learn to find the volume of such a solid. Now, imagine slicing the solid into individual discs of height 1 unit. The volume of one of those discs is V = πr2h, or V = πr2. b Sum of all the discs = a f ( x) Axis of Rotation dx Volume of the Solid = 2 b f ( x) Axis of Rotation 2 dx a Notice that the axis of rotation is the x – axis and the bottom function of the region is also the x – axis. Imagine for a moment what the solid would look like if the axis of rotation were still the x – axis but the bottom function of the region was the line y = c. The solid would look similar except for the fact that there would be a cylinder that is cut out of the center. To find the volume of this solid, we would find the volume of the whole solid that we found previously and then subtract out the solid in the form of a cylinder. In order to do this, we use the formula below to find the volume of such a solid. Volume OuterFunct ion axis 2 InnerFunct ion axis 2 dx b a The “outer function” is defined to be the function that is farther from the axis of rotation. The “inner function” is defined to be the function that is closer to the axis of rotation. Daily Lessons and Assessments for AP* Calculus AB, A Complete Course Mark Sparks 2012 Page 639 If the axis of rotation is the x-axis or is parallel to the x-axis, the integrand needs to be in terms of x and the limits of integration need to be the x-values of the points of intersection of the curves that form the region being rotated. Consider the region pictured to the right that is bounded by the graphs of y = x2 and y = x + 2. Find the volume of the solid formed when the region is rotated about the x – axis. Find the volume when the region is rotated about the line y = 4. Find the volume when the region is rotated about the line y = –2. Daily Lessons and Assessments for AP* Calculus AB, A Complete Course Mark Sparks 2012 Page 640 If the axis of rotation is the y-axis or is parallel to the y-axis, the integrand needs to be in terms of y and the limits of integration need to be the y-values of the points of intersection of the curves that form the region being rotated. Consider the region pictured to the right that is bounded by the graphs y = x and y = x – 2. Find the volume when the region is rotated about the y – axis. Find the volume when the region is rotated about the line x = 4. Find the volume when the region is rotated about the line x = 7. Daily Lessons and Assessments for AP* Calculus AB, A Complete Course Mark Sparks 2012 Page 641 Name_________________________________________Date____________________Class__________ Day #62 Homework Let R be the region bounded by the graphs of y = ln x and the line y = x – 2 as shown below. Though you may use a calculator, show the integral that you found to arrive at your answer. 1. Find the coordinates of the points at which the two graphs intersect each other. Then, find the area of R. 2. Find the volume of the solid generated when R is rotated about the horizontal line y = –3. 3. Write and evaluate an integral expression that can be used to find the volume of the solid generated when R is rotated about the y-axis. Let f and g be the functions given by f ( x) 1 sin(x) and g ( x) 4 x . Let R be the region in the first 4 quadrant enclosed by the y-axis and the graphs of f and g, and let S be the region in the first quadrant enclosed by the graphs of f and g shown to the right. Though you may use a calculator, show the integral that you found to arrive at your answer. 4. Find the points of intersection of f and g. 5. Find the area of the region bounded by the graphs of f and g and the x – axis. Daily Lessons and Assessments for AP* Calculus AB, A Complete Course Mark Sparks 2012 Page 642 4. Find the volume of the solid generated when R is revolved about the horizontal line y = 8. 5. Find the volume of the solid generated when S is revolved about the horizontal line y = –1. Daily Lessons and Assessments for AP* Calculus AB, A Complete Course Mark Sparks 2012 Page 643 CALCULATOR NOT PERMITTED Daily Lessons and Assessments for AP* Calculus AB, A Complete Course Mark Sparks 2012 Page 644 Volumes of Solids with Known Cross Sections Calculus cannot only be used to find the volume of solids created by revolving two-dimensional shapes around an axis but also the volume of solids formed by cross sections that are geometric shapes. In this lesson, you will derive the formulas for finding volumes of solids given that their cross sections are squares, isosceles right triangles, equilateral triangles, and semi circles. Cross Sections that are Squares Find the area of the square above in terms of f(x) and g(x). If all of the cross sectional areas were added up, the total would be the volume of the solid pictured. How do we write this in calculus using an integral? Cross Sections that are Isosceles Right Triangles Find the area of the triangle above in terms of f(x) and g(x). If all of the cross sectional areas were added up, the total would be the volume of the solid pictured. How do we write this in calculus using an integral? Daily Lessons and Assessments for AP* Calculus AB, A Complete Course Mark Sparks 2012 Page 645 Cross Sections that are Equilateral Triangles Find the area of the triangle above in terms of f(x) and g(x). If all of the cross sectional areas were added up, the total would be the volume of the solid pictured. How do we write this in calculus using an integral? Cross Sections that are Semicircles Find the area of the semicircle above in terms of f(x) and g(x). If all of the cross sectional areas were added up, the total would be the volume of the solid pictured. How do we write this in calculus using an integral? What do you notice about the integral-defined formulas for finding the volume of solids with certain cross sections? Daily Lessons and Assessments for AP* Calculus AB, A Complete Course Mark Sparks 2012 Page 646 Region R is bounded by y = sin(x) and y = x3 – 4x. Find the volume of the solids formed whose cross sections are the shapes indicated below. The cross sections are perpendicular to the x – axis. a. Cross sections are equilateral triangles b. Cross sections are semi-circles c. Cross sections are isosceles right triangles d. Cross sections are squares. e. Cross sections are rectangles whose height is twice the length of the base. f. Cross sections are rectangles whose height is one-third the length of the base. Daily Lessons and Assessments for AP* Calculus AB, A Complete Course Mark Sparks 2012 Page 647 Name_________________________________________Date____________________Class__________ Day #63 Homework Let R be the region bounded by the graphs of y = ln x and the line y = x – 2 as shown below. Though you may use a calculator, show the integral that you found to arrive at your answer. 1. Find the volume of the solid whose base is region R that is formed by cross sections that are semi-circles that are perpendicular to the x – axis. 2. Find the volume of the solid whose base is region R that is formed by cross sections that are squares that are perpendicular to the x – axis. Let f and g be the functions given by f ( x) 1 sin(x) and g ( x) 4 x . Let R be the region in the first 4 quadrant enclosed by the y-axis and the graphs of f and g, and let S be the region in the first quadrant enclosed by the graphs of f and g shown to the right. Though you may use a calculator, show the integral that you found to arrive at your answer. 3. Find the volume of the solid whose base is the cross section area of region S and is formed by squares that are perpendicular to the x-axis. 4. Find the volume of the solid whose base is the cross section area of region S and is formed by equilateral triangles that are perpendicular to the x – axis. Daily Lessons and Assessments for AP* Calculus AB, A Complete Course Mark Sparks 2012 Page 648 Let f and g be the functions given by f ( x) 2 x(1 x) and g ( x) 3( x 1) x for 0 < x < 1. The graphs of f and g are shown in the figure to the right. Though you may use a calculator, show the integral that you found to arrive at your answer. 5. Find the volume of the solid whose base is the cross section of the region bounded by the graphs of f and g and is formed by squares that are perpendicular to the x-axis. 6. Find the volume of the solid whose base is the cross section of the region bounded by the graphs of f and g and is formed by semi-circles that are perpendicular to the x-axis. 7. Find the volume of the solid whose base is the cross section of the region bounded by the graphs of f and g and is formed by equilateral triangles that are perpendicular to the x-axis. 8. Find the volume of the solid whose base is the cross section of the region bounded by the graphs of f and g and is formed by right isosceles triangles that are perpendicular to the x-axis. Daily Lessons and Assessments for AP* Calculus AB, A Complete Course Mark Sparks 2012 Page 649 CALCULATOR PERMITTED Daily Lessons and Assessments for AP* Calculus AB, A Complete Course Mark Sparks 2012 Page 650 CALCULATOR PERMITTED Daily Lessons and Assessments for AP* Calculus AB, A Complete Course Mark Sparks 2012 Page 651 Daily Lessons and Assessments for AP* Calculus AB, A Complete Course Mark Sparks 2012 Page 652 Daily Lessons and Assessments for AP* Calculus AB, A Complete Course Mark Sparks 2012 Page 653 Daily Lessons and Assessments for AP* Calculus AB, A Complete Course Mark Sparks 2012 Page 654 Daily Lessons and Assessments for AP* Calculus AB, A Complete Course Mark Sparks 2012 Page 655 AP Calculus Quiz #13 Answer Key & Rubric Multiple Choice: * 1. C E 2. A C 3. D B 4. D A 5. D B 6. C E 7. E B Free Response Part A – 4 points total ____ 1 Correct integrand and limits for [0, 1] ____ 1 Correct integrand and limits for [1, 3] ____ 1 Correct answer Free Response Part B – 3 points total ____ 1 Correct integrand ____ 1 Correct limits and constant ____ 1 Correct answer Free Response Part B – 3 points total ____ 1 Correct constant and limits ____ 1 Correct integrand 2 1 1 Volume 3 x 2 2 x dx 8 a 2.436 ____ 1 Correct answer Daily Lessons and Assessments for AP* Calculus AB, A Complete Course Mark Sparks 2012 Page 656 AP CALCULUS QUIZ #13 Name____________________________________________________Date________________________ Calculator Permitted 1. 2. 3. 4. 5. 6. 7. MULTIPLE CHOICE − Graphing Calculator Permitted 1. Find the area of the region in the first quadrant enclosed by the graphs of y = cos x, y = x and the y – axis. A. 0.127 B. 0.385 C. 0.400 D. 0.600 E. 0.947 2. Find the volume of the solid formed by revolving the region bounded by the graphs of y = 4x – x2 and y = 0 about the x – axis. A. 107.233 B. 34.133 C. 33.510 D. 10.667 E. 129.322 Daily Lessons and Assessments for AP* Calculus AB, A Complete Course Mark Sparks 2012 Page 657 3. A solid is generated when the region in the first quadrant enclosed by the graph of y = (x2 + 1)3, the line x = 1, the x – axis, and the y – axis is revolved about the x – axis. Its volume is found by evaluating which of the following integrals? 3 2 x 1 dx 1 8 6 B. x 2 1 dx 1 1 2 3 C. x 1 dx 0 1 6 D. x 2 1 dx 0 1 6 E. 2 x 2 1 dx 0 A. 8 4. The region bounded by the graph of y = 2x – x2 and the x – axis is the base of a solid. For this solid, each cross section perpendicular to the x – axis is an equilateral triangle. What is the volume of this solid? A. 1.333 B. 1.067 C. 0.577 D. 0.462 E. 0.267 5. The base of a loud speaker is determined by the two curves y = x 2 10 yx 2 10 and for 1 < x < 4 as shown in the figures to the right. For this loud speaker, the cross sections perpendicular to the x – axis are squares. What is the volume of this speaker, in cubic units? A. 2.046 B. 4.092 C. 4.200 D. 8.184 E. 25.711 Daily Lessons and Assessments for AP* Calculus AB, A Complete Course Mark Sparks 2012 Page 658 6. The slope field pictured below represents all general solutions to which of the following differential equations? A. dy 2x dx B. dy 2 x dx C. dy y dx D. dy y dx E. dy x y dx 7. The graph of a function f, which consists of two line segments and a quarter circle, is pictured to the right. If H ( x) x 2f (t)dt , which of the following statements is true? A. H(4) < H ' (2) < H ' ' (3) B. H(4) < H ' ' (3) < H ' (2) C. H ' (2) < H(4) < H ' ' (3) D. H ' ' (3) < H(4) < H ' (2) E. H ' ' (3) < H ' (2) < H(4) Daily Lessons and Assessments for AP* Calculus AB, A Complete Course Mark Sparks 2012 Page 659 FREE RESPONSE Let R and S in the figure to the right be defined as follows: R is the region in the first and second quadrants bounded by the graphs of y = 3 – x2 and y = 2x. S is the shaded region in the first quadrant bounded by the two graphs, the x – axis, and the y – axis. a. Find the area of region S. b. Find the volume of the solid generated when R is rotated about the horizontal line y = –1. c. The region R is the base of a solid. For this solid, each cross section perpendicular to the x – axis is a semi-circle whose diameter lies on the base of the solid. Find the volume of this solid. Daily Lessons and Assessments for AP* Calculus AB, A Complete Course Mark Sparks 2012 Page 660 AP CALCULUS *QUIZ #13* Name____________________________________________________Date_____________________ Calculator Permitted 1. 2. 3. 4. 5. 6. 7. MULTIPLE CHOICE − Graphing Calculator Permitted 1. Find the area of the region in the first quadrant enclosed by the graphs of y = cos x, y = x and the y – axis. A. 0.127 B. 0.385 C. 0.947 D. 0.600 E. 0.400 2. Find the volume of the solid formed by revolving the region bounded by the graphs of y = 4x – x2 and y = 0 about the x – axis. A. 33.510 B. 34.133 C. 107.233 D. 10.667 E. 129.322 Daily Lessons and Assessments for AP* Calculus AB, A Complete Course Mark Sparks 2012 Page 661 3. A solid is generated when the region in the first quadrant enclosed by the graph of y = (x2 + 1)3, the line x = 1, the x – axis, and the y – axis is revolved about the x – axis. Its volume is found by evaluating which of the following integrals? 3 2 x 1 dx 1 1 6 B. x 2 1 dx 0 1 2 3 C. x 1 dx 0 8 6 D. x 2 1 dx 1 1 6 E. 2 x 2 1 dx 0 A. 8 4. The region bounded by the graph of y = 2x – x2 and the x – axis is the base of a solid. For this solid, each cross section perpendicular to the x – axis is an equilateral triangle. What is the volume of this solid? A. 0.462 B. 1.067 C. 0.577 D. 1.333 E. 0.267 5. The base of a loud speaker is determined by the two curves y = x 2 10 yx 2 10 and for 1 < x < 4 as shown in the figures to the right. For this loud speaker, the cross sections perpendicular to the x – axis are squares. What is the volume of this speaker, in cubic units? A. 2.046 B. 8.184 C. 4.200 D. 4.092 E. 25.711 Daily Lessons and Assessments for AP* Calculus AB, A Complete Course Mark Sparks 2012 Page 662 6. The slope field pictured below represents all general solutions to which of the following differential equations? A. dy 2x dx B. dy 2 x dx C. dy x y dx D. dy y dx E. dy y dx 7. The graph of a function f, which consists of two line segments and a quarter circle, is pictured to the right. If H ( x) x 2f (t)dt , which of the following statements is true? A. H(4) < H ' (1) < H ' ' (5) B. H ' (1) < H ' ' (5) < H(4) C. H ' ' (5) < H ' (1) < H(4) D. H ' (1) < H(4) < H ' ' (5) E. H(4) < H ' ' (5) < H ' (1) Daily Lessons and Assessments for AP* Calculus AB, A Complete Course Mark Sparks 2012 Page 663 FREE RESPONSE Let R and S in the figure to the right be defined as follows: R is the region in the first and second quadrants bounded by the graphs of y = 3 – x2 and y = 2x. S is the shaded region in the first quadrant bounded by the two graphs, the x – axis, and the y – axis. a. Find the area of region S. b. Find the volume of the solid generated when R is rotated about the horizontal line y = –1. c. The region R is the base of a solid. For this solid, each cross section perpendicular to the x – axis is a semi-circle whose diameter lies on the base of the solid. Find the volume of this solid. Daily Lessons and Assessments for AP* Calculus AB, A Complete Course Mark Sparks 2012 Page 664 AP Calculus Test #8 Answer Key & Rubrics Raw Score to Percentage Conversion Multiple Choice Calculator 1. A 2. D 3. C 4. D 5. B 6. E 7. C Calculator NOT Permitted Free Response Part A – 3 points total ____ 1 g (3) 3 0 f (t )dt 14 (2) 12 (1)(1) 12 2 ____ 1 g ' (3) = f(3) = –1 ____ 1 g ' ' (3) = undefined because g ' ( x) f ( x) is not differentiable at x = 3. Calculator NOT Permitted Free Response Part B – 3 points total ____ 1 g(x) has a point of inflection when g ' ' ( x) changes signs. ____ 1 g ' ' ( x) changes signs when the graph of g ' ( x) f ( x) has a relative maximum or minimum ____ 1 Thus, g has a point of inflection when x = 0 and x = 3. Calculator NOT Permitted Free Response Part C – 3 points total ____ 1 Correctly finds the values of g(−2) and g(5), as x = −2 and x = 5 are the endpoints of the interval. 2 g (2) f (t )dt f (t )dt 1 (2) 2 4 2 0 5 g (5) 0 f (t )dt 1 (2) 2 1 (1)(1) 1 0 4 2 2 ____ 1 Correctly finds the value of g(4), as x = 4 is the only relative minimum of g on the interval 4 g (4) f (t)dt (2) (2)(1) 1 0 1 4 2 1 2 ____ 1 According to the Extreme Value Theorem, the absolute minimum value of g on −2 < x < 5 is –π. Daily Lessons and Assessments for AP* Calculus AB, A Complete Course Mark Sparks 2012 Page 665 Multiple Choice Non Calculator 8. C 9. D 10. B 11. E 12. C 13. A 14. C Free Response – Non Calculator Part A – 2 points total _____ 1 Correctly drawn zero slopes _____ 1 Correctly drawn non-zero slopes Part B – 6 points total _____ 1 Correct separation of variables _____ 1 Correct anti-differentiation of y _____ 1 Correct anti-differentiation of x _____ 1 Includes constant of integration ln y 1 1 c _____ 1 Uses the initial condition ln 0 1 1 c ln 1 1 c 0 1 c c 1 1 dy y 1 1 x2 x 2 y 1 e _____ 1 Correctly solves for y dx 2 2 2 1x 12 y 1 e 1x 12 y 1 e 1x 12 or y 1 e 1x 12 Part C – 1 point total _____ 1 Since dy y 1 2 represents the slope of the tangent line, then it is only positive for points such dx x that y – 1 > 0, or such that y > 1, provided that x ≠ 0. Daily Lessons and Assessments for AP* Calculus AB, A Complete Course Mark Sparks 2012 Page 666 AP CALCULUS AB TEST #8 Unit #7 – Advanced Integration and Applications Name______________________________________________________Date_____________________ A GRAPHING CALCULATOR IS REQUIRED FOR SOME PROBLEMS OR PARTS OF PROBLEMS IN THIS PART OF THE EXAMINATION. (1) The exact numerical value of the correct answer does not always appear among the choices given. When this happens, select from among the choices the number that best approximates the exact numerical value. (2) Unless otherwise specified, the domain of a function f is assumed to be the set of all real numbers x for which f(x) is a real number. MULTIPLE CHOICE 1. Let R be the region in the first quadrant bounded by the graphs of y 2 sin x , y e x 3 , and the y – axis as shown in the figure above. Find the volume of the solid generated when R is rotated around the line y = 4. A. 115.380 B. 36.727 D. 23.052 E. 7.338 Daily Lessons and Assessments for AP* Calculus AB, A Complete Course Mark Sparks 2012 C. 67.036 Page 667 2. The graph of the piecewise linear function f is shown in the figure above. If g ( x) x 2f (t)dt , which of the following values is the greatest? A. g(–3) B. g(–2) C. g(0) D. g(1) E. g(2) 3. The base of a solid is a region in the first quadrant bounded by the x-axis, the y-axis and the graph of the line x + 2y = 8, as shown in the figure below. If the cross sections of the solid perpendicular to the x-axis are semicircles, what is the volume of the solid? A. 12.566 B. 14.661 C. 16.755 D. 67.021 E. 134.041 4. The region bounded by the graph of y = 2x – x2 and the x – axis is the base of a solid. For this solid, each cross section perpendicular to the x – axis is an equilateral triangle. What is the volume of this solid? A. 1.333 B. 1.067 C. 0.577 D. 0.462 E. 0.267 Daily Lessons and Assessments for AP* Calculus AB, A Complete Course Mark Sparks 2012 Page 668 5. The graph of a function f is shown in the figure below and has a horizontal tangent at x = 4 and x = 8. If g ( x) x 2 2x f (t) dt , what is the value of g ' (3) ? 0 A. –2.5 B. 10 C. –4 D. 13 E. 2 6. Which of the following is the solution to the differential equation condition y( ) 1. A. dy 2 sin x with the initial dx y 2 cos x 3 B. y 2 cos x 1 C. y 2 cos x 3 D. y 2 cos x 1 E. y 2 cos x 1 Daily Lessons and Assessments for AP* Calculus AB, A Complete Course Mark Sparks 2012 Page 669 7. The graph of the function f shown above has horizontal tangents at x = 2 and x = 5. Let g be the function defined by g ( x) x 0 f (t)dt . For what values of x does the graph of g have a point of inflection? A. 2 only B. 4 only C. 2 and 5 only D. 2, 4, and 5 E. 0, 4, and 6 FREE RESPONSE The graph of a function f, pictured above, consists of a semicircle and two line segments as shown to the x right. Let g be the function given by g ( x) f (t )dt . 0 a. Find the values of g(3), g ' (3) , and g ' ' (3) , if they exist. Show the computations that lead to your answers or give a reason for your answers. Daily Lessons and Assessments for AP* Calculus AB, A Complete Course Mark Sparks 2012 Page 670 b. Find the x-coordinate of each point of inflection of the graph of g on the open interval –2 < x < 5. Justify your answer. c. On the closed interval –2 < x < 5, what is the absolute minimum value of g. Show your work and justify your answer. Daily Lessons and Assessments for AP* Calculus AB, A Complete Course Mark Sparks 2012 Page 671 AP CALCULUS AB TEST #8 Test #8: Unit #7 – Advanced Integration and Applications Name______________________________________________________Date_____________________ A GRAPHING CALCULATOR IS NOT ALLOWED FOR THIS SECTION OF THE EXAM. (1) The exact numerical value of the correct answer does not always appear among the choices given. When this happens, select from among the choices the number that best approximates the exact numerical value. (2) Unless otherwise specified, the domain of a function f is assumed to be the set of all real numbers x for which f(x) is a real number. MULTIPLE CHOICE 4 x 8. Using the substitution u x , e dx is equal to which of the following? 1 16 x 4 A. 2 e u du B. 2 e u du 2 2 D. 1 e u du 1 C. 2 e u du 1 1 2 1 4 E. e u du 1 Daily Lessons and Assessments for AP* Calculus AB, A Complete Course Mark Sparks 2012 Page 672 1 9. e 4 x dx = 0 A. e 4 4 B. 4e 4 C. e 4 1 4 D. 1 e 4 4 E. 4 4e 4 10. Region R is the region in the first quadrant bounded by the graphs of f ( x) x , g(x) = 6 – x and the x – axis. Find the area of R. A. 4 B. 22 3 C. 14 3 D. 13 3 E. 6 Daily Lessons and Assessments for AP* Calculus AB, A Complete Course Mark Sparks 2012 Page 673 11. If f is the function given by f ( x) 2x t 2 t dt , then f ' (2) 4 A. 0 B. 7 2 12 C. 2 D. 12 E. 2 12 12. Shown above is a slope field for which of the following differential equations? A. dy x dx y dy x 3 B. dx y D. dy x 2 dx y 2 E. dy x 3 C. dx y 2 dy x 2 dx y x 13. The graph of a differentiable function f is shown at right. If h( x) f (t )dt , which of the following 0 is true? A. h(6) h' (6) h' ' (6) B. h(6) h' ' (6) h ' (6) C. h ' (6) h(6) h' ' (6) D. h' ' (6) h(6) h ' (6) E. h' ' (6) h ' (6) h(6) Daily Lessons and Assessments for AP* Calculus AB, A Complete Course Mark Sparks 2012 Page 674 14. x 4 dx x 2 A. B. C. 1 2 4x 2 4 1 2 x2 4 C C 1 ln x 2 4 C 2 D. 2 ln x 2 4 C E. 2 x2 4 C FREE RESPONSE Consider the differential equation dy y 1 , where x ≠ 0. dx x2 a. On the axes provided, sketch a slope field for the given differential equation at the nine points indicated. Daily Lessons and Assessments for AP* Calculus AB, A Complete Course Mark Sparks 2012 Page 675 b. Find the particular solution y = f(x) to the differential equation with the initial condition f(2) = 0. c. Describe all points in the x – y coordinate plane for which the slope of the tangent line would be positive. Give a reason for your answer. Daily Lessons and Assessments for AP* Calculus AB, A Complete Course Mark Sparks 2012 Page 676 x Let F(x) = f (t )dt , where the graph of f(t) is shown to the right. Answer the following questions. 3 1. Complete the following table for values of F(x). x 2 3 5 6 9 F(x) 2. On what interval(s) is f(t) positive? 3. On what interval(s) is f(t) negative? 4. On what interval(s) is F(x) increasing? Justify your answer. 5. On what interval(s) is F(x) decreasing? Justify your answer. 6. On what interval(s) is F(x) concave up? Justify your answer. 7. On what interval(s) is F(x) concave down? Justify your answer. Daily Lessons and Assessments for AP* Calculus AB, A Complete Course Mark Sparks 2012 Page 677 Pictured to the right is the graph of f, which consists of two semi-circles and one line segment on the interval [0, 17]. Let g ( x) f (t )dt . x 0 45. Find the values of g(8), g ' (8) and g ' ' (8) . 46. On what interval(s) is the graph of g(x) concave down? Justify your answer. 47. On what interval(s) is the graph of g(x) increasing? Justify your answer. 48. Find all values on the open interval (0, 17) at which g has a relative minimum. Justify your answer. 49. What are the x – coordinates of each point of inflection of g(x)? Justify your answer. Daily Lessons and Assessments for AP* Calculus AB, A Complete Course Mark Sparks 2012 Page 678 Test #8 Additional Free Response – Calculator NOT Permitted Consider the differential equation dy 2x . dx y a. On the axes provided, sketch a slope field for the given differential equation at the twelve points indicated. b. Write an equation of the tangent line to the graph of f at (1, –1) and use it to approximate f(1.1). Explain why the tangent line gives a good approximation of f(1.1). c. Find the particular solution y = f(x) to the given differential equation with the initial condition f(1) = –1. Daily Lessons and Assessments for AP* Calculus AB, A Complete Course Mark Sparks 2012 Page 679 Calculator NOT Permitted Free Response Part A – 2 points total ____ 1 Accurately drawn slope segments are provided for all six points above the x – axis as pictured ____ 1 Accurately drawn slope segments are provided for all six points below the x – axis as pictured. Calculator NOT Permitted Free Response Part B – 3 points total ____ 1 Equation of the tangent line is y = 2x – 3 or equivalent form. ____ 1 f(1.1) –0.8 or 4 5 ____ 1 At x = 1, the value of the function and the value of the tangent line are equivalent because they intersect each other. Thus, at x = 1.1, the tangent line would give a very close approximation of the function because the graph of the tangent line would be a very close under or over approximation depending upon the concavity of the function at x = 1. Calculator NOT Permitted Free Response Part C – 4 points total ____ 1 Separation of variables ____ 1 Correct anti-differentiation of y variable expression: ½y2 ____ 1 Correct anti-differentiation of x variable expression: –x2 + c ____ 1 Correct function y = f(x) = 2 x 2 3 Daily Lessons and Assessments for AP* Calculus AB, A Complete Course Mark Sparks 2012 Page 680
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