AMA INTERNATIONAL UNIVERSITY – BAHRAIN Salmabad, Kingdom of Bahrain COLLEGE OF ENGINEERING B Department of Math & Science 2nd Trimester SY 2019-2020 MATH 406 – Differential Calculus with Analytic Geometry Special Project Submission Date Leader Members Submission Deadline Section ID#: 1. ID#: 2. ID#: 3. ID#: April 22, 2020 at 5:00PM ( use this file to put all your answers) Activity 1 In “Hardball,” the man is searching for an amateur mathematician who has gone into hiding. It can be hypothesizes that people who disappear and are still alive are like satellites that have lost their planet. It explains that if the planet vanishes, the satellites would not have a focus to their orbit, and would travel in another direction, likely to another source of gravity. One such satellite orbiting the Earth is the Geostationary Operational Environmental Satellite (GOES). The GOES-10, also called the GOES-EAST, satellite is a high altitude geosynchronous orbiting satellite used for weather observations. If the Earth vanished, this satellite would maintain the same velocity but would travel at a path tangent to its orbit. In this activity we will focus on the direction of that travel by calculating the slope of the tangential path. The GOES-EAST appears to be stationary in the sky because its nearly circular orbit is located in the Earth’s equatorial plane and matches the rotation of the Earth. Because of this circular orbit, its path can be expressed as the equation of a circle with the center of the Earth as the origin. 1. What is the radius of the circle the GOES-EAST traces? 2. If the center of the Earth is the origin (0, 0), what is the equation for the orbit? If the Earth were to vanish, the path of the satellite would be a straight line tangent to the circular orbit of the satellite. To find the slope of this tangent line at any point on the orbit, calculate the derivative of the equation of the orbit. Rather than explicitly solving the equation for a variable before calculating the derivative, use the implicit form of the equation and find the derivative of the equation with respect to x (dy/dx). Prepared by : Dr. Lina S. Calucag Course Coordinator &Member Teachers Date : Reviewed by: Verified by: Approved by: Dr. Belen T. Lumeran Department head Dr. Noaman M. Noaman Associate Dean Dr. Beda T. Aleta Dean Date : Date : Date : AMA INTERNATIONAL UNIVERSITY – BAHRAIN Salmabad, Kingdom of Bahrain COLLEGE OF ENGINEERING B Department of Math & Science 2nd Trimester SY 2019-2020 MATH 406 – Differential Calculus with Analytic Geometry Special Project 3. Find the derivative of the equation for the GOES-EAST orbit. The National Oceanic and Atmospheric Administration (NOAA) operates polar orbiting weather satellites that travel in circular orbits around the Earth. One such satellite is NOAA-15. 4. Without knowing the altitude of the NOAA-15, calculate the derivative of the circular orbit. 5. Explain why the radius of the circular orbit does not affect the slope of the line tangent to the orbit. Circular orbits are special cases of elliptical orbits. In fact, the paths of all of the satellites around the Earth are elliptical orbits. 6. Find the derivative of this general case using implicit differentiation. Activity 2 Using the information from the activity 1, calculate the current linear velocity of the GOES-EAST satellite and express your answer in meters/second. If the Earth does not vanish, then in order for the GOES-EAST satellite to leave its orbit it would have to have a great enough velocity to overcome the pull of gravity back to the Earth. This velocity is called the escape velocity, Ve. This is represented by the equation where G is the gravitational constant ( 6.673 x 10-11 m3 /kg s2 ), M is the mass of the Earth (6 x 1024kg), and R is the radius of the satellite from the center of the Earth (in meters). 2. Calculate the escape velocity for the GOES-EAST satellite. 3. How much greater would the velocity of the GOES-EAST need to be to escape the Earth’s orbit? 4. What would happen to the GOES-EAST satellite if the velocity were less than the velocity required to maintain a circular orbit? Prepared by : Dr. Lina S. Calucag Course Coordinator &Member Teachers Date : Reviewed by: Verified by: Approved by: Dr. Belen T. Lumeran Department head Dr. Noaman M. Noaman Associate Dean Dr. Beda T. Aleta Dean Date : Date : Date : AMA INTERNATIONAL UNIVERSITY – BAHRAIN Salmabad, Kingdom of Bahrain COLLEGE OF ENGINEERING B Department of Math & Science 2nd Trimester SY 2019-2020 MATH 406 – Differential Calculus with Analytic Geometry Special Project 5. Describe the orbit of the GOES-EAST satellite if the velocity was greater than the velocity required for a circular orbit, but less than the escape velocity. Activity 3: Applications of the Derivative To determine the titles of the picture below, solve the 18 application of derivatives below. Then replace each numbered blank with the letter corresponding to the answer for that problem. Show all solutions on the answers given below. Here is the title right-side-up: " __ __ __ __ __ __ __ 16 10 18 4 16 18 7 __ __ __ __ __ __ __ __ __ __ __ __ __ __ __ __ 17 5 9 2 10 13 8 8 16 7 6 12 16 3 3 18 __ __ __ __ __" 14 13 11 6 5 Here is the title upside-down: "__ __ __ __ __ __ __ 17 4 13 8 9 11 12 __ __ 13 2 __ 18 __ __ __ __ __ __ __ 17 18 15 15 18 6 9 __ __ __ __ __ 12 18 10 17 5 __ __ __ ." 1 16 14 Derivative Application Problems: 1. Find the equation of the line normal to the curve f(x) = x3 – 3x2 at the point (1, -2). 2. Find the equation of the line tangent to the curve x2 y – x = y3 – 8 at the point where x = 0. 3. Determine the point(s) of inflection of f(x) = x3 – 5x2 + 3x + 6. 4. Determine the relative minimum point(s) of f(x) = x4 – 4x3. 5. A particle moves along a line according to the law s = 2t3 – 9t2 + 12t – 4, where t≥0 . Determine the total distance traveled between t = 0 and t = 4. 6. A particle moves along a line according to the law s = t4 – 4t3, where t≥0. Determine the total distance traveled between t = 0 and t = 4. 7. If one leg, AB, of a right triangle increases at the rate of 2 inches per second, while the other leg, AC, decreases at 3 inches per second, determine how fast the hypotenuse is changing (in feet per second) when AB = 6 feet and AC = 8 feet. Prepared by : Dr. Lina S. Calucag Course Coordinator &Member Teachers Date : Reviewed by: Verified by: Approved by: Dr. Belen T. Lumeran Department head Dr. Noaman M. Noaman Associate Dean Dr. Beda T. Aleta Dean Date : Date : Date : AMA INTERNATIONAL UNIVERSITY – BAHRAIN Salmabad, Kingdom of Bahrain COLLEGE OF ENGINEERING B Department of Math & Science 2nd Trimester SY 2019-2020 MATH 406 – Differential Calculus with Analytic Geometry Special Project 8. The diameter and height of a paper cup in the shape of a cone are both 4 inches, and water is leaking out at the rate of ½ cubic inch per second. Determine the rate (in inches per second) at which the water level is dropping when the diameter of the surface is 2 inches. 9. For what value of y is the tangent to the curve y2 – xy + 9 = 0 vertical? 10. For what value of k is the line y = 3x + k tangent to the curve y = x3 ? 11. Determine the slopes of the two tangents that can be drawn from the point (3, 5) to the parabola y = x2 . 12. Determine the area of the largest rectangle that can be drawn with one side along the x-axis and two vertices on the curve y = e-x2 13. A tangent drawn to the parabola y = 4 – x2 at the point (1, 3) forms a right triangle with the coordinate axes. What is the area of this triangle? 14. If the cylinder of largest possible volume is inscribed in a given sphere, determine the ratio of the volume of the sphere to that of the cylinder. 15. Determine the first quadrant point on the curve y2x = 18 which is closest to the point (2, 0). 16. Two cars are traveling along perpendicular roads, car A at 40 mph, car B at 60 mph. At noon when car A reaches the intersection, car B is 90 miles away, and moving toward it. At 1PM, what is the rate, in miles per hour, at which the distance between the cars is changing? 17. A 26-foot ladder leans against a building so that its foot moves away from the building at the rate of 3 feet per second. When the foot of the ladder is 10 feet from the building, at what rate is the top moving down (in feet per second)? 18. A rectangle of perimeter 18 inches is rotated about one of its sides to generate a right circular cylinder. What is the area, in square inches, of the rectangle that generates the cylinder of largest volume? Prepared by : Dr. Lina S. Calucag Course Coordinator &Member Teachers Date : Reviewed by: Verified by: Approved by: Dr. Belen T. Lumeran Department head Dr. Noaman M. Noaman Associate Dean Dr. Beda T. Aleta Dean Date : Date : Date : AMA INTERNATIONAL UNIVERSITY – BAHRAIN Salmabad, Kingdom of Bahrain COLLEGE OF ENGINEERING B Department of Math & Science 2nd Trimester SY 2019-2020 MATH 406 – Differential Calculus with Analytic Geometry Special Project Activity 4 – Prepare a comprehensive report of the activities you have done, in a paragraph form. Include the following: *how you start the activity *who participated in the activity *how many hours did you do per activity. *What did you learn from the activities you have done * Enumerate math topics you use in the activities *You may add photos of you doing the activity, label the photos per activity Project Evaluation Rubric Student NameID#_____________________ Project Title____________________________________ DIRECTIONS: All projects must be submitted/presented by the assigned deadline. This time needs to be scheduled well in advance to avoid conflicts. Planning and Time Management: This score indicates the amount of time spent planning the case/problem and the effectiveness of time management skills throughout the completion of the case/problem (meeting assigned deadlines, turning in paperwork, etc.). Indicator Level 0 Level 1 Level 2 Level 3 Level 4 Level 5 PLANNING & TIME MANAGEMEN T No evidenc e of planning Circle one 0 Procrastination lead to incomplete case/problem 2 2.5 Little planning or forethought Basic planning and time management needs necessary for case/problem completion met 6 6.5 Case/problem hastily completed for deadline 4 4.5 Planning and time management exhibited enhance the overall case/problem 8 Exhibits a professional level of planning and time managemen t 8.5 10 Comments: Points Possible: 10 Score Time and Effort: This score indicates the amount of time and effort the student expended completing the case/problem. Indicator Level 0 Level 1 Level 2 Level 3 Level 4 TIME AND EFFORT Circle one No evidence of effort Little or no “authentic” time spent on case/problem 0 Prepared by : Dr. Lina S. Calucag Course Coordinator &Member Teachers Date : 4 5 Minimal effort Met minimum time requirements and didn’t complete case/problem 8 9 Meets basic time and effort required to complete case/problemt 12 Time and effort expended on case/problem enhances the overall case/problem 13 16 Level 5 Exhibits a professional level of time and effort expended on case/problem 17 Reviewed by: Verified by: Approved by: Dr. Belen T. Lumeran Department head Dr. Noaman M. Noaman Associate Dean Dr. Beda T. Aleta Dean Date : Date : Date : 20 AMA INTERNATIONAL UNIVERSITY – BAHRAIN Salmabad, Kingdom of Bahrain COLLEGE OF ENGINEERING B Department of Math & Science 2nd Trimester SY 2019-2020 MATH 406 – Differential Calculus with Analytic Geometry Special Project Comments: Points Possible: 20 Score Evidence of Learning and Risk Factor: This score indicates the level of knowledge gained by the student evident through the case/problem, and the extent to which the student was “stretched” or took risks through the case/problem experience. Indicator Level 0 Level 1 Level 2 Level 3 Level 4 Level 5 EVIDENCE OF LEARNING AND RISK FACTOR No evidence of genuine learning Circle one Student never stretched their knowledge/cap abilities 0 4 Little demonstration of genuine learning; limited risks taken 5 8 Case/problem demonstrates genuine learning/risks were taken for expand-ing knowledge and skills 9 12 13 Case/Proble m and case/problem experience clearly “stretched” student knowledge and skills 16 17 Student took several risks to achieve a superior level of knowledge and skills through the case/problem process 20 Comments: Points Possible: 20 Score Degree of Difficulty: This score indicates the variety and complexity of the components to completing the project. Indicator Level 1 Level 2 Level 3 Level 4 Level 5 Level 0 Case/proble m DEGREE OF DIFFICULTY incomplete Circle one 0 Not ageappropriate difficulty Little degree of difficulty evident One dimensional case/problem 4 5 8 9 Case/problem comprised of more than one component of appropriate difficulty 12 13 Case/problem comprised of multiple components or components exhibit great difficulty 16 17 Case/proble m complexity approaches professional quality 20 Comments: Points Possible: 20 Score Portfolio Preparation: This score indicates the quality of the portfolio. Prepared by : Dr. Lina S. Calucag Course Coordinator &Member Teachers Date : Reviewed by: Verified by: Approved by: Dr. Belen T. Lumeran Department head Dr. Noaman M. Noaman Associate Dean Dr. Beda T. Aleta Dean Date : Date : Date : AMA INTERNATIONAL UNIVERSITY – BAHRAIN Salmabad, Kingdom of Bahrain COLLEGE OF ENGINEERING B Department of Math & Science 2nd Trimester SY 2019-2020 MATH 406 – Differential Calculus with Analytic Geometry Special Project Indicator Level 0 Level 1 Missing case/proble m report Incomplete case/problem report PROJECT REPORT PREPARATI ON Some required sections missing Circle one 0 2 2.5 Level 2 Case/problem report has major formatting and/or many spelling errors 4 4.5 Level 3 Level 4 Level 5 Case/probl em report complete with several minor errors Case/problem report complete with very few minor errors Case/proble m report clear, concise, accessible, with unique content 6 6.5 Contents concise and accessible 8 8.5 No formatting or spelling errors 10 Comments: Points Possible: 10 Score Quality of Final Project: This score indicates the actual quality of the physical product or quality of the case/problem experience, with “professional” quality being a score of 10.. Indicator QUALITY OF FINAL PROJECT Level 0 Level 1 No physical case/problem or documentation of case/problem experience Little concern for case/probl em quality or incomplete case/probl em 4 5 Circle one 0 Level 2 Case/problem completed but demonstrates low quality 8 9 Level 3 Level 4 Level 5 Case/probl em demonstrat es appropriate quality High quality case/problem illustrates student work ethic Professiona l quality product or case/proble m experience 12 13 16 17 20 Comments: Points Possible: 20 Score Total Score________ Exemplary Proficient Developing Emerging Needs Major Support Final Points________ Case/Problem Ranking Key Strong evidence of meeting or exceeding learning goals Evidence suggests adequate meeting of learning goals. Evidence suggests some learning goals are met. Evidence suggests some partially met learning goals. Evidence suggests failure to meet desired learning goals. Prepared by : Dr. Lina S. Calucag Course Coordinator &Member Teachers Date : 95-100% 85-94% 65-84% 50-64% Below 50% Reviewed by: Verified by: Approved by: Dr. Belen T. Lumeran Department head Dr. Noaman M. Noaman Associate Dean Dr. Beda T. Aleta Dean Date : Date : Date :