AP CALCULUS AB
~er Work and List of Topical Understandings~
Revere High Math Department
Intro-1
As instructors of AP Calculus, we have extremely high expectations of students
taking our courses. As stated in the district program planning guide, we expect a
certain level of independence to be demonstrated by anyone taking AP Calculus.
Your first opportunity to demonstrate your capabilities and resourcefulness to us
is through this summer work packet which will help you maintain/improve your
skills. This packet is a requirement for all AP Calculus students and is due on the
three different assigned dates. If it is not completed on the assigned dates, you may
not be allowed to attend the AP Calculus class starting in late August. Work on
as much of this packet on your own as you can, then get together with a friend,
e-mail one of the teachers or “google” the topic to find an appropriate solution.
SHOW US YOUR BEST WORK.
Requirements
The following are guidelines for completing the summer work packet…
There are many questions you must complete. You must show us all of your quality
work. There is enough room for you to show your work in the packet.
Be sure all problems are neatly organized and all writing is legible.
In the event that you are unsure how to perform functions on your calculator, you
may need to read through your calculator manual to understand the necessary syntax
or keystrokes. You must be familiar with certain built-in calculator functions such as
finding maximum and minimum values, intersection points, and zeros of a function.
You will also need to be able to do regression analysis on your calculators.
We expect you to come in with certain understandings that are
Intro-2
prerequisite to Calculus. A list of these topical understandings is below.
Please be familiar with all of these and ready to apply them to a higher level.
Topical understandings within summer work…
Factoring
Zeros/roots/x-intercepts of rational and polynomial functions
Unit Circle
Composite function and notation
Limits of functions
Solving trigonometric equations
Trigonometry Identities
Conic shapes and their equations
Domain/Range
Interpreting and comprehending word problems
Regression analysis
Graphing, simplifying expressions, and solving equations of
The following types: trigonometric, rational, piecewise, logarithmic,
exponential, polynomial/power, radical, polar, and parametric.
Finally, we suggest not waiting until the last moment to begin working on this packet. If you spread
it out, you will most likely retain the information much better. Once again this is due, completed with
quality, on the dates indicated inside this package. It is your ticket into the class.
NO TICKET…NO SEAT!! Best of luck and if you have Any questions, feel free to contact us.
We really hope you are showing all you work. Do not skip steps.
Please be extremely neat. The secret in successfully learning
calculus is to approach each step in solving a problem with
caution. Label all answers when it is required.
Helpful web-sites
http://www.coolmath.com/precalculus-review-calculus-intro/index.html
http://www.youtube.com/user/khanacademy
http://www.kutasoftware.com/free.html
http://www.algebra-class.com/algebra-cheat-sheet.html
http://www.ecalc.com/math-help/
http://regentsprep.org/REgents/math/ALGEBRA/math-ALGEBRA.htm
First impressions are lasting impressions…impress us!!
If I have seen farther than others it is because
I have stood on the shoulders of giants.
Intro-3
TWO
Mr Cifuni and Ms Szymanski will be available to give hints and answers to the questions that you
might be having difficult doing.
Email address: cszymanski29@comcast.net
Email address: geraldcifuni@comcast.net
If you feel the need to, Mr Cifuni or Ms Szymanski can schedule class meeting(s) at the high school,
the public library, or some place else. You just have to let us know if and when you would like to
Meet.
Good Luck and Welcome to Advanced Placement Calculus!
Intro-4
How to Succeed In AP Calculus
How to succeed in Mathematics By being prepared for class each day - with homework complete and all
required materials. Reason - puts us in position to succeed. By taking notes – write all teacher notes and
practice problems done in class in your notebook. Reason – writing things down helps us to remember
them and gives us examples to use if we have trouble doing homework. It also helps to keep us focused
in class. By asking questions to help gain understanding.
Reason - we need to ask questions when we do not understand – it leads to learning. By answering
questions. Reason - helps to check our understanding and make sure we are on the right track. By
doing the problems the way the teacher does them. Reason - this way works. Doing problems this way
will result in the correct answers. By reviewing your notes each night. Reason - reviewing helps us to
determine if we really understand the process and helps us identify the questions we need to ask. By
doing homework each night. Reason - practicing a skill helps us to become better.
It also helps us to determine if we really understand the process and helps us identify the questions
we need to ask. (Math is not a spectator sport. We learn math by doing it ourselves. To improve, we
need to practice.) By practicing problems for quizzes/tests. Reason - practice helps make us better
and more confident. We need to be our best on quiz/test days. By seeking help when it is needed.
Reason - it is important to keep up. New lessons build on previous lessons. By having a positive
attitude.
Attitude by Charles Swindoll "The longer I live, the more I realize the impact of attitude on life.
Attitude to me, is more important than facts. it is more important than the past, than education, than
money, than circumstances, than failures, than successes, than what other people think or say or do. it
is more important than appearance, giftedness, or skill. it will make or break a company . . . a church . . . a
home. The remarkable thing is we have a choice every day regarding the attitude we will embrace for that
day. We cannot change our past . . . we cannot change the fact that people will act in a certain way. We
cannot change the inevitable. The only thing we can do is play on the one string we have, and that is our
attitude. . . I am convinced that live is 10% what happens to me and 90% how I react to it. And so it is with
you . . . we are in charge of our attitudes."
Daily Check List I am prepared for class I am in my desk when the bell rings I have my homework finished
and out for review I have a “I want to learn” Attitude I participate in “my” learning I am listening when others
are talking I ask questions to help me understand I do “my” best - My success in My success Study Skills
(from a poster in the guidance office at CCC)
• Pay Attention in Class
• Take good notes
• Keep an organized notebook for each subject
• Know the purpose of each assignment
• Ask questions in class
• Review, Review, Review It can help you retain 80% of the information
• Plan a definite time and place for studying each day
• Study for awhile and take short breaks
• Don’t cram for hours the night before a test Study a little bit each day
• Think positive
• Do your best
Intro-5
How Your Summer Assignment
Will Be Graded.
The Rubric is shown below:
Intro-6
Intro-7
Intro-58
Intro-9
Of
2011
1
Time Due: Before 1:00 pm
27
Summer Assignment 1
Page 1
Ex1: During as camping trip in North Bay, a couple went one-third of the way by boat, 10 miles by foot,
and one sixth of the way by horse. How long was the trip?
Identify the question:
Set up an equation:
Solve the equation:
Check the solution:
Ex 2: The manual for your car suggests using gasoline that is 89 octane. In order to save money, you decide
to use some 87 octane and some 93 octane in combination with the 89 octane currently in your tank in
order to have an approximate 89 octane mixture. Assuming you have 1 gallon of 89 octane remaining in
your tank (your tank capacity is 16 gallons), how many gallons of 87 and 93 octane should be used to fill
up your tank to achieve a mixture of 89 octane?
Identify the question:
Set up an equation:
Solve the equation:
Check the solution:
Summer Assignment 1
Ex3: It takes eight hours to fly from Orlando to London and 9.5 hours to return. If an airplane
averages 550 mph in still air, what is the average rate of the wind blowing in the direction from
Orlando to London? Assume the wind speed for both legs of the trip.
Page 2
Identify the question:
Set up an equation:
Solve the equation:
Check the solution:
Ex4: From 1999 to 2001 the price of Abercrombie & Fitch’s (ANF) stock was approximately give by
P = 0.2t2 – 5.6t + 50.2, where P is the price of stock in dollars, t is in months, and t=1 corresponds
January 1999. When was the value of the stock worth $30?
Identify the question:
Set up an equation:
Solve the equation:
Check the solution:
Ex5: Solving Rational Equations:
Solve the equation:
Summer Assignment 1
1
1
1

 2
3x  18 2 x  12 x  6 x
Ex6: Solving an Equation Involving a Radical:
Solve the equation:
x  2  7x  2  6
Ex7: Solving an Equation Quadratic in Form with Negative Exponents:
Find the solutions to the equation:
x 2  x 1  12  0
Ex8: Solving an Equation Quadratic in Form with Negative Exponents:
Find the solutions to the equation:
x 2 / 3  3x1/ 3  10  0
Page 3
Ex9: Solving a Quadratic Absolute Value Equation
Solve the equation:
Summer Assignment 1
Page 4
5  x2  1
Ex10: Solving a Linear Inequality
Solve and graph the inequality:
5  3x  23
Ex 11: Two car rental companies have advertised weekly specials on full-size cars. Hertz is advertising
an $80 rental fee plus an additional $.10 per mile. Thrifty is advertising $60 and $.20 per mile. How
many miles must you drive for the rental car from Hertz to be the better deal?
Ex12: Finding the Center and Radius of a circle by Completing the Square
Solve the center and radius of the circle with the equation
x 2  8 x  y 2  20 y  107  0
Summer Assignment 1
Page 5
Ex13: Finding an Equation of a Line That is Perpendicular to Another Line
Find the equation of the line that passes through the point (3,0) and is perpendicular to the line y = 3x + 1.
Ex14: Evaluating Functions Sums
For the given function H(x) = x2 + 2x
(a.) H(x+1)
(b) H(x) + H(1)
Ex15: Evaluating the Difference Quotient
For the function
f ( x)  x 2  x, find
f ( x  h)  f ( x )
.
h
Page 6
Summer Assignment 1
Ex16: Determine Whether a Function Is Even, Odd, or Neither.
(a) f ( x)  x 2  3
(c) h( x)  x 2  x
(b) g ( x)  x 5  x 3
Ex17: Determine Average Rate of Change.
f (b)  f (a)
ba
average  rate  of  change 
Key: The slope of the secant line is used to represent
the average rate of change of a function.
Find the average rate of change of f(x) = x4 from x = -1 to x= 0.
Ex18: Graphing Piecewise-Defined Functions.
Graph the piecewise-defined function, and state the domain, range, and interval when the function is
increasing, decreasing, or constant.
You must graph piecewise function here!
G(x) =
{
x  1
x
1 1  x  1
x 1
x
y
2




x












Ex19: Sketching the Graph of a Function Using Both Shifts and Reflections.
Sketch the graph of the function
f ( x)  2  x  1
You must graph function here!
y





x













Summer Assignment 1
Ex20: Determine a Composite Function.
(a) f (g (1))
(b)
f ( g (2))
(c)
Page 7
g ( f (4))
Ex21: Determine the Domain of a Composite Function.
Given the function
f ( x) 
1
x 1
and
g ( x) 
1
, determine f
x
g and state its domain.
Ex22: Determine the Inverse Function.
The function
f ( x) 
2
, x  3
x3
is a one-to-one function. Find its inverse
Ex23: Determine the Inverse Function.
You have just bought a puppy and want to fence in an area in the backyard for her. You buy 100 linear
feet of fence from Home Depot and have decided to make a rectangular fenced-in area using the back
of your house as one side. Determine the dimensions of the rectangular pen that will maximize the area
in which your puppy may roam. What is the maximum area of the rectangular pen?
Do you know why dogs do not go to heaven?
Dog
Pen
House
Summer Assignment 1
Ex24: Identifying Polynomial and Their Degrees.
(a) f ( x)  3  2 x
(b) F ( x) 
5
(c) h( x)  3x  2 x  5
(c) g ( x )  2
x 1
(d) H ( x)  4 x (2 x  3)
2
5
Page 8
2
(e) G( x)  2 x 4  5 x 3  4 x 2
Ex25: Identify the Real Zero’s of a Polynomial Function.
Find the zeros of the polynomial function
Ex26: Long Division of Polynomial
with “Missing” Terms.
Divide
3x 4  2 x 3  x 2  4
by
f ( x)  x3  x 2  2 x.
Ex27: Synthetic Division
Divide
x2 1
3x 5  2 x 3  x 2  7
by
x2
Page 9
Summer Assignment 1
Ex28: Find Slant Asymptotes
4 x3  x 2  3
Determine the slant asymptote of the rational function f ( x) 
x2  x 1
Ex29: Graphing Rational Function with a Hole in the Graph
x2  x2  6
Graph the rational function f ( x) 
. Find the domain, x and y intercept, asymptotes, and hole .
x2  x  2
y






x












Ex30: Graphing Rational Function with a Hole in the Graph
Find the exact value of
1
(a) log 3 81
(b) log 169 13
(c) log 5 ( )
5






Summer Assignment 1
Ex31: Application of Logarithms
The magnitude of an earthquake is measured using the Richter scale.
Where M= is the magnitude
E is the seismic energy released by the earthquake (in joules)
E0 is the energy released by a reference earthquake E0= 104.4 joules
Page 10
2 E
M  log  
3  E0 
On October 17, 1989, just moments before game 3 of the World Series between the Qakland A’s and the
San Francisco Giants was about to start - with 60,000 fans in Candlestick Park – a devastating earthquake
erupted. Parts of interstates and bridges collapsed and President George W Bush declared the area a disaster
zone. The earthquake released approximately 1.12 x 1015 joules. Calculate the magnitude of the earthquake
using the Richter scale.
Ex32: Combining Logarithmic Expressions into a Single Logarithm
Write the expression
3 log b x  log b (2 x  1)  2 log b 4
as a single logarithm.
Ex33: Expanding Logarithmic Expressions into a Sum or Difference of Logarithms
Write the expression
 x2  x  6 

ln  2
x

7
x

6


as a sum and difference of logarithms.
Ex34: Solving a Simple Logarithmic Equation
Solve the equation
log 4 (2x  3)  log 4 ( x)  log 4 ( x  2).
Summer Assignment 1
Page 11
Ex35: Calculating How Many Years It Will Take for Money to Double.
Recall the compound continuous interest formula:
A  PE rt
You save $1,000 from a summer job and put it in a CD earning 5% compounding continuously. How
many years will it take for your money to double? Round to the nearest year.
Ex36: Calculating World Population Projections.
The world population is the total number of humans on Earth at a given time. In 2000 the world
population was 6.1 billion and in 2005 the world population was 6.5 billion. Find the annual growth
rate and determine what year the population will reach 9 billion.