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2018-19 Are you Ready for AP Calculus AB? 1
Welcome to Mrs. Dodds AP Calculus AB Class. [email protected]
For many people, "Calculus" is synonymous with "difficult." However, anyone who has taken Calculus
will tell you that the reason students are not successful in Calculus, is not the Calculus itself. The
problem is their algebra & trigonometry skills are weak. There is often only one step in the problem that
actually involves calculus, the rest is simplifying, by using algebra & trigonometry.
The idea of summer assignment is not to ruin your summer or burden you with an excessive amount of
work in the days leading up to the start of school. There is an expectation that each of you comes into
this course with a specific set of skills. The summer assignment is meant to keep your skills sharp, and
to strengthen any skills that may not be as strong as you need to be successful. This assignment is
designed to help you review, practice, & deepen your understanding & fluency (done easily without
much thought) of the pre-requisite skills that are NECESSARY for success in Calculus. It is imperative
to confidently know what, when, why, & how you can apply each rule.
Time is of the essence on the AP Calculus Exam. I encourage and challenge you to take a few
seconds strategizing an effective and efficient plan, before approaching & solving the problems in your
summer assignment. You can arrive at a simple answer in only a few short steps—or choose the wrong
strategy—and your few short steps; turn into half a page of unnecessary, complicated, & dead-end
endeavors. So, the strategy you choose to serve any particular problem becomes of the upmost
importance when time, simplicity, & accuracy count the most. Calculus is easy—your choice of strategy
can make it difficult & exhausting.
I strongly suggest that you take this assignment seriously. If you have difficulty with a question,
DO NOT SKIP IT – RESEARCH IT. That research can be in the form of a textbook, a parent, a friend,
free on-line help, or your old math notebook. You may also refer to the websites below as a resource.
In addition, you can study the examples offered in the last few pages of this summer assignment.
Virtual Math Lab http://www.wtamu.edu/academic/anns/mps/math/mathlab/
Algebra Help https://www.wyzant.com/resources/lessons/math/algebra
Purple Math http://www.purplemath.com/
Khan Academy https://www.khanacademy.org/
MathBits.com http://mathbits.com/
The Math Page http://www.themathpage.com/
Math Open Reference http://www.mathopenref.com/
You must offer orderly, legible, and legitimate work that supports your answer. Neatness is important in
Math, so take your time & use a pencil. Calculators are NOT permitted, unless specified.
The Summer Assignment will be collected & graded based on completion & correctness on the 1st
day of school, August 14th. Points will be deducted if the packet is turned in late. You will be tested on
the topics from the Summer Assignment on Friday, August 17th. There will be NO RETAKES for a Poor
Test Score, so seek out all the help that you require prior to this test. The Test will assess mastery and
retention of the prerequisite math skills needed to succeed in AP Calculus AB.
You DO NOT Need to print the last pages; they are for your reference only!
2018-19 Are you Ready for AP Calculus AB? 2
Simplify using rules, properties, & mathematical skills learned in previous math classes.
NO Parenthesis, mix fractions, & negative exponents in your answer. And rationalize radical expressions.
1]
h
x h
h
2]
1 1
1 1
x y
x y
3]
52
2
2
xx
x
4]
1 1
2 24 42
4
xx x
x
5]
2
2 2
7 5 5
1 6
x x x
x x
6]
x y x z z y
xy xz yz
7]
2
2
2
2
1
21
x
y
x x
y y
8]
2
1
9
3
x
x
9]
2
1 1
51 1
25
x
x
10]
3
2
9
7 12
x x
x x
11]
2
3 2
2 8
2
x x
x x x
12]
3 2 4
4 3 2 4
45 75
28 8
a b a b
c d c d
2018-19 Are you Ready for AP Calculus AB? 3
13]
3
4
4 16
4
x
x
14 ]
1 2
2 1 1 2
1 4 1
x x y y
15]
83 1
4 2
1 4
x y
x
16]
3
21 1
1
x x
x
17]
2
1 sin cos
2sin
x x
x
18]
3 2
4
1 2 3 1
1
x x x
x
Rationalize the denominator using conjugates:
19] 23
2
20]
51
4
Rationalize & write each complex number in standard form, a bi . Meaning 5 3
4 2i and NOT
5 6
4
i
21] 7
2i 22]
6
3 i
23] i
i
42
3
2018-19 Are you Ready for AP Calculus AB? 4
Function Notation: Note: f g x represents product & f g x represents composition.
Use the given function notations for three functions 23 2f x x , 2 1g x x , 2 5h x x x to find:
1] f g x 2] f h x 3] h g x
4] g h x 5] f h x 6] h
xg
7] f g x 8] g h x
9] 3h 10] 1
2g
11] 2h g
12] 0f
h
13] 3f g 14] 1f h
If 2,3 , 4,5 , 7,2f x ; 2,1 , 3,2 , 7,4h x ; 23 1g x x ; 2k x x then determine:
15] 2f h = 16] 2k g 17] 7f h
18] 7g k 19] 1f x 20] 1g x
21]
4
2
f
g 22] gk x
23] Tom uses an internet server, which charges $12.50 per month plus $0.60 for each hour over 20 hours
that he uses it during the month. a. Write this relation in function notation.
b. Use the function notation to find how much will he be charged for using the service for 38 hrs in April?
2018-19 Are you Ready for AP Calculus AB? 5
Find the difference quotient, f x h f x
h
where 0h , for the following:
1] 28 1f x x 2] 3
4 2
xf x
x
3] 22 5 3f x x x 4] 3f x x
5] Solve for P . nrPPA 6] Solve for x. 02
1
4
2
xx
7] Solve for y . 1xy y y 8] Solve for dy
dx. 2 2 6 0
dy dyx x y y
dx dx
Find the inverse, 1f x of the given functions & determine the domain of 1f x
.
9] 15
2)(
x
xxf 10] 2 2 1 , 0f x x x x 11]
3 6
2 5
xg x
x
, find 1 3g
.
2018-19 Are you Ready for AP Calculus AB? 6
You MUST, write an Equation for each situation. Then use the Equation to Solve the problem.
1] 4] On a coordinate grid where each unit represents 1 mile, Isabel’s house is located at (3, 0) and a mall
is located at (0,4). What is the distance between Isabel’s house and the mall?
2] Write the equation for the line parallel to the line 4 3 9x y and through the point (3, -1).
Write the answer in slope intercept form.
3] Given a line through points (-2, 4) and (8, -1), find the equation of the line perpendicular to the line
through the midpoint of those points. Write the answer in slope intercept form.
4] Find the equation of the line that is perpendicular to the line 2 3 8x y at the point (1, 2).
5] A line with slope 5 passes through the point (-1, 3) intersects the x-axis at a point. What are the
coordinates of this point?
6] What are the coordinates of the point at which line passing through point (1, -3) and (-2, 4) intersects
the y-axis?
7] A seven foot ladder, leaning against a wall, touches the wall x feet above the ground.
Write an expression (in terms of x) for the distance from the foot of the ladder to the base of the wall.
8] Joe can drive 4 miles in m minutes. At this rate, how many minutes will it take him to drive 19 miles?
Leave your answer in terms of m.
9] Given 3 5f x x , find the average rate of change of the function on [1, 5].
10] The minimum value of 322 xx is ?
2018-19 Are you Ready for AP Calculus AB? 7
11] During an experiment, a ball is dropped from a height of 196 ft. The formula 2
016h t h can be used
to approximate the number of seconds t it takes the ball to reach height h from an initial height of 0h in ft.
Find the time it takes the ball to reach the ground.
12] The cheerleaders at Bulls games launch T-shirts into the crowd every time the Bulls win. The height of
the T-shirt can be modeled by the function 216 48 6h x x x , where h x represents the height in feet
of the T-shirt after x seconds.
a) At what height was the T-shirt launched? b) What is the maximum height of the T-shirt?
c) When was the maximum height reached?
13] Molly is designing a poster. The length of the poster is 6 inches longer than the width. If the poster
requires 616 square inches of poster board, find the width w of the poster.
(Write an equation in terms of w; then use the equation to solve the problem.)
14] Find the surface area of a box of height h whose base dimensions are p and q, and that satisfies the
following condition:
a. The box is closed. b. The box has an open top.
c. The box has an open top and a square base with side length p.
15] A piece of wire 5 inches long is to be cut into two pieces. One piece is x inches long and is to be bent
into the shape of a square. The other piece is to be bent into the shape of a circle. Find an expression for
the total area made up by the square and circle as a function of x.
2018-19 Are you Ready for AP Calculus AB? 8
16] Car A travels 20 miles per hour faster than car B. Car A can travel 100 miles in the same time it takes
car B to travel 60 miles. Find the speed of each car.
17] A group of students wish to split the $100 cost of reserving a room for a party. When 5 more students
join the group, the cost per student is reduced by $10. How many students were in the original group?
What was the original cost per student?
18] An open box is made from a 10 inch piece of cardboard by cutting a square from each corner and
folding up the edges. If the area of the bottom of the box is 60 square inches, what is the length of the
sides of the squares removed?
19] The speed of a boat in still water is 10 mph. It travels 15 miles upstream and 15 miles downstream in a
total time of 4 hours. What is the speed of the current?
20] A rectangular garden is 12 feet long and 10 feet wide. Part of the garden is torn up to install a sidewalk
of uniform width around the garden. If the area of the new garden is 48 square feet, find the width of the
sidewalk.
21] A water tank has the shape of a cone. The tank is 10m high and has a radius of 3m at the top. If the water is 5m deep, what is the surface area of the top of the water?
22] Two cars start moving from the same point. One travels south at 100 km/hr, the other west at
50 km/hr. How far apart are they, two hours later?
2018-19 Are you Ready for AP Calculus AB? 9
23] A kite is 100m above the ground. If there are 200m of string out, what is the angle between the
string and the horizontal? (Assume the string is perfectly straight).
24] The second angle in a triangle is 3 less than twice the first angle. The third angle measure 8 more
than twice the first angle. a) Write an equation using one variable x to represent this problem.
b) Use your equation to find the measure of each angle.
25] Roger is looking up a guard-tower that is 45 feet tall in the forest. The angle of depression from the top
of the guard-tower is 40 . How far is Roger from the guard-tower?
26] Using the diagram below, find the distance between CB if AB is 20.
27] Use figure A to find the ratio of the area inside the square but outside the circle to the area of the
square.
b) Use figure B to find a formula for it’s: a) perimeter b) area of the Norman Window.
2018-19 Are you Ready for AP Calculus AB? 10
Write the exact value of each without using a calculator.
1] 2 ln x
e
2] ln3e 3] 2ln e 4]
1 ln2
x
e
5] 4
1log
4 6]
0.5log 8 7]
3
29 8] 1ln
2
9] 2
4 2log 9 log 3 10] 20 15
4log 4 ln e 11] 3
8
log 81
log 9
3
8 12]
8 4log 2 log 2
13] 3 3log 5 log 43
14] 3log9log2 24 15]
53log23
Use properties of logs to rewrite each expression as a simplified single logarithm.
16] 2 2
1log log
3x x 17] 3log 4log 2logk m p
18] 2 24 4 4log 1 log 1 log 2 1x x x x 19] 21
ln 2ln ln 11
x xx
x x
20] 3/11010 log3log2 xx 21] 2
2 2 2log 5 log 1 log 1x x
Solve each equation for x. Do not use a calculator.
22] 24 21 18 0x x 23]
22 3 3 0x x
24] 4 29 8 0x x 25] 96124 4
3
x
26] 01
1
x
x
x
x 27]
1 5 1
3 2 6 2
x x
x x
2018-19 Are you Ready for AP Calculus AB? 11
Solve each equation for x. Do not use a calculator.
1] 3lne x 2] ln 4xe 3] 5xe
4] ln5e x 5] ln1 ln e x 6] ln ln 0x x
7] 35 2 1xe 8] ln 6 ln ln 2 3x 9] ln 5 ln 1 ln 1x x x
10] 2 5 1
2256
x (don’t use log’s) 11] 1
1 1
81 3x (don’t use log’s) 12]
3 232 8x x (don’t use log’s)
a) Re-write each function as a piecewise function and then b) Graph.
13] 2f x x 14] 1
1
xf x
x
15]
3
3
xf x
x
2018-19 Are you Ready for AP Calculus AB? 12
You MUST be Very Comfortable with these Graphs & their features. (note 0a )
a. f x x b. 1
f xx
c. 2
1f x
x d.
2 2
1f x
x a
e. 2f x x
f. f x x g. xf x e h. lnf x x i. xf x a j. 2 2 2x y r
k. sinf x x l. cosf x x m. tanf x x n. secf x x o. cscf x x
p. cotf x x q. 1sinf x x r. 1cosf x x s. 1tanf x x t. 2 2f x a x
u. intf x x or f x x
Do not use a calculator. Graph neatly & show all the important details of each specific graph.
Calculate x-intercept & y-intercept Algebraicaly & make sure they agree with what your graph displays.
Write x & y-intercepts as points (x, 0) and (0, y).
Write domain, range, increasing or decreasong intervals in Interval Notations. Example: , 4
Min/Max value refer to the Highest or Lowest y-value on the entire graph.
1] 2 3f x x 2] 2
2 1 2f x x
3] 2 4f x x 4] 1
2f x
x
x-intercept: .
y-intercept: .
Domain: .
Range: .
Increasing Intervals:
.
Decreasing Intervals:
.
Max/Min Values:
.
End Behaviors:
as , yx
as , yx
x-intercept: .
y-intercept: .
Domain: .
Range: .
Increasing Intervals:
.
Decreasing Intervals:
.
Max/Min Values:
.
End Behaviors:
as , yx
as , yx
x-intercept: .
y-intercept: .
Domain: .
Range: .
Increasing Intervals:
.
Decreasing Intervals:
.
Max/Min Values:
.
End Behaviors:
as , yx
as , yx
x-intercept: .
y-intercept: .
Domain: .
Range: .
Increasing Intervals:
.
Decreasing Intervals:
.
Max/Min Values:
.
End Behaviors:
as , yx
as , yx
2018-19 Are you Ready for AP Calculus AB? 13
5] 2
1f x
x 6]
2
3
xf x
x
7] 2f x x 8] 3xf x e
9] ln 3f x x 10] 2 1xf x
x-intercept: .
y-intercept: .
Domain: .
Range: .
Increasing Intervals:
.
Decreasing Intervals:
.
Max/Min Values:
.
End Behaviors:
as , yx
as , yx
x-intercept: .
y-intercept: .
Domain: .
Range: .
Increasing Intervals:
.
Decreasing Intervals:
.
Max/Min Values:
.
End Behaviors:
as , yx
as , yx
x-intercept: .
y-intercept: .
Domain: .
Range: .
Increasing Intervals:
.
Decreasing Intervals:
.
Max/Min Values:
.
End Behaviors:
as , yx
as , yx
x-intercept: .
y-intercept: .
Domain: .
Range: .
Increasing Intervals:
.
Decreasing Intervals:
.
Max/Min Values:
.
End Behaviors:
as , yx
as , yx
x-intercept: .
y-intercept: .
Domain: .
Range: .
Increasing Intervals:
.
Decreasing Intervals:
.
Max/Min Values:
.
End Behaviors:
as , yx
as , yx
x-intercept: .
y-intercept: .
Domain: .
Range: .
Increasing Intervals:
.
Decreasing Intervals:
.
Max/Min Values:
.
End Behaviors:
as , yx
as , yx
2018-19 Are you Ready for AP Calculus AB? 14
11] 2 2 25x y 12] 216f x x
13] 2
2 , 2
1 1 , 2 1
3 , 1
x
f x x x
x
14] 2
2 , 1
2 , 1 2
3 , 2
x x
f x x x
x x
15] Graph one cycle of sin 26
f x x
16] Graph one cycle of 2cos 13
f x x
x-intercept: .
y-intercept: .
Domain: .
Range: .
Increasing Intervals:
.
Decreasing Intervals:
.
Max/Min Values:
.
End Behaviors:
as , yx
as , yx
x-intercept: .
y-intercept: .
Domain: .
Range: .
Increasing Intervals:
.
Decreasing Intervals:
.
Max/Min Values:
.
End Behaviors:
as , yx
as , yx
x-intercept: .
y-intercept: .
Domain: .
Range: .
x-intercept: .
y-intercept: .
Domain: .
Range: .
Amplitude: Domain:
Period: Range:
Amplitude: Domain:
Period: Range:
2018-19 Are you Ready for AP Calculus AB? 15
17] Graph one cycle of csc 2f x x 18] Graph one cycle of 1
sec2
f x x
19] Graph one cycle of 3cot 1f x x 20] Graph one cycle of tan 2f x x
21] a. Graph siny & cosy on 0,2 using two different colors on the same plane given below.
b. Identify the x-intercept(s) of each graph. c. Identify their intersection point(s).
Amplitude: Domain:
Period: Range:
Amplitude: Domain:
Period: Range:
Amplitude: Domain:
Period: Range:
Amplitude: Domain:
Period: Range:
2018-19 Are you Ready for AP Calculus AB? 16
Evaluate without a calculator.
1] cos 2] tan2
3]
3csc
2
4]
4
5sin
= 5]
1 3sin
2
=
6] 1tan 3 7] cot2
8] 1 2
tan2
9] 5
cos6
10] arcsin( 1)
11]1tan ( 1) 12] 1cos ( 1) 13]
3cot arcsin
2
14] 1 4cos sin
3
15] Verify sin coscsc
1 cos sin
. 16] Verify
22
2
sec 1sin
sec
.
Solve each equation for x in the given restricted intervals. Check for extraneous solutions, if necessary.
17] xx 22 cossin3 ; 20 x 18] xxx sinsincos 22 ; x 19] xxx cos2sectan ; 0 2x
Solve each equation on 0,2 . Check for extraneous solutions, if necessary.
20] 22sin 1 sin 21]
22tan sec 0 22] sin 2 sin 0
23] tan sec 2 tanx x x 24] 29tan 3 0x 25]
2sin 2cos 2 0x x
26] 2 sin3 cos sin32
xx x
2018-19 Are you Ready for AP Calculus AB? 17
You DO NOT need to print these pages; they are for your reference only!
You are advised to use the next few pages of examples to help refresh your memory or relearn the
essential mathematical skills that you may have forgotten or not fully mastered.
Use Function Notation to Add, Subtract, & Multiply
Composition of Functions
2018-19 Are you Ready for AP Calculus AB? 18
Difference Quotient f x h f x
h
Find & simplify the difference quotient for 4f x x .
To rationalize radical expression, multiply by
conjugate.
2018-19 Are you Ready for AP Calculus AB? 19
Without calculator
With calculator
Without calculator
With calculator
2018-19 Are you Ready for AP Calculus AB? 20
Without calculator
With calculator
Without calculator
With calculator
Without calculator
With calculator
2018-19 Are you Ready for AP Calculus AB? 21
Graphing a Piecewise Function
Greatest Integer Funtion ior ntf x x f x x
Solution graph
