Name: ______________________________________________
Due August 29, 2015
AP CALCULUS SUMMER PACKET 2015
I would like to congratulate you for making the decision to take AP Calculus. It is fabulous that you want to
expand your mathematical knowledge and better prepare for college. I am very happy to have you! Let me
impress upon you how important it is that you have solid Algebra skills as you pursue Calculus. It is often poor
Algebra skills that determine a student’s success in Calculus. Often students understand the Calculus concepts
but will get a problem incorrect because of an algebraic error. I will assume that you have the algebra skills
taught to you over the years; there will be no time to review basic skills in AP Calculus.
 This summer packet is designed as a review of concepts you have already learned but need to brush-up
on, or even re-learn if necessary, before you begin your Calculus course. Work these problems
carefully and completely. Algebraic skills like solving equations, simplifying algebraic expressions and
factoring should be second nature to you by now and are skills used continually in AP Calculus. You will
also be expected to know basic Geometric formulas such as perimeters and areas for two dimensional
objects as well as surface areas and volume from three dimensional objects.
 If you find you are having trouble with certain concepts YouTube is a good place to go for video
explanations, www.purplemath.com is also a good site to go for information as well as
www.khanacademy.org . You may also email me at [email protected] and I will try to get back to
you within the week.
 It is important that you spend some quality time on this packet. Please do not wait until summer is
almost over to begin your packet, however you do not want to start right away either, but instead
begin about mid-summer so the concepts are fresh when school begins. Have a great summer and I’ll
see you August 29, 2015.
NOTE: Challenge yourself and find out what you truly know; Calculators should not be used!
Directions for packet: Show all your work in a composition book, write neatly, and separate each section.
You will use this composition book as your homework notebook. Your work will be turned in with this packet.
Write your answers on the provided answer sheet. This packet will be graded for completion as well as
accuracy. If you should lose your packet I will have a PDF file at my webpage. This can be found at the EVHS
staff directory site. You can print out a new one. There will be a test on the material covered in the packet
shortly after the packet is due. This will be your first grade; a good opportunity for you to start the semester
off strong.
PACKET DUE DATE: August 29, 2015
Remember to show all work in a composition book!
No calculators should be used!
Linear Functions
1. Write the equation of the line described below in point-slope, slope-intercept and standard form.
a. slope: 
3
and y-intercept: –5
2
b. slope: 3 through  2, 4  c. through the points  1,3 &  2, 4 
d. through  3, 2  and parallel to x  2 y  8
e. through  1, 2  and perpendicular to 2 x  3 y  5  0
Polynomial Functions and Factoring
1. Factor each expression completely.
a. 16 x 2  2 x
b. 6 x 5  54 x 3
c. x 2  10 x  16
d. 5 x 2  40 x  80
f. 6  x  x2
h. 8 x3  125
i. 4 x2  49
g. x 3  27
k. x  xy  x y  y
3
j. 2 x 2 e x  xe x  e x
2
2
l. 4x  8x  25x  50
3
3
3
2
n. 3x  x  7   4 x  x  7 
3
m. ac  cd  ab  bd
e. 2 x 2  7 x  3
2
o.  x  3  2 x  1   x  3  2 x  1
4
2
3
3
2
Rational Functions
1. Simplify each expression.
a.
x3  9 x
x 2  7 x  12
4
2
b.
x2  2 x  8
x3  x 2  2 x
c.
1
x
1
x
b.
x 1 x 1

1
3
2
c.
x 1
x

0
x
x 1
d.
x
1
y
x
y
2. Solve each equation.
a.
2 5 1
 
3 6 x
d. x 
6
5
x
3. Find any holes as well as any horizontal, vertical or slant asymptotes of the function.
x4
b. y  2
x 1
x
a. y 
x3
x3
c. y  2
x 4
x2  2 x  1
d. y  2
x  3x  4
Exponential and Logarithmic Functions
1. Simplify each expression. Give exact answers.
a. ln  0 
b. ln 1
f. ln e x 3
j. 4log
4
d. ln e5
g. 2  4ln e2
h. log 10
i. 32 log
k. 2log 4 9  log 4 3
1/ 3
l. 2log10 x  3log10 x
 
 x1
 
c. ln  e 
3
 
e. ln e4
5


2
m. log 2 5  log 2 x  1  log 2  x  1
2. Expand each logarithm.

2
a. ln x  y  2 
3

 100 x3 
b. log  2 
 y z 
3. Solve each equation.
 32 x 2
d. log 2 x  3
a. 5x1  625
b. 23 x1  42 x
c.
e. ln x  0
f. ln x  2
2
g. log 3 x  2log 3 4  4log 3 5
- Pg 1 -
1
3
Remember to show all work in a composition book!
No calculators should be used!
Domain of Functions
1. Find the domain of each function below. Give in interval notation.
a. y 
3x  2
4x  1
b. y 
x2  5x  6
x 2  3x  18
c. y 
g. f  x  
f. y  x  2
e. y  2  1  x
22 x
x
d. y  2e  3
x
3x  1
x  x2
2
h. g  x   ln 4  x 2
Trigonometry
1. Find the other five trig functions given each of the following.
a. cos  
2
3
   2
and
5
2
b. csc  3 and    
2. Solve each equation on the interval 0, 2  . Give exact answers.
a. sin x 
1
b. cos x  1
2
c. 4sin 2 x  1
3. Evaluate. Give exact answers (in radians when necessary).
 5 
 4 
 3 
 7 
a. cos  
b. csc  
c. tan   
d. sin 


 6 
 3 
 3 
 2 
 
 3 
 7 
 5 
g. sin  
h. tan   
i. csc 
j. cos 


4
 4 
 4 
 6 
 11 
m. tan 

 6 
 
s. csc  
3
 7 
n. cot  

 4 
o. sec  2 
p. tan  0 
t. cos  
 
y. sin   
 2
 3 
z. sec  
 2 
 2 
u. sin 

 3 
 
aa. cot  
6
 
v. sec   
 6
 5 
ab. csc  
 6 
1
ae. sin 1  
2

2
af. sin 1  
 2 


ag. cos1  0 
 3
ah. cos 1 
 2 


2 3
al. csc 1 
 3 


am. sec1  1
an. tan 1
ak. csc1
 2
3
2
 3
d. 2cos x  3  0
 2 
e. sec  

 3 
 4 
k. cot 

 3 
 
f. cot  
2
 3 
l. sec  
 4 
 
q. sin   
 3
 5 
w. tan  

 3 
 1 
ac. cos  

 6 
 5 
r. cos  

 4 
 
x. cot   
 4
ai. cot 1  0 
aj. tan 1  1
ad. csc   
 3
ao. cot 1 
ap. sec1  2 
 3 


Function Composition
3
1. If f  x   x  2 x  1 , find f  2  f  2
2
x
2. Let f  x   x , g  x   2 x  1 and h  x   2 . Find the following:
a. g  t  3

b. f h  x 

d. f  h  1 
 
f  g  x   and g  f  x   .
c. f g  2 
2
3. Given f  x   x  1 and g  x   cos x . Find
- Pg 2 -

e. g f  h  1 2  

Remember to show all work in a composition book!
No calculators should be used!
Piecewise Functions
Answer the following questions: State the (a) domain, (b) find the x-intercepts, (c) the y-intercept, and (d) the
indicated function values. Then (e) graph the function (without a calculator). (f) Is the function continuous?
1.
2.
 49  x 2


f  x   7

14

x  1
1

f  x   x
1  x  3

2 x  4 x  3
Find f  3 , f  1 , f  3
0 x7
7  x  14
x  14
Find f  2 , f  7  , f 10
Basic Functions
For each function (a) sketch the graph, (b) state its domain, (c) identify any asymptotes, (d) state how many
zeros it has, (e) state the end behavior limits.
1. f  x   c
2. f  x   x
3. f  x   x 2
4. f  x   x3
5. f  x   x
6. f  x   ln x
7. f  x   x
8. f  x  
1
x
1
10. f  x   e x
11. f  x   e x
2
x
12. f  x   sin x;  2 , 2 
13. f  x   cos x;  2 , 2 
14. f  x   tan x;  2 , 2 
9. f  x  
15. f  x   sec x;  2 , 2 
16. f  x   csc x;  2 , 2 
17. f  x   cot x;  2 , 2 
Problem Solving
1. A hot-air balloon rising straight up from a level field is tracked by a range finder located 5000 ft from the
liftoff point. Express the balloon’s height as a function of the angle of elevation θ.
2. The opposite sides of a rectangle, with a perimeter of 16, are of length x. Express the area as a function of x.
3. An open box is formed by cutting congruent squares, of length x, from the corners of a 10in-by-16in piece of
paper and folding up the sides to form a box with no lid. Express the volume of this box as a function of x.
4. A cylindrical can has a volume of 2 in3. Express the surface area of the can as a function of its radius, r.
5. A rectangular piece of paper contains 12 square inches of printed material. If the top and bottom margins are
each 1 inch and the left and right margins are each 2 inches, express the total area of the paper as a function of
the width of printed material, w.
6a. Find the ratio of the area inside the square but outside the circle to the
area of the square.
6b. Find a formula for the perimeter of a window of the shape given.
- Pg 3 -
Remember to show all work in a composition book!
No calculators should be used!
ANSWERS
Name_____________________________
Linear Functions
1a: _________________________
_________________________
_________________________
1b: _________________________ _________________________
_________________________
1c: _________________________
_________________________
_________________________
1d: _________________________ _________________________
_________________________
1e: _________________________
_________________________
_________________________
Polynomial Functions and Factoring
1a: ________________________
1b: ________________________
1c: ________________________
1d: ________________________
1e: ________________________
1g: ________________________
1h: ________________________
1i: ________________________
1j: ________________________
1k: ________________________
1l: ________________________
1m: ________________________
1n: ________________________
1o: ________________________
2a: ________________________
2b: ________________________
2c: ________________________
2d: ________________________
Rational Functions
1a: __________
1b: __________ 1c: ___________
1d: __________
2a: __________________
2b: ___________________ 2c: __________________
2d: ______________
3a: Holes: __________
3b: Holes: __________
3c: Holes: __________
3d: Holes: _________
HA: ___________
HA: ___________
HA: ___________
HA: ___________
VA: ___________
VA: ___________
VA: ___________
VA: ___________
SA: ___________
SA: ___________
SA: ___________
SA: ________
Exponential and Logarithmic Functions
1a: _____________
1b: _____________ 1c: _____________
- Pg 4 -
1d: _____________ 1e: _____________
Remember to show all work in a composition book!
1f: _____________
No calculators should be used!
1g: _____________ 1h: _____________
1i: _____________ 1j: _____________
1k: _____________ 1l: _____________ 1m:_____________ 1n:______________
2a: __________________________ 2b: __________________________
3a: _______________
3b: _______________
3c: _______________
3e: _______________
3f: _______________
3g: _______________
3d. ______________
Domain of Functions
1a: ___________________
1b: ___________________ 1c: ___________________
1e: ___________________
1f: ___________________
1d: ______________
1g: ___________________ 1h: ______________
Trigonometry
1a.sin   ______
tan   ______
csc   ______
sec   ______
cot   ______
1b.sin   ______
tan   ______
cos   ______
sec   ______
cot   ______
2a: __________________
3a: __________
2b: __________________
2c: __________________
2d: ______________
3b: __________
3c: __________
3d: __________ 3e: __________
3g: __________ 3h: __________
3i: __________
3j: __________
3m: __________ 3n: __________
3o: __________
3p: __________ 3q: __________ 3r: ________
3s: __________
3u: __________
3v: __________ 3w: __________ 3x: ________
3y: __________ 3z: __________
3aa: __________
3ab: __________ 3ac: __________ 3ad: ________
3ae: __________ 3af: __________
3ag: __________
3ah: __________ 3ai: __________ 3aj: ________
3ak: __________ 3al: __________
3am: __________
3an: __________ 3ao: __________ 3ap: ________
3t: __________
3f: ________
3k: __________ 3l: ________
Function Composition
1: _______________
2a: _____________


2b: _____________ 2c: _____________
3: f g  x   __________________________
2d: _____________ 2e: _____________
g  f  x    _______________________________
Piecewise Functions
- Pg 5 -
Remember to show all work in a composition book!
No calculators should be used!
1a. _____________
1b. _____________
1c. _____________
1d. _____________
1f. _____________
2a. _____________
2b. _____________
2c. _____________
2d. _____________
2f. _____________
Basic Functions
1b._____________
1c._____________
1d. _____________
1e. _____________
2b._____________
2c._____________
2d. _____________
2e. _____________
3b._____________
3c._____________
3d. _____________
3e. _____________
4b._____________
4c._____________
4d. _____________
4e. _____________
5b._____________
5c._____________
5d. _____________
5e. _____________
6b._____________
6c._____________
6d. _____________
6e. _____________
7b._____________
7c._____________
7d. _____________
7e. _____________
8b._____________
8c._____________
8d. _____________
8e. _____________
9b._____________
9c._____________
9d. _____________
9e. _____________
10b._____________ 10c._____________ 10d. _____________ 10e. _____________
11b._____________ 11c._____________ 11d. _____________ 11e. _____________
12b._____________ 12c._____________ 12d. _____________ 12e. _____________
13b._____________ 13c._____________ 13d. _____________ 13e. _____________
14b._____________ 14c._____________ 14d. _____________ 14e. _____________
15b._____________ 15c._____________ 15d. _____________ 15e. _____________
16b._____________ 16c._____________ 16d. _____________ 16e. _____________
17b._____________ 17c._____________ 17d. _____________ 17e. _____________
Problem Solving
1: ________________________________________
2: _______________________________________
3: ________________________________________
4: _______________________________________
5: ________________________________________
6a: ________________________________________
- Pg 6 -
6b: ______________________________________