Name:_____ _________________
Precalculus Teacher:
____
AP CALCULUS AB
~ (Σer) ( Force × Distance ) and ( L1 , L 2 ,... ) of Topical Understandings ~
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. We
expect you stay current on your skills as well as improve upon them. Therefore, this packet is a
requirement for Calculus AB. It will be your first major grade of the new school year. SHOW
US YOUR BEST WORK. It needs to be completed when handed in on the first day of class.
Requirements
The following are guidelines for completing the summer work packet…
 There are 50 questions you must complete.
 SHOW ALL WORK ON THE FOLLOWING PAGES. Please include all the steps used to answer
each question. Your work should lead to the correct answer. Please circle or box each answer!!
 Be sure all problems neatly organized, and all writing is legible. If you have trouble answering any
question, be resourceful. This might include looking back to your notes from previous years or even
finding a book or website with formulas or explanations.
 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.
 We expect you to come in with certain understandings that are prerequisite to Calculus. A list of
these topical understandings can be found below. Please be familiar with all of these and ready to
apply them to a higher level.
Topical understandings within summer work…





Factoring
Limits of Functions
Prove Trig Identities
Graphing Piecewise Functions
Graphing, simplifying expressions, and solving equations of the following types:
Trigonometric, rational, logarithmic, exponential, Polynomial/Power, Radical.
Finally, we suggest not waiting until the last two weeks of summer to begin 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 first day of class, and it counts as a grade. It is your ticket
into the class. Best of luck and if you have any questions, feel free to contact us.
Mr. Johnston
dave_johnston@ipsd.org
Calc AB
Mr. Koos
joe_koos@ipsd.org
Calc AB
Mrs. Jessie Lavin
jessie_lavin@ipsd.org
Calc AB
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"--Newton
"In most sciences one generation tears down what another has built. In
mathematics, each generation builds a new story to the old structure"
AP CALCULUS SUMMER PACKET
1.
a)
Use an appropriate procedure to simplify each of the following. No negative exponents allowed in final
answer.
 x 2 + 2 x − 3  x 2 + 2 x 

 2

 x + 2  x − 1 
1
t
c)
1
1−
t
 5t 2 


8 

b)
 15t 2 


12


1+
e)
3
5
−
1− x 1+ x
g)
x 4 (− 2 x ) 6 x 0
3
( )
−2
d)
1 2 3
+ −
2 x x2
f)
y
1+ z
+
1+ z
y
h)
− 8 y 4 (2 y )
−2
(− 3 y )−5
2.
Find the domain of the following functions. Give answers in interval notation. Give algebraic justifications
for each domain.
2 2− x
x
A)
y=
B)
y=
C)
y = log(2 x − 12 )
D)
y = tan x
E)
f ( x) =
2x − 9
2x + 9
x+5
x+2
3.
Determine the range of: f ( x) = 13 − 20 x − x 2 − 3 x 4 . Also, find the max and min values of f (x ) , and state
where they occur.
4.
Solve the double inequality, − 12 <
5.
Solve the equation 4 x − 3 = 5 x + 4 graphically and
5x − 6
≤ −7 .
2
Express your answer in interval notation.
algebracially:
6.
Write the following absolute value expression as a piecewise function.
y = 6 + 2x + 1
7.
Three sides of a fence and an existing wall form a rectangular enclosure. The total length of a fence used
for the three sides is 240 feet. Let x be the length of the two sides perpendicular to the wall as shown.
?
X
X
Existing wall
Write the area A of the enclosure as a function of the length x of the rectangular area as shown in the above
2
figure. Then find the value(s) of x for which the area is 5500 ft .
8.
Rewrite the expression log 5 ( x + 3) into an equivalent expression using only natural logarithms.
9.
Solve by completing the square: x 2 − 4 x = 7
10.
Solve the inequality, x − 8 > 4 . Express your answer in interval notation.
11.
Let f ( x) =
x − 3,
12.
Is this function one-to-one? Justify your answer.
13.
The following three transformations are applied (in the order given) to the graph of y = x 2 .
I.
A vertical stretch by a factor of 3
II.
A horizontal shift right 5 units
III.
A vertical shift down 6 units
and
g ( x) = x 2 + 1 . Compute ( g  f )( x), and state its domain in interval notation.
y = x+7 −2
Which of the following is an equation for the graph produced as a result of applying these transformations?
14.
A.
y = 5x 2 − 1
B.
y = 3( x − 5) 2 − 6
C.
y = 3( x + 5) 2 − 6
D.
y = 3x 2 − 1
E.
y = 3( x − 6) 2 + 5
Let y = f ( x) =
3x + 7
. Find a rule for f
x−2
−1
. In other words… find the inverse function.
15.
Find an equation for the parabola whose vertex is (2, -5) and passes through (4, 7). Express your answer in
the standard form for a quadratic function.
16.
Which of the following could represent a complete graph of f ( x) = ax − x 3 , where a is a real number?
17.
Find a degree 3 polynomial with leading coefficient 4 and zeros -2, 1, and 5.
18.
Let g (x ) be a sinusoidal function with a min at (3π ,5) and the next max at (5π ,8) .
Write an equation for g (x ) . State the amplitude and the period of g (x ) .
19.
20.
Solve the following by factoring and making appropriate sign charts.
A)
x 2 + 6 x − 16 > 0
B)
x3 + 4x 2 − x ≥ 4
C)
2 sin 2 x ≥ sin x
0 ≤ x < 2π
MULTIPLE CHOICE: Solve the inequality x 2 − x − 12 > 0 .
A. (−∞,−4 ) ∪ (3, ∞ )
B. x = 4, x = −3 C. (− 3,4 )
D. (−∞,−3) ∪ (4, ∞ )
21.
The graph of y = 2 − a x +3 for a > 1 is best represented by which graph?
22.
Describe the transformations that can be used to transform the graph of
log x to a graph of
f ( x) = 4 log( x + 2) − 3.
23.
Arturo invests $2700 in a savings account that pays 9% interest, compounded quarterly. If there are no
other transactions, when will his balance reach $4550?
24.
State whether the following relations are functions. Write “yes” or “no”. Then for each one that is a
function, state the domain and range.
a) {(-1, 2), (3, 10), (-2, 20), (3, 11)}
b) {(0, 2), (13, 6), (2, 2), (3, 1)}
25.
26.
Show work to determine if the relation is even, odd or neither.
A)
f ( x) = −4 x 3 − 2 x
B)
f ( x) = x − x 2 + 1
C)
f ( x) = e x −
Simplify (csc x − tan x) sin x cos x .
sin x − cos 2 x
sin 2 x + cos x
A.
C.
27.
1
ex
B.
D.
cos x − sin 2 x
cos 2 x − sin x
Evaluate all six trigonometric functions of the angle θ for the triangle given below.
θ
8
17
28.
29.
Solve each equation.
A)
4 x( x − 2 ) − 5 x( x − 1) = 2
B)
2 x 2 − ( x + 2 )( x − 3) = 12
C)
x+
D)
x 4 − 9x 2 + 8 = 0
1 13
=
x 6
Which transformation was not performed on
A.
Horizontal shift left by
π
B.
units
9
Horizontal stretch by a factor of 3
C.
Vertical stretch by a factor of 2
D.
Reflection through the x-axis
π
y = sin x to obtain y = −2 sin(3x + ) ?
3
B
30.
Solve the right triangle ∆ABC for all its unknown parts,
if β = 38  and b = 4.5.
β
c
A
a
α
b
C
31.
Use a half-angle identity to find the exact value of cos(67.5  ).
32.
Find the equations of both the vertical asymptote(s) and horizontal asymptotes (if they exist). State your
answers as equations.
A)
y=
2x
x−3
B)
y=
x 2 − 2x + 1
x 2 − 3x − 4
C)
y=
2x 2 + 6x
x 3 − 3x 2 − 4 x
(
)
33.
Without using a calculator, find the exact value of cos −1 cos 175π . Justify your answer.
34.
Determine lim
35.
2 students are 180 feet apart on opposite sides of a telephone pole. The angles of elevation from the
students to the top of the pole are 35  and 23 . Find the height of the pole.
36.
Graph the piecewise function.
− x 2
− 2 ≤ x <1

f ( x ) = − 2
x =1
3 x + 5
1< x ≤ 3

37.
Using
x →0
sin x
x2 − x
if possible
f ( x ) from #38 above,
at what points, c, in the domain of
f ( x ) does lim f ( x ) exist?
x→ c
38.
Solve the equation 2 sin 2 x cos x = cos x by factoring.
39.
Find all the exact solutions of the equation on the interval [0, 2π ) .
40.
For the function f (x) graphed, evaluate lim f ( x).
x→3
A. lim f ( x) = 10.
B. lim f ( x) = 3.
C. lim f ( x) = 2.
D. lim f ( x) does not exist
x →3
x →3
x →3
x→3
2 sin 2 x + 3 sin x − 2 = 0
41.
Solve for x: e 2 x = 3x 2 .
42.
Solve
43.
Graph the function y = x 3 − 9 x 2 + 23 x + 1. Find the local
maximum/minimum values, and all the x-intercepts.
Sketch the graph and state the window dimensions.
44.
Use a graphing calculator to approximate all of the function’s real zeros. Round to 3 decimal places.
f ( x) = 3x 6 − 5 x 5 − 4 x 3 + x 2 + x + 1.
(sin 2 x)(cos 3 x) + (cos 3 x)(cos 2 x) =
1
2
for x over [0, 2π ) .
45.
Simplify the following.
3
−
x
B)
4
−
x
1
+4
x
A)
1
−2
x
4
y
3
y
x
1+ x
+
x
1− x
C)
x
1− x
+
x
1+ x
46.
a)
Completely factor the following
4 x 2 + 5x − 6
b)
8 x 3 − 27 y 6
c)
3 x 3 − 15 x + 2 x 2 y − 10 y
47.
48.
Solve each equation for x.
6
=5
x
A)
x+
B)
60
60
2
−
=
x x−5 x
Find an equation for the following lines with the given properties. Express your answer using slope
intercept form.
a) x-intercept = 2, y-intercept = -1
b) parallel to the line y = 2x; containing the point (-1, 2)
49. Use the following functions.
f ( x) = 4 x 2 − 4 x + 3
g ( x) = 2 x − 2
h( x ) = 0
m( x ) =
4x
x −9
2
Find the domain and range of the following. Give answers in interval notation.
a)
f (x)
b)
h(x)
c)
m(x)
Find each of the following:
d)
( f g )(x)
e)
f
 ( x )
h
f)
( f  g )(x )
g)
g −1 ( x )
h)
f (−5)
i)
g (0)
j)
1
m 
2
k)
(f
l)
( f  g )(2)
+ g )(− 3)
50.
a.
Solve.
3
b5 ⋅ 4 b3 = b x
c. 27 x
−3
4
= 125
b. 8 3 x +1 = 128 x − 2
d. log 2 ( x − 2 ) + log 2 x = 3
e. (5.07) x −1 = 100
Simplify.
f. log 8 5 32 4
g. log 7
h. log 3 48 − 2 log 3 4
i. e 3 ln x
1
5
+ log 7
5
49