Calculus Summer Review Packet
DUE THE FIRST DAY OF SCHOOL
The problems in this packet are designed to help you review topics that are
important to your success in Calculus. All work must be shown for each
problem. The problems should be done correctly, not just tried. You are
expected to get each problem correct. Please DO NOT use your calculators to
solve these problems. You must know how to do all these problems WITHOUT a
calculator.
Note: Starred problems are for AP Calculus BC students only.
It is recommended that you work with one or more people, but each person
must submit his/her own work. Before you leave school, write down the names,
phone numbers, and/or email addresses for at least two people who are also
taking calculus in the fall.
Name _________________________ Phone _____________
Email _________________________
Name _________________________ Phone _____________
Email _________________________
If you need help with any of the problems, check the Poolesville web site for
links to on-line classes at Montgomery College, which are available to you free of
charge. You may also e-mail Mrs. Loomis at [email protected]
with questions.
All work should be completed to the best of your ability on
the first day of school. After you have an opportunity to ask
questions, you will be quizzed on this material during the
first week of school!!
Have fun with the problems!
AP Calculus Summer Review
I. Simplify. Show the work that leads to your answer.
x−4
x3 − 8
1. 2
2.
x − 3x − 4
x−2
3.
5− x
x 2 − 25
4.
x 2 − 4 x − 32
x 2 − 16
II. Complete the following identities.
1. sin2x + cos2x = __________
3. cot2x + 1 =
__________
5. sin 2x =
__________
2. 1 + tan2x = __________
4. cos 2x =
__________
III. Simplify each expression.
1
1
1.
−
x+h x
2
2
2. x
10
x5
1
1
−
3. 3 + x 3
x
4.
2x
1
8
−
−
x2 − 6x + 9 x + 1 x2 − 2x − 3
1
AP Calculus Summer Review
IV. Solve for z:
2. y2 + 3yz – 8z – 4x = 0
1. 4x + 10yz = 0
V. If
x −3
f(x) = {(3,5), (2,4), (1,7)}
g(x) =
h(x)= {(3,2), (4,3), (1,6)}
k(x) = x2 + 5
determine each of the
following:
1. (f + h)(1) = _______________
2. (k – g)(5) = _______________
3. (f ◦ h)(3) = _______________
4. (g ◦ k)(7) = _______________
5. f -1(x) =
_______________
6. k-1(x) =
1
=
f ( x)
_______________
7.
8. (kg)(x) =
_______________
_______________
VI. Miscellaneous: Follow the directions for each problem.
1. Evaluate
f ( x + h) − f ( x )
and simplify if f(x) = x2 – 2x.
h
2. Expand (x + y)3
3. Simplify:
3
2
5
2
x ( x + x − x2 )
* 4. Eliminate the parameter and write a rectangular equation for
x = t2 + 3
y = 2t
2
AP Calculus Summer Review
VII. Expand and simplify
4
n2
* 1. ∑
n =0 2
3
1
∑n
* 2.
n =1
3
VIII. Simplify
1.
3.
x
x
e(1+ ln x )
_________________ 2.
eln 3
_________________
4. ln 1
_________________
_________________
5. ln e7
_________________ 6. log3(1/3)
_________________
7. log 1/2 8
_________________
_________________
9.
e3ln x
8. ln
1
2
4 xy −2
_________________ 10.
−
1
3
12 x y
_________________
−5
11. 272/3
_________________ 12. (5a2/3)(4a3/2)
13. (4a5/3) 3/2
_________________
* 14.
3(n + 1)!
5n !
_________________
_________________
3
AP Calculus Summer Review
IX. Using the point-slope form y – y1 = m(x – x1), write an equation for the line
1. with slope –2, containing the point (3, 4)
1. __________________________
2. containing the points (1, -3) and (-5, 2)
2. __________________________
3. with slope 0, containing the point (4, 2)
3. __________________________
4. perpendicular to the line in problem #1,
containing the point (3, 4)
4. __________________________
* X. Given the vectors v = -2i + 5j and w = 3i + 4j, determine
1.
1
v
2
____________
2. w – v
3. length of w
4. the unit vector
for v
_____________
______________
_____________
XI. Determine the exact value of each expression.
1. sin 0
4. cos π
7. tan
7π
4
________
________
________
10. cos(Sin –1
1
)
2
2. sin
5. cos
8. tan
________
π
2
3π
4
π
6
________
3. sin
________
6. cos
________
9. tan
11. Sin –1 (sin
7π
)
6
3π
4
π
3
________
________
2π
________
3
________
4
AP Calculus Summer Review
XII. For each function, determine its domain and range.
Function
1. =
y
x−4
2. =
y
x2 − 4
3. =
y
4 − x2
4. =
y
x2 + 4
Domain
Range
_________________
_________________
_________________
_________________
_________________
_________________
_________________
_________________
XIII. Determine all points of intersection.
1. parabola y = x2 + 3x –4 and
line y = 5x + 11
2. r = cos θ and r = sin θ for 0 ≤ θ ≤ 2π
XIV. Solve for x, where x is a real number. Show the work that leads to your
solution.
x4 − 1
2
2.
=0
1. x + 3x – 4 = 14
x3
3. (x – 5)2 = 9
4. 2x2 + 5x = 8
5
AP Calculus Summer Review
Solve for x, where x is a real number. Show the work that leads to your
solution.
5. (x + 3)(x – 3) > 0
6. x2 – 2x - 15 ≤ 0
7. 12x2 = 3x
8. sin 2x = sin x , 0 ≤ x ≤ 2π
9. |x – 3| < 7
10. (x + 1)2(x – 2) + (x + 1)(x – 2)2 = 0
11. 272x = 9x-3
12. log x + log(x – 3) = 1
6
AP Calculus Summer Review
XV. Graph each function. Give its domain and range.
1. y = sin x
2. y = ex
Domain_________________
Domain_________________
Range _________________
Range _________________
3. y =
4. y =
x
3
x
Domain_________________
Domain_________________
Range _________________
Range _________________
7
AP Calculus Summer Review
Graph each function. Give its domain and range.
5. y = ln x
6. y = |x + 3| - 2
Domain_________________
Domain_________________
Range _________________
Range _________________
7.
1
y=
x
8.
x 2

y = x + 2
4

if x < 0
if 0 ≤ x ≤ 3
if x > 3
Domain_________________
Domain_________________
Range _________________
Range _________________
8
AP Calculus Summer Review
XVI. Identify, by name, each polar graph. Give at least one characteristic of
each graph (e.g. radius, location, length of petal, point (other than the pole) on
the graph, etc.) Then sketch a graph of each.
1. r = 2
____________________________________________________
2. r = 3sec θ
____________________________________________________
3. r = 1 + sin θ ____________________________________________________
4. r = 2cos 3θ ____________________________________________________
* XVII. Compute each of the following limits:
cos x
=
x
2.
x


lim 
=
x →∞
 x + sin x 
3.
lim
x3 + 1
=
x → −1 x + 1
4.
lim−
5.
lim
25 − 4 x 3
=
x →∞ 5 x 3 − x 2
6.
1.
lim
x →−π
* XVIII.
1.
Let f(x) =
1

,

x


 x2 − 4

,
 x−2


3,




x ≤ −2
lim−
x →3
x
x −3
x −3
=
Compute each of the following limits:
−2< x < 2
a)
x≥2
c)
2. Consider the function h defined below:
 tan 5 x
 4x , x ≠ 0

h( x ) = 
 k, x = 0


x →0
[x] =
e)
lim f ( x) =
b)
lim f ( x) =
d)
x →−∞
x→2−
lim f ( x ) =
x→2
lim f ( x) =
x → −2 −
lim f ( x ) =
x→2+
f) lim f ( x) =
x →∞
What value should be
assigned to k so that h
will be continuous?
9