AP Calculus Summer Review Packet
This packet is a review of the entering objectives for AP Calculus and is due on
the first day back to school. It is to be done NEATLY and on a SEPARATE
sheet of paper. Have a great summer.
I. Simplify. Show the work that leads to your answer.
1.
x4
2
x  3x  4
2.
x3  8
x2
3.
5 x
x 2  25
4.
x 2  4 x  32
x 2  16
II. Trigonometric Identities.
1. Pythagorean = ___________________
___________________
___________________
3. sin 2x =
2. cos 2x = ____________________
____________________
____________________
___________________
III. Simplify each expression.
2
2
2. x
10
x5
1
1

1.
xh x
1
1

3. 3  x 3
x
IV. Solve for z:
1. 4x + 10yz = 0
2x
1
8


x2  6x  9 x  1 x2  2x  3
2. y2 + 3yz – 8z – 4x = 0
V. If; f(x) = {(3,5), (2,4), (1,7)} g(x) =
determine each of the following:
1. (f + h)(1) =
2. (k – g)(5) =
5. f -1(x) =
4.
6. k-1(x) =
x 3
h(x)= {(3,2), (4,3), (1,6)}
3. (f ◦ h)(3) =
7.
1
=
f ( x)
k(x) = x2 + 5
4. (g ◦ k)(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
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AP Calculus Summer Review Packet
VII. Expand and simplify
3
n2

n 0 2
4
1.
2.
1
n
n 1
3
VIII. Simplify
1.
x
x
eln 3
2.
4. ln 1
5. ln e7
7. log 1/2 8
8. ln
4 xy 2
10.

1
3
e (1 ln x )
3.
6. log3(1/3)
1
2
9.
11. 272/3
e 3ln x
12. (5a2/3)(4a3/2)
12 x y 5
13. (4a5/3) 3/2
3(n  1)!
5n !
14.
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. parallel to 2x – 3y = 7 and passes through (5 , 1)
4. __________________________
5. perpendicular to the line in problem #1, containing
the point (3, 4)
5. __________________________
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. Without a calculator, determine the exact value of each expression.
1. sin 0
2. sin

2
5. cos
7
6
6. cos
9. tan
2
3
10. tan

3

2
3. sin
3
4
4. cos 
7. tan
7
4
8. tan
11. cos(Sin-1
1
)
2

6
12. Sin-1(sin
7
)
6
XII. For each function, determine its domain and range.
1. y 
x4
2. y 
x2  4
3. y  4  x 2
4. y 
x2  4
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AP Calculus Summer Review Packet
XIII. Determine all points of intersection.
1. parabola y = x2 + 3x –4 and
line y = 5x + 11
2. y = cos x and y = sin x in the
first quadrant
XIV. Solve for x, where x is a real number. Show the work that leads to your solution.
x 4 1
0
x3
1. x + 3x – 4 = 14
2.
4. 2x2 + 5x = 8
5. (x + 3)(x – 3) > 0
6. x2 – 2x - 15  0
8. sin 2x = sin x , 0  x  2
9. |x – 3| < 7
2
7. 12x2 = 3x
3. (x – 5)2 = 9
10. (x + 1) (x – 2) + (x + 1)(x – 2) = 0
11. 272x = 9x-3
12. log x + log(x – 3) = 1
14. ln y = 2t - 3
2
2
13. e3 k = 5
XV. Graph each function. Give its domain and range.
1. y = sin x
2. y = cos x
3. y = tan x
4. y = x3 – 2x2 – 3x
5. y = x2 – 6x + 1
6. y =
7. y =
10. y =
13.
x2  4
x2
3
y
x
x 4
x 1
8. y = ex
9. y =
11. y = ln x
12. y = |x + 3| - 2
1
x
14.
x 2

y = x + 2
4

x
if x < 0
if 0  x  3
if x > 3
XVI. Graph each of the following. Then, dentify, 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.)
1. r = 2
___________________
_________________________________
2. r = 3 sin θ
___________________
_________________________________
3. r = 1 + sin θ
___________________
_________________________________
4. r = 2cos 3θ
___________________
_________________________________
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AP Calculus Summer Review Packet
XVII.

2
1.   
n 1  3 
Sequences and series. Evaluate each of the following.
n
5
2)
 ( 2 n  4)
4
3)
n 1
n
2
n
n 1
50
4)
 ( 2 n  4)
n 1
5. Find the 131st term in the arithmetic sequence in which t 3  7 & t 9  61 .
6. Find the first two terms of the geometric sequence for which t 5  36 2 & r 
2.
XVIII. Growth, decay, and logistics.
1. Write a logistic model with the following characteristics…
Limit to growth = 30
Initial Value = 12
Passes through ( 2 , 20 )
2. Write the equation for the exponential model that passes through….


2b)   5,
2a) ( -2 , 12 ) and ( 0 , 3 )
243 
3

 and   1, 
2 
2

3. Fill in the blanks…
Fill in the blanks for
f ( x)  2(5)
Continuous ( yes/no) _____
lim f ( x) 
x 
lim f ( x) 
x  
H.A.
_______
_______
___________
x
.
Domain: _______________
Range: ________________
Increasing: _______________
Decreasing: _______________
Upper bound: __________________
Lower bound: __________________
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