AP Calculus Summer Review
1. Sketch the following without a graphing calculator. Determine the domain and range of each:
3 − x, x −3
f ( x) = ( x − 2 ) , −3 x 2
2
x3 , x2
2. Determine a formula for the linear function f that satisfies the conditions that f(2) = 0 and f(0) = 4
3. Solve the following inequality algebraically.
3x + 6
0
x+2
4. Determine whether the following function is one-to-one, and find an equation for the inverse
relation of each function. Is the inverse a function? If so, find a function rule y = f −1 ( x) for the
inverse. Draw a graph of f and its inverse relation.
f ( x) = 2 x − 4
5. Draw a complete graph of f ( x) = x 3 + 7 x 2 + 6 x − 8 . To do so confirm that there is at least one
rational root (hint: it should be at x = -2). Use synthetic division to find the other quadratic factors and
then determine all real-number zeros. Classify each real-number zero as rational or irrational.
6. Find the domain of each function by algebraic means.
x +1
a. f ( x) =
( x − 1)( x + 2 )
3
b. f ( x) =
x − 2x − 3
2
5
c. f ( x) = 2
x − 3x + 1
7. Do not use a calculator to answer any of the following:
Given the following point on the terminal side of an angle in standard position, give
the smallest positive angle measure in both degrees and radians.
a. (-1, 1) b. (-1, 3 ) c. ( 3 , 1)
8. Use a calculator to solve cot = 1/3 if 0 90
9. Use a calculator, if necessary, to find all solutions for each of these trig equations:
a. sin 2x = 0.5
b. tan x = -1
c. 2 cox 3x = 0.45
d. 2 sin −1 x = 2
10. Find the equation of the line with slope 3, through the point (-4, -1)
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11. Find the equation of the line that passes through the points (3, 1) and (-5, 4)
12. Find the equation of the line through the point (1, 4) and parallel to the line whose equation is
2x – 5y + 7 = 0
13. Find the equation of the line with slope –2 and x-intercept 4.
14. Find the center and radius of the circle: 2 x 2 + 2 y 2 − 2 x + 2 y − 7 = 0 and sketch the graph
15. Sketch the following graphs without using a graphing calculator:
a. y = 4 x − 3 b. y = 9 − x 2
c. y = x − 1 d. xy = −1
e. y = − x − 1 f. xy = 9
g. y 2 = x − 1 h. y = 2 x and y = − 2 x
i. y = x + 2
2
16. Given g ( x) = 3x 2 − 4 , find:
[g(x + h) – g(x)]/h
17. Given that f(x) = 3 – 2x, and g(x) = 6 – 3x, define the following functions and determine the
domain of the composite function:
a. f g b. g f c. f f d. g g
18 Determine the domain and range of each function and sketch the graph (do not use a graphing
calculator)
a. g ( x) = 6 − 2 x
b. G(x) = 5 − x 2
c. f ( x) = 3x − 6
d. H ( x) = x − 1
4 x2
f. F ( x) =
2x +1
g. 4𝑥 + 9𝑥 2 = 36
2
19. Simplify the following by factoring out the greatest common factor:
a. 4 ( 2 x + 1) ( 3x − 1) + 3 ( 2 x + 1) (3x − 1)
3/ 2 1/ 3 1/ 2 4/ 3
b. 9 ( 6 x + 7 ) ( 4 x − 5) + 10 ( 6 x + 7 ) ( 4 x − 5)
1/ 2 5/ 2 3/ 2 3/ 2
( 2 x + 1) (3x − 2) + ( 2 x + 1) ( 3x − 2 )
1/ 2 −2/ 3 −1/ 2 1/ 3
c.
d. 2 ( 4 x − 1) ( 3x + 5 ) + 2 ( 4 x − 1) ( 3x + 5 )
1/ 2 −1/ 3 −1/ 2 2/ 3
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20. Use the table below to evaluate the following:
x 1 2 3 4 5 6 7
f(x) -2 8 9 0 -5 -6 4
g(x) 5 4 2 -19 -11 3 1
(a) f(g(3) (b) (g ○g)(2) (c) 𝑓 −1 (4)
21. Use the given graph of f and g to evaluate the expressions:
(a) f(g(5)) (b) (𝑓 ○ 𝑔)(0) (c) f(f(5))
−4𝑥 2 −35𝑥+9
22. Find the horizontal and vertical asymptotes: 𝑓(𝑥) = 𝑥 2 −81
23. Graph the following:
1
(a) 𝑦 = 𝑒 𝑥 (b) 𝑦 = ln(𝑥) (c) 𝑦 = sin(2𝑥) (d) 𝑦 = 2 𝑡𝑎𝑛 x (e) 𝑦 = 𝑡𝑎𝑛−1 𝑥
𝑓(𝑥+ℎ)−𝑓(𝑥)
24. If 𝑓(𝑥) = 𝑥 2 − 𝑥 − 5, find ℎ
25. Describe a quadratic function whose vertex is at (-1, 3) and has a y-intercept at (0, 5) numerically,
graphically, algebraically and verbally
26. Use the Law of Logarithm to write as a single Log: 𝑦 = 𝑙𝑜𝑔(𝑥 − 2) + log(2𝑥 + 3) − log 7𝑥
27. Write 𝑙𝑛𝑦 = 3𝑥 − 1 in exponential form
28. Use a graphing utility to find:
(a) zeros: 𝑦 = −4𝑥 3 + 16𝑥 2 − 11
(b) maxima/minima: 𝑦 = 5𝑥 4 − 10𝑥 2 − 3
(c) intersection: 𝑦 = 𝑒 𝑥 − 1𝑎𝑛𝑑𝑦 = √5𝑥 + 8
29. Find the exact value of the following without a calculator:
𝜋 3𝜋 1
(a) 𝑠𝑖𝑛 4 (b) 𝑡𝑎𝑛−1 (0) (c) cos 2 (d) 𝑠𝑖𝑛−1 (2)
30. Factor: 8𝑥 3 − 1
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