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AP Calculus AB
Summer Assignment
ATM Math Department
The purpose of the summer assignment is to prepare you with the necessary Pre-Calculus skills required for AP Calculus AB. Next year we will be starting off the year with Calculus and will not be reviewing any Pre-Calculus topics at the beginning of the year. Instead we will review the Pre-Calculus topics as needed.
However, to make sure you have the pre-requisites required to handle Calculus you will be given a 50 question multiple choice Pre-Calculus test during the first week of school. The date of the test will be announced on the first day of school.
To prepare for this test, you should complete the review provided. The answers are included for you to check to see if you are answering the questions correctly. The review will not be collected. We will spend some time of the first day of school going over any questions you have on the review.
In additon to the review, complete the following:
• Join AP Calc AB Remind for important updates over the summer. Join by clicking this link: http://rmd.at/atmcalcab or texting 81010 with “@ATMCalcAB”.
• Memorize the unit circle (radians only) and complete the attached table.
• Obtain a TI-nspire CX CAS graphing calculator and bring it to first day of school. If you are taking and or plan to take AP Statistics, you will need this calculator in that class as well. You may elect to use another calculator; however ONLY the TI-nspire CX CAS will be taught in class. Learning to use another calculator will be your own responsibility. A limited number of TI-nspire calculators will be available to rent from the school. Starting Monday August 12th. Details will be sent via Remind.
If you have any questions please use Remind.
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AP Calculus AB Summer Assignment Study Guide
Name___________________________________ Period: Date:
SHORT ANSWER. Write the word or phrase that best completes each statement or answers the question.
Evaluate the exponential expression.
1) (-5)-2 1)
2) -3-4 2)
Evaluate the expression without using a calculator.
3) 811/4 3)
4) 274/3 4)
Simplify the exponential expression.
5)x-6
y-95)
6) (10x7)2
6)
7) -4y0 7)
8)18x7y11
6x3y88)
9) x2y-5 9)
10)3x2y3
z4
310)
11)xy6
x6y
-211)
Rationalize the denominator.
12)81
712)
13)3
7 - 313)
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14)5
6 + 1114)
Find the product.
15) (x - 4)(x2 + 4x + 16) 15)
16) (x + 4)(3x2 + 6x + 5) 16)
17) (9x3 - 2)(x2 + 2) 17)
Solve the problem.
18) Write a polynomial in standard form that represents the volume of the open box.
x
7 - 12x
2 - 12x
18)
19) Write a polynomial in standard form that represents the area of the shaded region.
x + 7
x + 4
x + 4 x + 1
19)
Factor out the greatest common factor.
20) 18x4 - 6x3 + 10x2 20)
21) x(x + 8) + 15(x +8) 21)
22) x2(x - 15) - (x - 15) 22)
Factor by grouping. Assume any variable exponents represent whole numbers.
23) x3 + 5x2 + 3x + 15 23)
24) x3 + 7x - 4x2 - 28 24)
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Factor the trinomial, or state that the trinomial is prime.
25) x2 - 5x + 6 25)
26) x2 + 9x - 22 26)
27) 10x2 + 21x + 9 27)
28) 25x2 - 30x + 8 28)
29) 3x2 + 2xy - 16y2 29)
30) 6x2 - 5xy - 6y2 30)
Factor the difference of two squares.
31) 64x2 - 25 31)
32) 9x2 - 4y2 32)
33) (81x4 - 1) 33)
Factor the perfect square trinomial.
34) 4x2 - 4x + 1 34)
Factor using the formula for the sum or difference of two cubes.
35) x3 - 8 35)
36) 64x3 + 1 36)
37) 125x3 - 64 37)
Factor completely, or state that the polynomial is prime.
38) 2x3 - 288x 38)
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Solve the problem.
39) Write an expression for the area of the shaded region and express it in factored form.
2 2
2 2
7y
2 2
2 2
7y
39)
Factor and simplify the algebraic expression.
40) x5/6 - x1/6 40)
41) (x + 3)1/4 + (x + 3)3/4 41)
42) (x + 2)2/5 - (x + 2)12/5 42)
43) (x + 6)-1/5 + (x + 6)-6/5 43)
Simplify the rational expression. Find all numbers that must be excluded from the domain of the simplified rational
expression.
44)4x + 4
20x2 + 28x + 844)
45)x2 + 11x + 18
x2 + 16x + 6345)
Add or subtract as indicated.
46)2
x2 - 3x + 2 +
7
x2 - 146)
47)6x
x + 1 +
7
x - 1 -
12
x2 - 147)
Simplify the complex rational expression.
48)
x
7 - 1
x - 748)
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49)
1 - 10
x
1 + 10
x
49)
50)
x - x
x - 3
x - 450)
Simplify the expression.
51)
x - 1
2 x
x51)
First, write the value or values of the variable that make a denominator zero. Then solve the equation.
52)9
x =
3
2x + 75 52)
53)27
x - 1 + 3 =
21
x - 153)
Solve the formula for the specified variable.
54) V = 1
3Bh for B 54)
55) S = 2πrh + 2πr2 for h 55)
Solve the quadratic equation by the method of your choice.
56) (3x + 7)2 = 9 56)
57) 2x2 = 7x + 9 57)
58) x2 + 8x = 3 58)
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Use the graph to find the indicated function value.
59) y = f(x). Find f(2).
Find the domain.
Find the range.
Intervals in which the function is increasing.
Intervals in which the funciton is decreasing.
Intervals in which the function is constant.
The minimum value.
The maximum value.
x-8 -7 -6 -5 -4 -3 -2 -1 1 2 3 4 5 6 7 8
y8
7
6
5
4
3
2
1
-1
-2
-3
-4
-5
-6
-7
-8
x-8 -7 -6 -5 -4 -3 -2 -1 1 2 3 4 5 6 7 8
y8
7
6
5
4
3
2
1
-1
-2
-3
-4
-5
-6
-7
-8
59)
Evaluate the piecewise function at the given value of the independent variable.
60) g(x) =x2 - 8
x - 6 if x ≠ 6
x + 6 if x = 6
; g(3) 60)
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Graph the function.
61) f(x) = x + 1 if -7 ≤ x < 4-6 if x = 4
-x + 5 if x > 4
x-10 -5 5 10
y
10
5
-5
-10
x-10 -5 5 10
y
10
5
-5
-10
61)
Find and simplify the difference quotient f(x + h) - f(x)
h, h≠ 0 for the given function.
62) f(x) = 6x + 7 62)
63) f(x) = 2x2 63)
Use the given conditions to write an equation for the line in point-slope form.
64) Slope = 8
9, passing through (5, 6) 64)
65) Passing through (-2, -4) and (-5, -2) 65)
66) Passing through (1, -3) with x-intercept = -1 66)
Begin by graphing the standard quadratic function f(x) = x2 . Then use transformations of this graph to graph the given
function.
67) h(x) = (x - 4)2 - 2
x-10 -8 -6 -4 -2 2 4 6 8 10
y10
8
6
4
2
-2
-4
-6
-8
-10
x-10 -8 -6 -4 -2 2 4 6 8 10
y10
8
6
4
2
-2
-4
-6
-8
-10
67)
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Graph the given functions on the same rectangular coordinate system. Describe how the graph of g is related to the
graph of f.
68) f(x) = x, g(x) = x - 1
x-6 -5 -4 -3 -2 -1 1 2 3 4 5 6
y6
5
4
3
2
1
-1
-2
-3
-4
-5
-6
x-6 -5 -4 -3 -2 -1 1 2 3 4 5 6
y6
5
4
3
2
1
-1
-2
-3
-4
-5
-6
68)
Begin by graphing the standard square root function f(x) = x . Then use transformations of this graph to graph the
given function.
69) g(x) = x + 4 - 1
x-10 -8 -6 -4 -2 2 4 6 8 10
y10
8
6
4
2
-2
-4
-6
-8
-10
x-10 -8 -6 -4 -2 2 4 6 8 10
y10
8
6
4
2
-2
-4
-6
-8
-10
69)
Find the zeros of the polynomial function.
70) f(x) = x3 + x2 - 30x 70)
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Begin by graphing the standard absolute value function f(x) = x . Then use transformations of this graph to graph the
given function.
71) g(x) = 1
4x + 5 + 2
x-10 -8 -6 -4 -2 2 4 6 8 10
y10
8
6
4
2
-2
-4
-6
-8
-10
x-10 -8 -6 -4 -2 2 4 6 8 10
y10
8
6
4
2
-2
-4
-6
-8
-10
71)
Find the domain of the function.
72) f(x) = x
x2 + 472)
73) h(x) = x - 4
x3 - 36x73)
74) f(x) = 2 - x 74)
75)x
x - 375)
Find the domain of the logarithmic function.
76) f(x) = log6 (x + 1) 76)
For the given functions f and g , find the indicated composition.
77) f(x) = 7
x - 3, g(x) =
4
3x
(f∘g)(x)
77)
78) f(x) = x - 9
10, g(x) = 10x + 9
(g∘f)(x)
78)
Find functions f and g so that h(x) = (f ∘ g)(x).79) h(x) = (-6x - 18)3 79)
Find the inverse of the one-to-one function.
80) f(x) = (x + 3)3 80)
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81) f(x) = x - 4 81)
Determine whether the given quadratic function has a minimum value or maximum value. Then find the coordinates of
the minimum or maximum point.
82) f(x) = x2 + 2x - 4 82)
83) f(x) = x2 + 2x - 7 83)
Solve the problem.
84) A developer wants to enclose a rectangular grassy lot that borders a city street for parking.
If the developer has 356 feet of fencing and does not fence the side along the street, what is
the largest area that can be enclosed?
84)
Find the x-intercepts of the polynomial function. State whether the graph crosses the x -axis, or touches the x-axis and
turns around, at each intercept.
85) f(x) = (x + 1)(x - 6)(x - 1)2 85)
Find the zeros for the polynomial function and give the multiplicity for each zero. State whether the graph crosses the
x-axis or touches the x-axis and turns around, at each zero.
86) f(x) = 5(x - 2)(x - 7)4 86)
Sketch the polynomial function using the zeros, multiplicity of each zero, and end behavior.
87) f(x) = x(x + 1)(x + 4)
x-6 -5 -4 -3 -2 -1 1 2 3 4 5 6
y654321
-1-2-3-4-5-6
x-6 -5 -4 -3 -2 -1 1 2 3 4 5 6
y654321
-1-2-3-4-5-6
87)
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88) f(x) = x4 - 9x2
x-10 -8 -6 -4 -2 2 4 6 8 10
y10
8
6
4
2
-2
-4
-6
-8
-10
x-10 -8 -6 -4 -2 2 4 6 8 10
y10
8
6
4
2
-2
-4
-6
-8
-10
88)
89) f(x) = x4 - 3x3 - x2 + 3
x-10 -8 -6 -4 -2 2 4 6 8 10
y10
8
6
4
2
-2
-4
-6
-8
-10
x-10 -8 -6 -4 -2 2 4 6 8 10
y10
8
6
4
2
-2
-4
-6
-8
-10
89)
Find the y-intercept of the polynomial function.
90) f(x) = (x + 1)(x - 6)(x - 1)2 90)
Use the Leading Coefficient Test to determine the end behavior of the polynomial function.
91) f(x) = -4x4 - 4x3 + 5x2 - 2x + 1 91)
92) f(x) = (x + 1)(x + 2)(x + 3)2 92)
Find the zeros of the polynomial function.
93) f(x) = x3 + x2 - 6x 93)
94) f(x) = x3 + 7x2 - x - 7 94)
Use the Intermediate Value Theorem to determine whether the polynomial function has a real zero between the given
integers.
95) f(x) = 6x3 + 7x2 + 7x + 7; between -2 and -1 95)
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Divide using synthetic division.
96)3x3 - 14x2 - 22x - 12
x - 696)
97)x5 + x2 - 3
x - 297)
Solve the polynomial equation. In order to obtain the first root, use synthetic division to test the possible rational roots.
98) x3 + 6x2 - x - 6 = 0 98)
99) x3 + 5x2 + 3x - 1 = 0 99)
Solve the problem.
100) A box with an open top is formed by cutting squares out of the corners of a rectangular
piece of cardboard and then folding up the sides. If x represents the length of the side of
the square cut from each corner, and if the original piece of cardboard is 10 inches by 9
inches, what size square must be cut if the volume of the box is to be 56 cubic inches?
100)
Use the graph of the rational function shown to complete the statement.
101)
x-10 -8 -6 -4 -2 2 4 6 8 10
y10
8
6
4
2
-2
-4
-6
-8
-10
x-10 -8 -6 -4 -2 2 4 6 8 10
y10
8
6
4
2
-2
-4
-6
-8
-10
As x→-1+, f(x)→ ?
101)
Find the vertical asymptotes, if any, of the graph of the rational function.
102) f(x) = x
x2 + 1102)
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103) f(x) = x - 5
x(x - 5)103)
Find the horizontal asymptote, if any, of the graph of the rational function.
104) f(x) = 20x
5x2 + 1104)
105) g(x) = 10x2
5x2 + 1105)
106) h(x) = 6x3
2x2 + 1106)
Graph the function by making a table of coordinates.
107) f(x) = 4x
x-6 -4 -2 2 4 6
y6
4
2
-2
-4
-6
x-6 -4 -2 2 4 6
y6
4
2
-2
-4
-6
107)
Graph the function.
108) Use the graph of f(x) = 4x to obtain the graph of g(x) = 4x - 1.
x-6 -4 -2 2 4 6
y6
4
2
-2
-4
-6
x-6 -4 -2 2 4 6
y6
4
2
-2
-4
-6
108)
Write the equation in its equivalent exponential form.
109) log2 8 = 3 109)
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Write the equation in its equivalent logarithmic form.
110)3343 = 7 110)
Evaluate the expression without using a calculator.
111) log4 4 111)
112) log9 1
9112)
Find the domain of the logarithmic function.
113) f(x) = log4 (x + 1) 113)
114) f(x) = log8 (x - 3)2 114)
Use properties of logarithms to expand the logarithmic expression as much as possible. Where possible, evaluate
logarithmic expressions without using a calculator.
115) log3x + 7
x2115)
116) logb xy4
z5116)
Use properties of logarithms to condense the logarithmic expression. Write the expression as a single logarithm whose
coefficient is 1. Where possible, evaluate logarithmic expressions.
117) 5 logb q - logb r 117)
118)1
3[5ln (x + 10) - ln x - ln (x2 - 7)] 118)
Use common logarithms or natural logarithms and a calculator to evaluate to four decimal places
119) log2 25 119)
Solve the exponential equation. Express the solution set in terms of natural logarithms.
120) 62x = 3.6 120)
121) ex + 6
= 4 121)
Solve the exponential equation. Use a calculator to obtain a decimal approximation, correct to two decimal places, for the
solution.
122) e3x - 1 - 4 = 1017 122)
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Solve the logarithmic equation. Be sure to reject any value that is not in the domain of the original logarithmic
expressions. Give the exact answer.
123) log6 (x + 3) = -1 123)
124) ln x + 5 = 4 124)
125) log9 (x + 2) - log
9 x = 2 125)
Use the Pythagorean Theorem to find the length of the missing side.Then find the indicated trigonometric function of
the given angle. Give an exact answer with a rational denominator.
126) Find sin θ.
7
6
126)
127) Find sec θ.
8
3
127)
128) Find tan θ.
2
9
128)
A point on the terminal side of angle θ is given. Find the exact value of the indicated trigonometric function of θ.
129) (21, 28) Find cos θ. 129)
130) (6, 8) Find csc θ. 130)
131) (3, 7) Find cot θ. 131)
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The graph of a function is given. Use the graph to find the indicated limit and function value, or state that the limit or
function value does not exist.
132) a. limx→0
f(x) b. f(0)
x-10-9 -8 -7 -6 -5 -4 -3 -2 -1 1 2 3 4 5 6 7 8 9 10
y
1
-1
x-10-9 -8 -7 -6 -5 -4 -3 -2 -1 1 2 3 4 5 6 7 8 9 10
y
1
-1
132)
133) a. limx→0
f(x) b. f(0) 133)
134) a. limx→2
f(x) b. f(2)
x1 2 3 4 5
y5
4
3
2
1
x1 2 3 4 5
y5
4
3
2
1
134)
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135) a. limx→3
f(x) b. f(3)
x1 2 3 4 5
y5
4
3
2
1
x1 2 3 4 5
y5
4
3
2
1
135)
136) a. lim
x→3-f(x) b. f(3)
x1 2 3 4 5
y5
4
3
2
1
x1 2 3 4 5
y5
4
3
2
1
136)
Use properties of limits to find the indicated limit. It may be necessary to rewrite an expression before limit properties
can be applied.
137) limx→0
(x2 - 5) 137)
138) limx→1
(x2 - 2)3
138)
139) limx→5
x2 - 2x - 15
x + 3139)
A piecewise function is given. Use the properties of limits to find the indicated limits, or state that the limit does not
exist.
140) f(x) = -6x + 5 if x < 1
5x - 6 if x > 1
a. lim
x→1-f(x) b. lim
x→1+f(x) c. lim
x→1f(x)
140)
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Answer KeyTestname: 2019_SUMMERASSIGNMENTREVIEW
1)1
25
2) - 1
81
3) 3
4) 81
5)y9
x6
6) 100x14
7) -4
8) 3x4y3
9)x2
y5
10)27x6y9
z12
11)x10
y10
12)9 7
7
13)21 + 3 3
46
14) 11 - 6
15) x3 - 64
16) 3x3 + 18x2 + 29x + 20
17) 9x5 + 18x3 - 2x2 - 4
18) 144x3 - 108x2 + 14x
19) 6x + 24
20) 2x2(9x2 - 3x + 5)
21) (x +8)(x + 15)
22) (x - 15)(x2 - 1)
23) (x + 5)(x2 + 3)
24) (x - 4)(x2 + 7)
25) (x - 3)(x - 2)
26) (x + 11)(x - 2)
27) (2x + 3)(5x + 3)
28) (5x - 2)(5x - 4)
29) (3x + 8y)(x - 2y)
30) (2x - 3y)(3x + 2y)
31) (8x + 5)(8x - 5)
32) (3x + 2y)(3x - 2y)
33) (9x2 + 1)(3x + 1)(3x - 1)
34) (2x - 1)2
35) (x - 2)(x2 + 2x + 4)
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Answer KeyTestname: 2019_SUMMERASSIGNMENTREVIEW
36) (4x + 1)(16x2 - 4x + 1)
37) (5x - 4)(25x2 + 20x + 16)
38) 2x(x + 12)(x - 12)
39) (7y + 4)(7y - 4)
40) x1/6(x2/3 - 1)
41) (x + 3)1/4 (1 + (x + 3)1/2)
42) (x + 2)2/5 (- x2 - 4x - 3)
43)(x + 7)
(x+ 6)6/5
44)1
5x + 2, x ≠ - 2
5, x ≠ - 1
45)x + 2
x + 7, x ≠ -7, -9
46)9x - 12
(x - 1)(x + 1)(x - 2)
47)6x - 5
x - 1
48)1
7
49)x - 10
x + 10
50)x
x - 3
51) 1 - 1
2x
52) 0; 1
10
53) 1; {-1}
54) B = 3V
h
55) h = S - 2πr2
2πr
56) - 10
3, -
4
3
57)9
2, -1
58) {-4 - 19, -4 + 19}
59) 1.25
60) - 1
3
19
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Answer KeyTestname: 2019_SUMMERASSIGNMENTREVIEW
61)
x-10 -5 5 10
y
10
5
-5
-10
(-7, -6)
(4, 5)
(4, -6)
(4, 1)
x-10 -5 5 10
y
10
5
-5
-10
(-7, -6)
(4, 5)
(4, -6)
(4, 1)
62) 6
63) 2(2x+h)
64) y - 6 = 8
9(x - 5)
65) y + 4 = - 2
3(x + 2) or y + 2 = -
2
3(x + 5)
66) y + 3 = - 3
2(x - 1) or y = -
3
2(x + 1)
67)
x-10 -8 -6 -4 -2 2 4 6 8 10
y10
8
6
4
2
-2
-4
-6
-8
-10
x-10 -8 -6 -4 -2 2 4 6 8 10
y10
8
6
4
2
-2
-4
-6
-8
-10
68) g shifts the graph of f 1 unit to the right
x-6 -5 -4 -3 -2 -1 1 2 3 4 5 6
y6
5
4
3
2
1
-1
-2
-3
-4
-5
-6
x-6 -5 -4 -3 -2 -1 1 2 3 4 5 6
y6
5
4
3
2
1
-1
-2
-3
-4
-5
-6
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Answer KeyTestname: 2019_SUMMERASSIGNMENTREVIEW
69)
x-10 -8 -6 -4 -2 2 4 6 8 10
y10
8
6
4
2
-2
-4
-6
-8
-10
x-10 -8 -6 -4 -2 2 4 6 8 10
y10
8
6
4
2
-2
-4
-6
-8
-10
70) x = 0, x = - 6, x = 5
71)
x-10 -8 -6 -4 -2 2 4 6 8 10
y10
8
6
4
2
-2
-4
-6
-8
-10
x-10 -8 -6 -4 -2 2 4 6 8 10
y10
8
6
4
2
-2
-4
-6
-8
-10
72) (-∞, ∞)73) (-∞, -6) ∪ (-6, 0) ∪ (0, 6) ∪ (6, ∞)74) (-∞, 2]75) (3, ∞)76) (-1, ∞)
77)21x
4 - 9x
78) x
79) f(x) = x3, g(x) = -6x - 18
80) f-1(x) = 3x - 3
81) f-1(x) = x2 + 4
82) minimum; - 1, - 5
83) minimum; - 1, - 8
84) 15,842 ft2
85) -1, crosses the x-axis;
6, crosses the x-axis;
1, touches the x-axis and turns around
86) 2, multiplicity 1, crosses x-axis; 7, multiplicity 4, touches x-axis and turns around
21
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Answer KeyTestname: 2019_SUMMERASSIGNMENTREVIEW
87)
x-6 -5 -4 -3 -2 -1 1 2 3 4 5 6
y654321
-1-2-3-4-5-6
x-6 -5 -4 -3 -2 -1 1 2 3 4 5 6
y654321
-1-2-3-4-5-6
88)
x-8 -6 -4 -2 2 4 6 8
y20
16
12
8
4
-4
-8
-12
-16
-20
x-8 -6 -4 -2 2 4 6 8
y20
16
12
8
4
-4
-8
-12
-16
-20
89)
x-10 -8 -6 -4 -2 2 4 6 8 10
y10
8
6
4
2
-2
-4
-6
-8
-10
x-10 -8 -6 -4 -2 2 4 6 8 10
y10
8
6
4
2
-2
-4
-6
-8
-10
90) -6
91) falls to the left and falls to the right
92) rises to the left and rises to the right
93) x = 0, x = - 3, x = 2
94) x = -1, x = 1, x = - 7
95) f(-2) = -27 and f(-1) = 1; yes
96) 3x2 + 4x + 2
97) x4 + 2x3 + 4x2 + 9x + 18 + 33
x - 2
98) {1, -1, -6}
99) {-1, -2 + 5, -2 - 5}
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Answer KeyTestname: 2019_SUMMERASSIGNMENTREVIEW
100) 1 in. by 1 in. square
101) +∞102) no vertical asymptote
103) x = 0 and x = 5
104) y = 0
105) y = 2
106) no horizontal asymptote
107)
x-6 -4 -2 2 4 6
y6
4
2
-2
-4
-6
x-6 -4 -2 2 4 6
y6
4
2
-2
-4
-6
108)
x-6 -4 -2 2 4 6
y6
4
2
-2
-4
-6
x-6 -4 -2 2 4 6
y6
4
2
-2
-4
-6
109) 23 = 8
110) log343 7 = 1
3
111)1
2
112) -1
113) (-1, ∞)114) (-∞, 3) or (3, ∞)115) log3 (x + 7) - 2 log3 x
116) logbx + 4logby - 5logbz
117) logbq5
r
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Answer KeyTestname: 2019_SUMMERASSIGNMENTREVIEW
118) ln 3 (x + 10)5
x(x2 - 7)
119) 4.6439
120)ln 3.6
2 ln 6
121) {ln 4 - 6}
122) 2.64
123) - 17
6
124) {e8 - 5}
125) {1
40}
126)7 85
85
127)73
8
128)2
9
129)3
5
130)5
4
131)3
7
132) a. limx→0
f(x) = 1
b. f(0) = 1
133) a. limx→0
f(x) = 1
b. f(0) does not exist
134) a. limx→2
f(x) = 2
b. f(2) = 1
135) a. limx→3
f(x) does not exist
b. f(3) = 5
136) a. lim
x→3-f(x) = 3
b. f(3) = 5
137) -5
138) -1
139) 0
140) a. -1
b. -1
c. -1
24
