ap calculus ab summer assignment mrs. berksonteachers.dadeschools.net/sdaniel/ap calculus ab summer...
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AP Calculus AB Summer Assignment Mrs. Berkson
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:
• Memorize the unit circle (radians only) and complete the table. You will have a speed quiz on the first quadrant of the unit circle during the 2nd week of school. You will have a speed quiz on the entire unit circle during the 3rd week of school. All quiz and tests dates will be posted on my website site in August.
• Purchase 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 Tuesday August 16th. Please email me to arrange pick up.
If you have any questions please email me at [email protected] or visit
http://teachers.dadeschools.net/lberkson
Unit Circle
0 6
4
3
2
2
3
3
4
5
6
7
6
5
4
4
3
3
2
5
3
7
4
11
6
2
sin t
cos t
tan t
csct
sect
cot t
AP Calculus AB Summer Review Packet
Name___________________________________
SHORT ANSWER. Write the word or phrase that best completes each statement or answers the question.
Evaluate the exponential expression.
1) (-5)-4 1)
2) -3-2 2)
Simplify the exponential expression.
3)x-8
y-73)
4) (11x3)2 4)
5) -4y0 5)
6)-28x12y9
7x9y46)
7) x2y-6 7)
8)2x4y4
z3
48)
9)xy3
x4y
-29)
Rationalize the denominator.
10)100
310)
11)2
8 - 311)
12)2
17 + 312)
13)5
10 + 1513)
1
Find the product.
14) (x - 3)(x2 + 3x + 9) 14)
15) (x + 7)(9x2 + 6x + 2) 15)
16) (8x3 + 3)(x2 + 6) 16)
Solve the problem.
17) Write a polynomial in standard form that represents the volume of the open box.
x
8 - 12x
5 - 12x
17)
18) Write a polynomial in standard form that represents the area of the shaded region.
x + 9
x + 6
x + 3 x + 1
18)
Factor out the greatest common factor.
19) 14x4 - 6x3 + 8x2 19)
20) x(x + 6) + 7(x +6) 20)
21) x2(x - 15) - (x - 15) 21)
Factor by grouping. Assume any variable exponents represent whole numbers.
22) x3 - 5x2 - 2x + 10 22)
23) x3 + 7x - 5x2 - 35 23)
Factor the trinomial, or state that the trinomial is prime.
24) x2 + 14x + 48 24)
2
25) x2 + 2x - 24 25)
26) 15x2 + 16x + 4 26)
27) 6x2 - 17x + 12 27)
28) 7x2 - 10xy - 8y2 28)
29) 12x2 + 7xy - 12y2 29)
Factor the difference of two squares.
30) 81x2 - 16 30)
31) 16x2 - 25y2 31)
32) (81x4 - 16) 32)
Factor the perfect square trinomial.
33) 4x2 + 4x + 1 33)
Factor using the formula for the sum or difference of two cubes.
34) x3 - 64 34)
35) 125x3 + 1 35)
36) 8x3 - 125 36)
Factor completely, or state that the polynomial is prime.
37) 9x3 - 576x 37)
38) 5x4 - 5 38)
Solve the problem.
39) Write an expression for the area of the shaded region and express it in factored form.
8 8
8 8
3y
8 8
8 8
3y
39)
3
Factor and simplify the algebraic expression.
40) x8/9 - x1/9 40)
41) (x + 4)1/4 + (x + 4)3/4 41)
42) (x + 3)2/5 - (x + 3)12/5 42)
43) (x + 1)-1/5 + (x + 1)-6/5 43)
Simplify the rational expression. Find all numbers that must be excluded from the domain of the simplified rational
expression.
44)3x + 2
15x2 + 19x + 644)
45)x2 + 5x + 6
x2 + 7x + 1245)
Add or subtract as indicated.
46)3
x + 4 -
6
x - 446)
47)4
x2 - 3x + 2 +
5
x2 - 147)
48)6x
x + 1 +
7
x - 1 -
12
x2 - 148)
Simplify the complex rational expression.
49)
x
2 - 1
x - 249)
50)
1 - 8
x
1 + 8
x
50)
51)
x - x
x - 7
x - 851)
4
Simplify the expression.
52)
x - 1
3 x
x52)
First, write the value or values of the variable that make a denominator zero. Then solve the equation.
53)8
x =
1
2x + 30 53)
54)4
x - 5 + 1 =
2
x - 554)
Solve the formula for the specified variable.
55) V = 1
3Bh for B 55)
56) S = 2πrh + 2πr2 for h 56)
Solve the quadratic equation by the method of your choice.
57) (8x + 7)2 = 36 57)
58) 2x2 = 5x + 7 58)
59) x2 + 4x = 9 59)
5
Use the graph to find the indicated function value.
60) y = f(x). Find f(5).
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
60)
Evaluate the piecewise function at the given value of the independent variable.
61) g(x) =x2 - 7
x - 8 if x ≠ 8
x - 5 if x = 8
; g(2) 61)
6
Graph the function.
62) f(x) = x + 2 if -9 ≤ x < 5
-5 if x = 5
-x + 9 if x > 5
x-10 -5 5 10
y
10
5
-5
-10
x-10 -5 5 10
y
10
5
-5
-10
62)
Find and simplify the difference quotient f(x + h) - f(x)
h, h≠ 0 for the given function.
63) f(x) = 7x - 6 63)
64) f(x) = 5x2 64)
Use the given conditions to write an equation for the line in point-slope form.
65) Slope = 8
9, passing through (2, 5) 65)
66) Passing through (-3, -3) and (-5, -7) 66)
67) Passing through (1, -7) with x-intercept = -1 67)
Begin by graphing the standard quadratic function f(x) = x2 . Then use transformations of this graph to graph the given
function.
68) h(x) = (x + 2)2 - 6
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)
7
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 + 4
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)
Begin by graphing the standard absolute value function f(x) = x . Then use transformations of this graph to graph the
given function.
70) g(x) = 1
2x + 2 - 5
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)
Find the domain of the function.
71) f(x) = x2
x2 + 1871)
72) h(x) = x - 2
x3 - 36x72)
73) f(x) = 23 - x 73)
74)x
x - 574)
8
For the given functions f and g , find the indicated composition.
75) f(x) = 6
x - 3, g(x) =
3
2x
(f∘g)(x)
75)
76) f(x) = x - 3
2, g(x) = 2x + 3
(g∘f)(x)
76)
Find functions f and g so that h(x) = (f ∘ g)(x).
77) h(x) = (9x + 2)6 77)
Find the inverse of the one-to-one function.
78) f(x) = (x - 6)3 78)
79) f(x) = x + 2 79)
Determine whether the given quadratic function has a minimum value or maximum value. Then find the coordinates of
the minimum or maximum point.
80) f(x) = x2 + 3x - 8 80)
81) f(x) = x2 + 2x + 2 81)
Solve the problem.
82) A developer wants to enclose a rectangular grassy lot that borders a city street for parking.
If the developer has 304 feet of fencing and does not fence the side along the street, what is
the largest area that can be enclosed?
82)
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.
83) f(x) = (x + 1)(x - 5)(x - 1)2 83)
84) x4 + 2x3 - 99x2 = 0 84)
Find the y-intercept of the polynomial function.
85) f(x) = (x + 1)(x - 7)(x - 1)2 85)
Use the Leading Coefficient Test to determine the end behavior of the polynomial function.
86) f(x) = -5x4 + 5x3 - 3x2 - 4x - 4 86)
87) f(x) = (x - 1)(x + 2)(x + 3)2 87)
Find the zeros of the polynomial function.
88) f(x) = x3 + x2 - 42x 88)
9
89) f(x) = x3 + 5x2 - x - 5 89)
Use the Intermediate Value Theorem to determine whether the polynomial function has a real zero between the given
integers.
90) f(x) = 9x3 - 4x2 - 5x - 3; between 1 and 2 90)
Divide using synthetic division.
91)3x3 - 18x2 + 27x - 12
x - 491)
92)x5 + x3 + 5
x - 292)
Solve the polynomial equation. In order to obtain the first root, use synthetic division to test the possible rational roots.
93) x3 - 6x2 - x + 6 = 0 93)
94) x3 + 6x2 + 10x + 4 = 0 94)
Solve the problem.
95) 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 15 inches by 9
inches, what size square must be cut if the volume of the box is to be 110 cubic inches?
95)
Use the graph of the rational function shown to complete the statement.
96)
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→-2-, f(x)→ ?
96)
10
Find the vertical asymptotes, if any, of the graph of the rational function.
97) f(x) = x
x2 + 397)
98) f(x) = x + 5
x(x - 5)98)
Find the horizontal asymptote, if any, of the graph of the rational function.
99) f(x) = 10x
5x2 + 199)
100) g(x) = 10x2
2x2 + 1100)
101) h(x) = 8x3
2x2 + 1101)
Graph the function by making a table of coordinates.
102) 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
102)
Graph the function.
103) Use the graph of f(x) = 3x to obtain the graph of g(x) = 3x + 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
103)
11
Write the equation in its equivalent exponential form.
104) log4 16 = 2 104)
Write the equation in its equivalent logarithmic form.
105)364 = 4 105)
Evaluate the expression without using a calculator.
106) log12
12 106)
107) log3 1
3107)
Find the domain of the logarithmic function.
108) f(x) = log9(x + 7) 108)
109) f(x) = log4(x - 3)2 109)
Use properties of logarithms to expand the logarithmic expression as much as possible. Where possible, evaluate
logarithmic expressions without using a calculator.
110) log6
x - 5
x8110)
111) logb xy4
z7111)
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.
112) 4 logbx - log
by 112)
113)1
6[3ln (x + 9) - ln x - ln (x2 - 1)] 113)
Use common logarithms or natural logarithms and a calculator to evaluate to four decimal places
114) log53 114)
Solve the exponential equation. Express the solution set in terms of natural logarithms.
115) 4 2x = 4.5 115)
116) e x + 8 = 5 116)
Solve the exponential equation. Use a calculator to obtain a decimal approximation, correct to two decimal places, for the
solution.
117) e2x - 6 - 3 = 1041 117)
12
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.
118) log5(x - 1) = -3 118)
119) ln x + 3 = 3 119)
120) log6 (x + 2) - log
6 x = 2 120)
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.
121) Find sin θ.
3
2
121)
122) Find sec θ.
6
7
122)
123) Find tan θ.
8
5
123)
A point on the terminal side of angle θ is given. Find the exact value of the indicated trigonometric function of θ.
124) (18, 24) Find cos θ. 124)
125) (9, 12) Find csc θ. 125)
126) (-9, 5) Find cot θ. 126)
13
Answer KeyTestname: 2016_SUMMERASSIGNMENTREVIEW
1)1
625
2) - 1
9
3)y7
x8
4) 121x6
5) -4
6) -4x3y5
7)x2
y6
8)16x16y16
z12
9)x6
y4
10)10 3
3
11)16 + 2 3
61
12)34 - 3 2
8
13) 15 - 10
14) x3 - 27
15) 9x3 + 69x2 + 44x + 14
16) 8x5 + 48x3 + 3x2 + 18
17) 144x3 - 156x2 + 40x
18) 5x + 21
19) 2x2(7x2 - 3x + 4)
20) (x +6)(x + 7)
21) (x - 15)(x2 - 1)
22) (x - 5)(x2 - 2)
23) (x - 5)(x2 + 7)
24) (x + 6)(x + 8)
25) (x + 6)(x - 4)
26) (3x + 2)(5x + 2)
27) (2x - 3)(3x - 4)
28) (7x + 4y)(x - 2y)
29) (4x - 3y)(3x + 4y)
30) (9x + 4)(9x - 4)
31) (4x + 5y)(4x - 5y)
32) (9x2 + 4)(3x + 2)(3x - 2)
33) (2x + 1)2
34) (x - 4)(x2 + 4x + 16)
14
Answer KeyTestname: 2016_SUMMERASSIGNMENTREVIEW
35) (5x + 1)(25x2 - 5x + 1)
36) (2x - 5)(4x2 + 10x + 25)
37) 9x(x + 8)(x - 8)
38) 5(x2 + 1)(x + 1)(x - 1)
39) (3y + 16)(3y - 16)
40) x1/9(x7/9 - 1)
41) (x + 4)1/4 (1 + (x + 4)1/2)
42) (x + 3)2/5 (- x2 - 6x - 8)
43)(x + 2)
(x+ 1)6/5
44)1
5x + 3, x ≠ -
3
5, x ≠ -
2
3
45)x + 2
x + 4, x ≠ -4, -3
46)-3x - 36
(x + 4)(x - 4)
47)9x - 6
(x - 1)(x + 1)(x - 2)
48)6x - 5
x - 1
49)1
2
50)x - 8
x + 8
51)x
x - 7
52) 1 - 1
3x
53) 0; 1
4
54) 5; {3}
55) B = 3V
h
56) h = S - 2πr2
2πr
57) - 13
8, -
1
8
58)7
2, -1
59) {-2 - 13, -2 + 13}
60) 2
61)1
2
15
Answer KeyTestname: 2016_SUMMERASSIGNMENTREVIEW
62)
x-10 -5 5 10
y
10
5
-5
-10(-9, -7)
(5, 7)
(5, -5)
(5, 4)
x-10 -5 5 10
y
10
5
-5
-10(-9, -7)
(5, 7)
(5, -5)
(5, 4)
63) 7
64) 5(2x+h)
65) y - 5 = 8
9(x - 2)
66) y + 3 = 2(x + 3) or y + 7 = 2(x + 5)
67) y + 7 = - 7
2(x - 1) or y = -
7
2(x + 1)
68)
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)
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
16
Answer KeyTestname: 2016_SUMMERASSIGNMENTREVIEW
70)
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) (-∞, ∞)
72) (-∞, -6) ∪ (-6, 0) ∪ (0, 6) ∪ (6, ∞)
73) (-∞, 23]
74) (5, ∞)
75)12x
3 - 6x
76) x
77) f(x) = x6, g(x) = 9x + 2
78) f-1(x) = 3x + 6
79) f-1(x) = x2 - 2
80) minimum; - 3
2, -
41
4
81) minimum; - 1, 1
82) 11,552 ft2
83) -1, crosses the x-axis;
5, crosses the x-axis;
1, touches the x-axis and turns around
84) 0, touches the x-axis and turns around;
-11, crosses the x-axis;
9, crosses the x-axis
85) -7
86) falls to the left and falls to the right
87) rises to the left and rises to the right
88) x = 0, x = - 7, x = 6
89) x = -1, x = 1, x = - 5
90) f(1) = -3 and f(2) = 43; yes
91) 3x2 - 6x + 3
92) x4 + 2x3 + 5x2 + 10x + 20 + 45
x - 2
93) {1, -1, 6}
94) {-2, -2 + 2, -2 - 2}
95) 2 in. by 2 in. square
96) -∞
97) no vertical asymptote
98) x = 0 and x = 5
99) y = 0
17
Answer KeyTestname: 2016_SUMMERASSIGNMENTREVIEW
100) y = 5
101) no horizontal asymptote
102)
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
103)
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
104) 42 = 16
105) log64 4 =
1
3
106)1
2
107) -1
108) (-7, ∞)
109) (-∞, 3) or (3, ∞)
110) log6(x - 5) - 8 log
6x
111) logbx + 4logby - 7logbz
112) logb
x4
y
113) ln 6 (x + 9)3
x(x2 - 1)
114) 0.6826
115)ln 4.5
2 ln 4
116) {ln 5 - 8}
18
Answer KeyTestname: 2016_SUMMERASSIGNMENTREVIEW
117) 6.48
118)126
125
119) {e6 - 3}
120) {2
35}
121)3 13
13
122)85
6
123)8
5
124)3
5
125)5
4
126) - 9
5
19