AP Calculus Summer Assignment Answer Key

I. Algebra A. Exponents:

1. x y z4 3

1134

B. Factor Completely:

2. (3x+1)(3x-y) 3. (2x -1)(4 ( )(4x )x2 + 2 + 1 x2 + 1 x ) x2 - 2 + 1 4. (2x )(3x ) 7 2 - 1 2 + 4 5. (5x )(3x ) x 2

1 + 6 - 4 6. x (x )(x )-3 - 2 - 1

C. Rationalize Denominators/Numerators:

7. 1 +√x - 2 8. 1

-1√x+1

D. Simplify the Rational Expression:

9.

(x+1)2x -x+12

E. Solve Algebraic Equations and Inequalities:

10. p(x) = (x-1)(x+2)(x+3) 11. p(x) = (x+1)(2x-7)(3x-1) 12. PRZ= ±1 ±5

±1±2±4 List: , , , , , , , , , 1, , - 5 - 2

5 - 45 - 1 - 2

1 - 41 5 2

5 45 2

1 41

3/2 is not one of the numbers on the list of possible rational zeros. 13. Since f(0) = -5 and f(1) = 4 and the value of the function changes from negative to positive, there has to be at least one number between 0 and 1 that the function crosses the x-axis at. 14. - , ) - , ∞) x ∈ ( ∞ - 5 ∪ ( 1 15. - , 3) x ∈ [ 5 16. - , ] 0, 5] x ∈ ( ∞ - 3

1 ∪ [ 17. - , 0) 0, 1) x ∈ ( 1 ∪ (

18. - , ) 3, ) x ∈ [ 3 - 1 ∪ [ + ∞ 19. 6, ) x ∈ ( + ∞ 20. - , 2) x ∈ ( 2

1. x - 8, 8) 2 ∈ ( 5/ - 3/

F. Solve the System:

22. x = 3 and x = -2 23. x = 3 and x = 2

II. Graphing and Functions:

A. Linear Graphs: Write the equation of the line described below

24. - x y = 3

1 - 31

25. x y = 32 - 3

17 26. x y = 5

3 - 57

B. Conic Sections: Write the equation in standard form and identify the conic

28. Point (2, -3)

C. Functions: Find the domain and range of the following

29. D: OR D: = x / 2 - , 2) 2, ) x ∈ ( ∞ ∪ ( + ∞ 30. D: OR D: x > 3 3, ) x ∈ ( + ∞ 31. D: OR D: RI - , ) x ∈ ( ∞ + ∞ 32. D: OR D: x ≥ 2

3 , ) x ∈ [ 23 + ∞

33. D: OR D: RI - , ) x ∈ ( ∞ + ∞ 34. D: OR D: =- , 1 and x - x / 1 > 1 - , 1) 1, ) x ∈ ( 1 ∪ ( + ∞ 35. R: OR R: - 5 ≤ x ≤ 3 - , 3] y ∈ [ 5 36. (x) x g-1 = 4 - 3 37. (x) h-1 = ex 38. (x) w-1 = x2 + 4 39. (g(x)) 6x 2x f = 1 2 - 1 - 2 40. (g(f (1))) n5h = l 41. No, because it’s a parabola which it fails Horizontal Line Test. It is not “one-to-one”

Function. 42. ( )(x) x f ° g = - 2 43. ( )(x) f ° g ° h = - 8 44. - , 2] x ∈ [ 2

D. Basic Shapes of Curves.

45.

46.

47.

48.

49.

50.

51.

52.

53.

E. Even, Odd, Tests for Symmetry.

54. Odd and f (- ) x x = - x3 - 3 f (x) x - = - x3 - 3 55. Even and f (x) x = x4 - 6 2 + 3 f (- ) x x = x4 - 6 2 + 3 56. Odd and f (- ) x = x2

-x +x3 f (x) - = x2-x +x3

57. Odd and f (- ) sin(- x) - in2x x = 2 = s f (x) in2x - = - s 58. Neither and and f (x) = x2 + x f (x) - = - x2 - x (- ) f x = x2 - x 59. Odd and f (- ) (x ) x = - x 2 - 1 f (x) (x ) - = - x 2 - 1 60. Neither and and f (x) = x2

1+ x| | f (x) - = x2-1- x| | (- ) f x = x2

1+ -x| | 61. Even in both cases. 62. Y-axis symmetry 63. Origin symmetry 64. Y-axis symmetry 65. X-axis symmetry 66. Y-axis

IV. Graphing and Functions:

A. Simplify expressions:

67. -2 68. -1 69. 3/2 70. -⅕ 71. 45 72. 1 73. 0 74. 2

B. Simplify expressions:

75. x = -1, -6 76. ± x = 1

10 77. 15 x = - 1 + log3

V. Trigonometry:

A. Unit Circle:

78.

a. D: OR D: RI - , ) x ∈ ( ∞ + ∞ b. D: OR D: RI - , ) x ∈ ( ∞ + ∞ c. D: =± x / 2

nπ

B. Identities:

79. = = = = = 1cscx tan x sinx2tan x csc x - 12 2

tan x 1sinx

21sinx

- 1sin x2

cos x21

sin x 2

sin x2cos x2

-11cos x2

sin x2cos x2

cos21-cos x2

sin x2cos x2

sin x2cos x2

80. xsin2 81. 1 82. Left Side = = 11 x)(1 x) x sec x x ( - sin2 + tan2 = cos2 2 = cos2 1

cos x2

C. Solve the Equations:

83. x = - π 84. ,x = 6

π , ,3π 6

7π 34π

85. - x = 23π

D. Inverse Trig Functions:

86. 2π

87. - 4π

88. 6π

89. 21

90. [- , 1] D arc sin x : 1 [- , ] R arc cos x : 2

π 2π

[- , 1] D arc cos x : 1

[- , ] R arc cos x : 2π 2

π

D (- , ∞) arc tan x : ∞ (- , ) R arc cos x : 2

π 2π

E. Right Triangle Trig:

91. x = 7.66 92. x = 14.62 93.

a. 76 feet b. AH = 196 feet and HC = 97 feet c. 8.3 ft/sec d. 34.6 ft/sec

F. Graphs:

94. IaI =2 Period = π

Horizontal shift: none Vertical shift: none 95. IaI = π Period = 4 Horizontal shift: ← 2 Vertical shift: none

VI. Functions and Models:

101. a. 3 b. -2, 2 c. -2

d. 0, ) ( + ∞ e. D: IR and R: - , 3) ( ∞ f. D: IR and R: 1 2, ) ( / + ∞ 102. a.

b.

1994 600 → 1996 1500 →

103. (2) 2f = 1 (- ) 6 f 2 = 1 (a) a f = 3 2 - a + 2

(- ) a f a = 3 2 + a + 2 (a ) a af + 1 = 3 2 + 5 + 4 f (a) a a 2 = 6 2 - 2 + 4

(a ) a f 2 = 3 4 - a 2 + 2 ( (a)) 3a ) f 2 = ( 2 - a + 2 2 (a ) a ah h f + h = 3 2 + 6 + 3 2 - a - h + 2

104.

a. D: = 3 x / 1/ b. D: 0, ] x ∈ [ 4

105. y = 1 -√- x

106.

- 0w A = w 2 + 1

108.

a. cy = 61 + 6

307 b. changes 6 chirps a minute per 1 degree m = 6

1 c. 6.2 T = 6

457 ≈ 7

109.

a. +15 .434dP = 0 = b. d = 195.6 ft 110.

a. Root b. Root c. Polynomial d. Rational

111. a. h , parabola b. g, odd degree polynomial, by leading coefficient test: left arm down, right arm up c. f, even degree polynomial, by leading coefficient test: both arms up 112.

a. (x) (x)h = f + 3 b. (x) (x) h = f - 3 c. (x) (x ) h = f - 3 d. (x) (x )h = f + 3 e. (x) - (x) h = f

f. (x) (- ) h = f x g. (x) f (x)h = 3 h. (x) f (x)h = 3

1 114.

115.

116. x f + g = x 3 + 5 2 - 1 RD : I f - g = x 3 - x 2 + 1 RD : I

g x x x f = 3 5 + 5 4 - x 3 - 2 2 RD : I fg =

3x -12x +2x 3 2

D =± : x / 3√3

117. in(1 ) f ° g = s -√x RD : I g ° f = 1 -√sin x D 0 nπ, π nπ] : x ∈ [ ± 2 ± 2 in(sinx) f ° f = s D R : I g ° g = 1 - √1 -√x 0, 1] D : x ∈ [ 118. (x) f = x 10 (x) g = x 2 + 1

_____________________________________________________________________

Part A

1. a. nt none x - i :

nt 1 y - i : RD : I

0, ) R : y ∈ ( + ∞

b. nt none x - i : nt 2 y - i : RD : I

1, ) R : y ∈ ( + ∞

c. nt none x - i : nt 4 y - i : RD : I

0, ) R : y ∈ ( + ∞

d. nt none x - i : nt 625 y - i : RD : I

0, ) R : y ∈ ( + ∞

e. nt none x - i : nt 1 2 y - i : / RD : I

0, ) R : y ∈ ( + ∞

f. nt 1 x - i : nt none y - i :

0, ) D : x ∈ ( + ∞ RR : I

g. nt 1 x - i : nt none y - i :

0, ) D : x ∈ ( + ∞ RR : I

h. nt x - i : - 1 nt y - i : - 1 - , ) D : x ∈ ( 2 + ∞

RR : I

i. nt 10 x - i : nt none y - i :

1, ) D : x ∈ ( + ∞ RR : I

j. nt 1 8 x - i : / nt none y - i :

0, ) D : x ∈ ( + ∞ RR : I

2. a. (not a solution) , xx = 1 = 6 b. - (not a solution) , x - 4 x = 1 = 3/ c. x = 4 d. (not a solution)x = 1 e. x = 1 f. og 3x = l 1 2 / g. - , x x = 1 = 2

3.

a. 8x < log4 b. - x < 2 c. - x > 1 d. - 2 x > 3/ e. - x < 3 f. x < 2 g. - 1 16 x > 3 / h. x < 3 1

100 i. x < e -3 j. 9 x < 1/ k. 1, 2) x ∈ ( l. - , 1] 2, ) x ∈ ( ∞ ∪ [ + ∞ m. 0, 1) x ∈ ( n. 0, 2) 4, ) x ∈ ( ∪ ( + ∞

4.

a. x 1, ) D : ∈ ( + ∞ b. R D : I c. x - , 1 2) D : ∈ ( ∞ / d. x 1, ) D : ∈ ( + ∞ e. = , D : x / 0 ± 3

2 f. x - ,- 2) 2, ) D : ∈ ( ∞ ∪ ( + ∞ g. x - , 1) D : ∈ ( 6 h. x - , ) - , 0] 1, ) D : ∈ ( ∞ - 1 ∪ ( 1 ∪ [ + ∞ i. x - , 3) D : ∈ ( ∞

5.

a. a a8 3 + 8

b. 7a 2a - 2 3 - 1 2 c. 4a 44a 24a 9 - 6 3 + 1 2 - 1 + 3

6.

a. √x + 2 + 2 b. (x ) x 2 - 1 c. x+4

x(3x-1) d. 3

4a 2

e. -a2x(2x+h)

f. x h6 + 3 g. X

7.

a. T b. F c. F d. F e. F

8.

a. x = 0 b. y = -1

9.

a. x )(x )(x ) ( - 1 - 2 + 2 b. - x )(x )(x )(x ) ( - 2 - 2 + 2 2 + 4

10.

a. 1 b. 2

-√2 c. 2

-√2 - 4π

d. 6.57 degrees7

11.

a. x = 2π

b. , ,x = 23π 6

π 65π

12.

a. Even b. Odd c. Neither

13.

a. - x y = 49 + 4

7 b. - x y = 2

3 + 437

c. x y = 32 - 4

5

14.

x- int (5/4, 0) y- int (0, -5)

15.

a. (g(x))f = 1

x -32

b. (f (x))g =(x-1) 2

-2x +4x-12

c. (f (x)) f = x-1-x+2

d. (g(x)) x g = x 4 - 4 2 + 2

16.

a. A wl wh lhS = 2 + 2 + 2

b. A 2wl h h S = 1/ + w + l + √l 2 + w 2 c. A xhS = x 2 + 4

17.

a. D : x ≥ 2 R : y ≥ 1

b. = D : x / ± 2 = , 1 4 (hole at x 2) R : y / 0 / = 1/

c. RD : I - R : y ≥ 2

d. - , 2] D : x ∈ [ 2 R 0, 2] : y ∈ [

e. RD : I R R : I

f. RD : I R R : I

18.

a. - x 40x A = 2 2 + 2 b. 0 x = 6 ± 5√34 c. 200A = 7

Part B

1.

3.

a. x a 2 2 - 2 2 b. -(x-2)

(x-3)(x+1)

5. x = 2

8. No, parabolas are not “one-to-one” functions, unless you restrict the domain.

9. No, parabolas are not “one-to-one” functions, unless you restrict the domain.

a. Yes, the domain was restricted, so the parabola passess the horizontal line test.

b. No, the restriction of the domain still did not allow the parabola to have an inverse.

10.

a.

b. (x) x f -1 = 3 - log2

c.

and (f (x)) f -1 = x (f (x)) f -1 = x

12.

- , 6] D : x ∈ [ 2 - , ] R : y ∈ [ 4 4

13.

a. - , 0) 3, 5) x ∈ ( ∞ ∪ ( b. 0, 3) 5, ) x ∈ ( ∪ ( + ∞

14. m = 2 and b = 2

15. y =

e sinx3e -2sinx

16. x D : ∈ - ,( 2√5 2

√5)

17. 0 π .72 A = 1 - 2 ≈ 3

19.

a. 0, 3) x ∈ ( b. 3, 5) x ∈ (

20. x ∈ (3, 9 )

21.

a.

b.

c.

d.

e.

22.

a.

[-5, 5] D f (x) : x ∈ [0, 5] R f (x) : x ∈

b.

[-5, 5] D g(x) : x ∈ [0, 12] R f (x) : x ∈

23.

25.