East Mecklenburg High School

Summer Packet for AP Calculus Students (AB & BC)

Teacher: Mr. Rego

Email: davidj.rego@cms.k12.nc.us

Directions: It is important that you carefully work all of the problems in this packet for two reasons. The first is to

keep you fresh on pre-calculus topics that you will need from day one in calculus class. The second is to expose any

weak content areas that you may have. For those problems that you find difficult, be sure to consult online resources

such as Khan Academy and PatrickJMT Math. You can also contact me throughout the summer if you need some

help. I may not be checking my email every day, but I will eventually see your email and get back to you. If there is

enough interest, we can set up our Google Classroom during the summer as an added resource and means of

communication. Finally, you can help each other figure out the various problems. When we get back to school, we

will take a day or two to go over problems that seemed to pose difficulty to all. You will also have a test on the

material once questions are answered.

If you can, it would be best to print this packet in black ink.

Please take this packet seriously. Please check with your friends that you know have signed up to take calculus and

make sure they have this packet. By the way, the answers to the odd problems are at the back of the packet.

Fine Print Disclosure: There is no guarantee that all answers are correct. I might have made a mistake or a typo.

Thank you for signing up for AP Calculus. It is my favorite class to teach. I look forward to the journey with all of you.

Have a great summer!

Mr. Rego

Helpful websites:

http://www.khanacademy.org/#browse

http://patrickjmt.com/

http://www.analyzemath.com/

AP CALCULUS AB SUMMER PACKET Name: _________________________

Topic 1: Fractional & Negative Exponents – simplify using only positive exponents.

1. −3𝑥−3 2. −5 (3

2) (4 − 9𝑥)

−1

2 (−9)

3. (16𝑥2𝑦)3

4⁄ 4. √4𝑥−16

√(𝑥−4)34

5. 1

2(2𝑥+5)

−32⁄

3

2

6. (1

𝑥−2+

2

𝑥−1𝑦−1+

1

𝑦−2)−1

2⁄

Topic 2: Domain – find the domain of the following functions:

7. 𝑦 =3𝑥−2

4𝑥+1 8. 𝑦 =

𝑥2−5𝑥−6

𝑥2−3𝑥−18 9. 𝑦 =

22−𝑥

𝑥

10. 𝑦 = √𝑥 − 3 − √𝑥 + 3 11. 𝑦 = √𝑥2 − 5𝑥 − 14

12. 𝑦 =√2𝑥+9

2𝑥−9 13. 𝑦 = ln(2𝑥 − 12) 14. 𝑦 =

𝑥

cos 𝑥

Topic 3: Write the following absolute value expression as a piecewise expression:

15. 𝑦 = |2𝑥 − 4| 16. 𝑦 = |𝑥2 + 𝑥 − 12|

Topic 4: Solve the following absolute value inequalities:

17. |𝑥 − 3| > 12 18. 2|𝑥 + 4| ≤ 10

19. |3𝑥 − 4| > −2 20. |𝑥 − 6| < −8

Topic 5: Solve the quadratic inequalities by factoring and using sign charts:

21. 𝑥2 − 16 > 0 22. 𝑥2 − 3𝑥 < 10

23. 𝑥3 + 4𝑥2 − 𝑥 ≥ 4 24. 2 sin2 𝑥 ≥ sin 𝑥 , 0 ≤ 𝑥 < 2𝜋

Topic 6: Special Factorization – factor expression completely:

25. 𝑥3 + 8 26. 27𝑥3 − 125𝑦3 27. 𝑥4 + 11𝑥2 − 80

28. 2𝑥2 − 20𝑥𝑦 + 50𝑦2 29. 𝑥2 + 12𝑥 + 36 − 9𝑦2

30. 𝑥3 − 𝑥𝑦2 + 𝑥2𝑦 − 𝑦3 31. (𝑥 − 3)2(2𝑥 + 1)3 + (𝑥 − 3)3(2𝑥 + 1)2

Topic 7: Transformation of functions

For #32-37, given the function 𝒚 = 𝒇(𝒙), describe the transformations applied in each part.

32. 𝑦 = 𝑓(𝑥) − 4 33. 𝑦 = 𝑓(𝑥 − 4) 34. 𝑦 = −𝑓(𝑥 + 2)

35. 𝑦 = 5𝑓(𝑥) + 3 36. 𝑦 = 𝑓(2𝑥) 37. 𝑦 = |𝑓(𝑥)|

For #38-43, the graph of 𝒚 = 𝒈(𝒙) is given. On each coordinate plane, sketch the graph of the indicated

transformation.

38. 𝑦 = 2𝑔(𝑥) 39. 𝑦 = −𝑔(𝑥) 40. 𝑦 = 𝑔(𝑥 − 1)

41. 𝑦 = 𝑔(𝑥 + 2) 42. 𝑦 = |𝑔(𝑥)| 43. 𝑦 = 𝑔(|𝑥|)

Topic 8: Use the Rational Roots Theorem (p/q) and synthetic division to factor each polynomial. Then solve 𝑷(𝒙) =

𝟎.

44. 𝑃(𝑥) = 𝑥3 + 5𝑥2 − 2𝑥 − 24

45. 𝑃(𝑥) = 𝑥3 + 2𝑥2 − 19𝑥 − 20

46. 𝑃(𝑥) = 2𝑥4 − 4𝑥3 − 5𝑥2 − 2𝑥 − 3

Topic 9: Even & Odd Functions – show work to determine if the function is even, odd, or neither.

47. 𝑓(𝑥) = 2𝑥2 − 7 48. 𝑔(𝑥) = −4𝑥3 − 2𝑥 49. ℎ(𝑥) = 4𝑥2 − 4𝑥 + 4

50. 𝐹(𝑥) = 𝑥 −1

𝑥 51. 𝐺(𝑥) = |𝑥| − 𝑥2 + 1 52. 𝐻(𝑥) = 𝑒𝑥 −

1

𝑒𝑥

Topic 10: Solve each quadratic equation by factoring or the quadratic formula (find all Complex solutions).

53. 7𝑥2 − 3𝑥 = 0 54. 𝑥2 + 6𝑥 + 4 = 0

55. 2𝑥2 − 3𝑥 + 3 = 0 56. 𝑥4 − 9𝑥2 + 8 = 0

57. 𝑥 − 10√𝑥 + 9 = 0 58. 1

𝑥2−

1

𝑥= 6

Topic 11: Asymptotes: For each function, find the equations of both the vertical and horizontal asymptotes if they

exist.

59. 𝑦 =𝑥

𝑥−3 60. 𝑦 =

𝑥+4

𝑥2−1 61.

𝑥+4

𝑥2+1

62. 𝑦 =𝑥2−2𝑥+1

𝑥2−3𝑥−4 63. 𝑦 =

𝑥2−9

𝑥3+3𝑥2−18𝑥

64. 𝑦 =2𝑥3

𝑥3−1 65. 𝑦 =

√𝑥

2𝑥2−10

Topic 12: Complex Fractions – simplify each fraction.

66.

3

𝑥−

4

𝑦4

𝑥−

3

𝑦

67. 1−

2

3𝑥

𝑥−4

9𝑥

68.

𝑥2−𝑦2

𝑥𝑦𝑥+𝑦

𝑦

69. 𝑥−3−𝑥

𝑥−2−1 70.

𝑥

1−𝑥+

1+𝑥

𝑥1−𝑥

𝑥+

𝑥

1+𝑥

Topic 13: Composition of Functions – find the following given the three functions below.

𝑓(𝑥) = 𝑥2 𝑔(𝑥) = 2𝑥 − 1 ℎ(𝑥) = 2𝑥

71. 𝑓(𝑔(2)) 72. 𝑓(ℎ(−1)) 73. 𝑔 (𝑓 (ℎ (1

2)))

74. 𝑓(𝑔(𝑥)) 75. 𝑔(𝑔(𝑥)) 76. 𝑓(ℎ(𝑥))

Topic 14: Rational Equations – solve each equation.

77. 𝑥−5

𝑥+1=

3

5 78.

60

𝑥−

60

𝑥−5=

2

𝑥 79.

𝑥

𝑥−2+

2𝑥

4−𝑥2=

5

𝑥+2

Topic 15: Trigonometry – solve or simplify each group of problems.

For #80-81, point P is on the terminal side of angle 𝜃. Find all six trig functions of 𝜃. Draw a graph.

80. 𝑃(−2, 4) 81. 𝑃(√5, −2)

82. If cos 𝜃 =−5

13 , 𝜃 in quadrant II, find sin 𝜃 and tan 𝜃. 83. If cot 𝜃 =

5

13 , 𝜃 in quadrant III, find sin 𝜃 and cos 𝜃.

For #84-86, find the exact value without using a calculator.

84. sin2 5𝜋

4− cos2 5𝜋

3 85. (6 sec 𝜋 − 4 cot

𝜋

2)

2

86. (4 cos𝜋

6− 6 sin

2𝜋

3)

−2

For #87-88, solve each triangle (3 decimal accuracy).

87. 𝐴 = 𝑎 = 21.7 88. 𝐴 = 𝑎 = 6 𝑓𝑡

𝐵 = 16° 𝑏 = 𝐵 = 𝑏 =

𝐶 = 90° 𝑐 = 𝐶 = 90° 𝑐 = 95"

Topic 16: Trigonometric Equations – solve each equation algebraically on the interval [𝟎, 𝟐𝝅). Use exact answers.

89. 2 cos 𝑥 + √3 = 0 90. cos2 𝑥 = cos 𝑥

91. 4 sin2 𝑥 = 1 92. cos2 𝑥 + 2 cos 𝑥 = 3

93. 2 sin 𝑥 cos 𝑥 + sin 𝑥 = 0 94. sin2 𝑥 − cos2 𝑥 = 0

Topic 17: Logarithms – simplify each expression.

95. log2 5 + log2(𝑥2 − 1) − log2(𝑥 − 1) 96. 2 log4 9 − log2 3

97. Rewrite log5(𝑥 + 3) using natural logarithms 98. 32 log3 5

Topic 18: Solve each equation for the indicated variable.

99. 𝑥

𝑎+

𝑦

𝑏+

𝑧

𝑐= 1, for 𝑎 100.

𝑥

2𝜋+

1−𝑥

2= 0, for 𝑥

101. 𝐴 = 2𝜋𝑟2 + 2𝜋𝑟ℎ , for 𝑟 > 0.

Topic 19: Equations of lines. Determine the equation of each line.

102. The line through (−1, 3) 𝑎𝑛𝑑 (2, −4)

103. The line through (−1, 2) and perpendicular to the line 2𝑥 − 3𝑦 = 0.

104. The line through (2, 3) and the midpoint of the line segment from (−1, 4) to (3, 2).

Topic 20: Inverse Functions – Find the inverse of each function.

105. 𝑦 = 2𝑥 + 3 106. 𝑓(𝑥) =𝑥+2

5𝑥−1

107. 𝑔(𝑥) = 𝑒𝑥 − 3

Topic 21: The Difference Quotient

Simplify 𝑓(𝑥+ℎ)−𝑓(𝑥)

ℎ , for each of the following:

108. 𝑓(𝑥) = 2𝑥 + 3 109. 𝑓(𝑥) = 𝑥2 110. 𝑓(𝑥) =1

𝑥

Topic 22: Area

111. Find the ratio of the area inside the square but outside the circle to the area of the square.

112. Find a formula for the perimeter of a window of the shape in the picture below.

113. A water tank has the shape of a cone (like an ice cream cone without the ice cream). The tank is 10m high and has

a radius of 3 m at the top. If the water is 5m deep (in the middle), what is the surface area of the top of the water?

Topic 23: Miscellaneous

114. Use long division to divide the following: 𝑥3−7𝑥2−8

𝑥2−3

115. Use a graphing calculator to solve the following: 𝑒2𝑥 = −3𝑥2 + 2

116. Sketch a graph of the piecewise function 𝑓(𝑥) = {𝑥2 − 5, 𝑥 < −10 , 𝑥 = −1

2𝑥 , 𝑥 > −1

Topic 24: Trigonometric Identities. You should know the following identities.

a. sin(−𝑥) = − sin 𝑥

b. cos(−𝑥) = cos 𝑥

c. tan(−𝑥) = − tan 𝑥

d. sin2 𝑥 + cos2 𝑥 = 1

e. tan2 𝑥 + 1 = sec2 𝑥

f. 1 + cot2 𝑥 = csc2 𝑥

g. sin 2𝑥 = 2 sin 𝑥 cos 𝑥

h. cos 2𝑥 = cos2 𝑥 − sin2 𝑥

i. cos 2𝑥 = 2 cos2 𝑥 − 1

j. cos 2𝑥 = 1 − 2sin2 𝑥

k. tan 2𝑥 =2 tan 𝑥

1−tan2 𝑥

l. sin𝑥

2= ±√

1−cos 𝑥

2

m. cos𝑥

2= ±√

1+cos 𝑥

2

n. sin(𝑥 ± 𝑦) = sin 𝑥 cos 𝑦 ± cos 𝑥 sin 𝑦

o. cos(𝑥 ± 𝑦) = cos 𝑥 cos 𝑦 ∓ sin 𝑥 sin 𝑦

Topic 25: Function Families. For each function given on the following pages, graph the function and fill in the blanks on

the chart.

BASIC FUNCTION: 𝑦 = 𝑥

Domain:

Range:

Continuity:

Increasing/decreasing:

Symmetry:

Bounded?

Local extrema?

Horizontal asymptotes?

Vertical asymptotes?

End Behavior?

BASIC FUNCTION: 𝑦 = 𝑥2

Domain:

Range:

Continuity:

Increasing/decreasing:

Symmetry:

Bounded?

Local extrema?

Horizontal asymptotes?

Vertical asymptotes?

End Behavior?

BASIC FUNCTION: 𝑦 = √𝑥

Domain:

Range:

Continuity:

Increasing/decreasing:

Symmetry:

Bounded?

Local extrema?

Horizontal asymptotes?

Vertical asymptotes?

End Behavior?

BASIC FUNCTION: 𝑦 = 𝑥3

Domain:

Range:

Continuity:

Increasing/decreasing:

Symmetry:

Bounded?

Local extrema?

Horizontal asymptotes?

Vertical asymptotes?

End Behavior?

BASIC FUNCTION: 𝑦 = √𝑥3

Domain:

Range:

Continuity:

Increasing/decreasing:

Symmetry:

Bounded?

Local extrema?

Horizontal asymptotes?

Vertical asymptotes?

End Behavior?

BASIC FUNCTION: 𝑦 = 𝑥2

3⁄

Domain:

Range:

Continuity:

Increasing/decreasing:

Symmetry:

Bounded?

Local extrema?

Horizontal asymptotes?

Vertical asymptotes?

End Behavior?

BASIC FUNCTION: 𝑦 =1

𝑥

Domain:

Range:

Continuity:

Increasing/decreasing:

Symmetry:

Bounded?

Local extrema?

Horizontal asymptotes?

Vertical asymptotes?

End Behavior?

BASIC FUNCTION: 𝑦 = 𝑒𝑥

Domain:

Range:

Continuity:

Increasing/decreasing:

Symmetry:

Bounded?

Local extrema?

Horizontal asymptotes?

Vertical asymptotes?

End Behavior?

BASIC FUNCTION: 𝑦 = ln 𝑥

Domain:

Range:

Continuity:

Increasing/decreasing:

Symmetry:

Bounded?

Local extrema?

Horizontal asymptotes?

Vertical asymptotes?

End Behavior?

BASIC FUNCTION: 𝑦 = sin 𝑥

Domain:

Range:

Continuity:

Increasing/decreasing:

Symmetry:

Bounded?

Local extrema?

Horizontal asymptotes?

Vertical asymptotes?

End Behavior?

BASIC FUNCTION: 𝑦 = cos 𝑥

Domain:

Range:

Continuity:

Increasing/decreasing:

Symmetry:

Bounded?

Local extrema?

Horizontal asymptotes?

Vertical asymptotes?

End Behavior?

BASIC FUNCTION: 𝑦 = tan 𝑥

Domain:

Range:

Continuity:

Increasing/decreasing:

Symmetry:

Bounded?

Local extrema?

Horizontal asymptotes?

Vertical asymptotes?

End Behavior?

Solutions to Odd Problems

1. −3

𝑥3 3. 8𝑥3

2⁄ 𝑦3

4⁄ 5. 1

3(2𝑥+5)3

2⁄ 7. (−∞,

−1

4 ) ∪ (

−1

4, ∞)

9. (−∞, 0) ∪ (0, ∞) 11. (−∞, −2] ∪ [7, ∞) 13. (6, ∞)

15. 𝑦 = {4 − 2𝑥, 𝑖𝑓 𝑥 < 22𝑥 − 4, 𝑖𝑓 𝑥 ≥ 2

17. (−∞, −9) ∪ (15, ∞) 19. (−∞, ∞)

21. (−∞, −4) ∪ (4, ∞) 23. [−4, −1] ∪ [1, ∞) 25. (𝑥 + 2)(𝑥2 − 2𝑥 + 4)

27. (𝑥2 + 16)(𝑥2 − 5) 29. (𝑥 + 6 + 3𝑦)(𝑥 + 6 − 3𝑦) 31. (𝑥 − 3)2(2𝑥 + 1)2(3𝑥 − 2)

33. Translate right 4 units 35. Vertical stretch by factor of 5; translate up 3 units.

37. Any points on 𝑓(𝑥) that lie below the x-axis will be reflected across the x-axis.

39. 41. 43.

45. (𝑥 + 1)(𝑥 + 5)(𝑥 − 4) 47. Even 49. Neither 51. Even

53. 𝑥 = 0 𝑜𝑟 3

7 55. 𝑥 =

3±𝑖√15

4 57. 𝑥 = 1 𝑜𝑟 81

59. VA: 𝑥 = 3; HA: 𝑦 = 1 61. VA: none; HA: 𝑦 = 0 63. VA: 𝑥 = 0, 𝑥 = −6; HA: 𝑦 = 0

65. VA: 𝑥 = √5; HA: 𝑦 = 0 67. 3

(3𝑥+2) 69.

1+𝑥2

𝑥 71. 9

73. 3 75. 4𝑥 − 3 77. 𝑥 = 14 79. 5±𝑖√15

2

81. sin 𝜃 =−2

3 ; csc 𝜃 =

−3

2 ; cos 𝜃 =

√5

3 ; sec 𝜃 =

3

√5 ; tan 𝜃 =

−2

√5 ; cot 𝜃 =

−√5

2

83. sin 𝜃 =−13

√194 ; cos 𝜃 =

−5

√194 85. 36 87. 𝐴 = 74° ; 𝑏 = 6.222 ; 𝑐 = 22.574

89. 𝑥 =5𝜋

6 𝑜𝑟

7𝜋

6 91. 𝑥 =

𝜋

6 ,

5𝜋

6 ,

7𝜋

6 ,

11𝜋

6 93. 𝑥 = 0, 𝜋 ,

2𝜋

3 ,

4𝜋

3

95. log2(5𝑥 + 5) 97. ln(𝑥+3)

ln 5 99. 𝑎 =

𝑏𝑐𝑥

𝑏𝑐−𝑐𝑦−𝑏𝑧 101.

−𝜋ℎ+√𝜋2ℎ2+2𝜋𝐴

2𝜋

103. 𝑦 =−3

2𝑥 +

1

2 105. 𝑦 =

𝑥−3

2 107. 𝑔−1(𝑥) = ln(𝑥 + 3) 109. 2𝑥 + ℎ

111. 4−𝜋

4 113.

9

4𝜋 𝑚2 115. 𝑥 = −0.7716 𝑜𝑟 0.2827