AP Calculus AB Summer Assignment Name:_______________

Welcome to AP Calculus. While you are enjoying your summer, you need to take time to

complete the attached assignment. These problems are designed to help you make the transition into this challenging course as smooth as possible. The more you do over the summer, the easier it will be when school starts and the more comfortable you will feel with the pace of the class.

If you need to, you may use reference materials to assist you and refresh your memory (old notes, textbooks, online resources, etc.). I would recommend you spend some time with the Algebra & Trigonometry review available at http://tutorial.math.lamar.edu/pdf/AlgebraTrig.pdf. While the graphing calculators will be used in class for some topics, there are no calculators allowed on this packet except where noted. You should be able to do almost all the problems below without a calculator. Where you cannot use the calculator, leave your answer in the simplest terms possible. If it is obvious to me that you used a calculator to compute your answer, it will be marked incorrect. You should know that a surprising amount of AP Calculus and about 60% of the AP exam is completed with no calculator. Your summer assignment consists of problems that cover the following areas: I. Algebra II. Logarithms III. Trigonometry IV. Graphing V. Problem Solving

It is very important that you complete the summer work and have a firm understanding of the prerequisite skills as we will be building on these concepts throughout the course. This assignment is DUE ON AUGUST 21, 2017 @ 2:00. The test on Chapter P (which consists of this material) will be during the first week we meet. Any updates or schedule changes will be posted at tmorrismath.weebly.com. You should check the website at least once a week starting in August. Please make sure you complete this summer work with care. If you fail the first test, your grade in AP Calculus is in serious jeopardy! The test and packet together will account for about 25% of your first quarter grade. I look forward to our year together! Mr. Morris

Name: ________________________________________________ Show all work – no credit will be awarded for answers missing appropriate work. You may use another sheet of paper if there is insufficient room provided. No calculators except where noted! Section I: Algebra Review Identify the following statements as true or false.

_____.6_____3

3.5_____3

3.4

_____2

2.3_____

111.2_____

222.1

22 babac

ba

c

ba

b

a

b

a

hx

k

hx

k

qpqp

yxyx

••

Identify the following statements as true or false over the set of real numbers. Give a counter example for any false statement.

3 3 3 3 2

2

7. 1 _____ 8. _____ 9. 0 _____

10. _____ 11. 2 _____ 12. 0 _____

113. 0 _____ 14. _____ 15. _____

x x x x x x

x x x x x

x x x xx

16. Solve '' 1 yyxy for 'y . 17.Solve kty ln for y . 16._____________

17._____________

18. Factor: 273 y 19. Factor: )1(4)1(2 xxx

18._____________ 19._____________

Simplify each expression.

32 1

3 67

3 23 2

20. _________________ 21. ___________________

5 5 3 322. _________________ 23. _________________

x xx x x

x

x h x x h x

h h

• •

2

2

3

1 1 4

24. __________________ 25. _______________________1 1

3

x

x x xx

x x

Simplify by rationalizing the numerator. Example:

24

1

2424

44

24

242424

•

xxx

x

xx

x

x

x

x

x

x

x

9 326. _____________________ 27. ________________________

x x h x

x h

Solve each equation or inequality for x over the set of real numbers.

4 3 2 2 7 228. 2 3 2 0 _____________________ 29. _________________________

1 4

x xx x x

x x

2

4

2

3 530. 0 _____________________ 31. 9 1 _________________________

( 1)( 7)

32. 2 3 14 ___________________________ 33. 2 8 0 _________________________

xx x

x x

x x x

Section II: Trigonometry Review 34. You must have the first quadrant of the unit circle (all 6 functions and arc-functions) memorized to the point where you can answer any question within 10 seconds. This part of the test will be a slide show with 10 seconds per slide. I suggest creating flash cards and using them MANY times (this is harder to master than you think). Although I only provided three examples in this review, you have a lot to remember!! examples

3a. tan ____________ . arcsin ____________ . csc ____________

6 2 4b c

Complete each of the following using trigonometric identities and formulas.

2 2 2 235. sin cos _________ 36. 1 cot ____________ 37. 1 tan ____________x x x x

Solve each trigonometric equation for 20 x .

2338. sin ____________ 39. tan 1 ___________

2x x

2240. cos ____________ 41. 2sin sin 1 0 ____________

2 2

xx x

Section III: Logarithm Review Solve each exponential or logarithmic equation.

3

1 442. 5 125 __________ 43. 8 16 __________ 44. 81 _________x x x x

2

32

4 3 3

145. 8 __________ 46. log 32 __________ 47. log 2 __________

9

48. log 3 __________ 49. log ( 7) log (2 1) ____________

xx x

x x x

Expand each of the following using the laws of logs. 2

3

2

50. log 5 ________________________

551. ln _______________________

x

x

y

Section IV: Graphing Review This section will also be timed to 10 seconds per problem. You must be able to go both ways (draw a rough graph given the function and write the function when shown a rough graph). Sketch the following functions. Draw and label your own axes.

52. ( )f x x 53. 2( )f x x

54. 3( )f x x 55. ( )f x x

56. 𝑓(𝑥) = 𝑖𝑛𝑡 𝑥 57. 1

( )f xx

58. 2

1( )f x

x 59.

2

1( )

1f x

x

60. ( )f x x 61. ( ) xf x e

62. ( ) lnf x x 63. 2( ) 1f x x

64. ( ) sinf x x 65. ( ) cosf x x

66. ( ) cscf x x 67. ( ) secf x x

68. ( ) tanf x x 69. ( ) cotf x x

Section V: Problem Solving 70. |𝑥|2 − 6|𝑥| + 8 = 0

71. Let 𝑓(𝑥) = 4𝑥, 𝑔(𝑥) = 𝑥2 + 1, 𝑎𝑛𝑑 ℎ(𝑥) = 1

𝑥. Solve the equations.

a. f(g(x)) = g(f(x))

b. h(f(g(x))) = 1

4

72. Let f(x) = 𝑥3, g(x) = √𝑥, h(x)= 𝑥 − 4, and k(x) = 2𝑥. Express each of the following as a composite of

three of these functions.

a. 2(𝑥 − 4)3

b. √(𝑥 − 4)3

c. (2𝑥 − 8)3

d. √𝑥3 − 4

73. A car leaves Dartmouth at 11:33 A.M. traveling south at 70 km/h. At the same time, another car

is 65 km west of Dartmouth traveling east at 90 km/h. Express the distance d between the cars as

a function of the time t after the first car left Dartmouth. Show that the cars are closest to each

other at noon.

74. Find the distance from the point (9,5) to the line 4x-3y = -4.

75. Show that (-1,-1), (9,4), (20,6) and (10,1) are the vertices of a rhombus. Find the area of this

rhombus.

76. If 𝑓(𝑥) = 𝑚𝑥 + 𝑘 and ℎ ≠ 0, show that the value of 𝑓(𝑥+ℎ)−𝑓(𝑥)

ℎ does not depend on x or h.

Interpret this result graphically.

77. Find the equation of the quadratic function for the parabola with x-intercepts 2 and -1 and y-

intercept 6.

78. Sketch the graph of 𝑦 = 𝑥(1 − 𝑥)(1 + 𝑥)(2 + 𝑥)

79. A cylinder is inscribed inside a sphere with radius 4. Show that the volume of the cylinder is

𝑉(𝑥) = 2𝜋𝑥(16 − 𝑥2).

Solve for x. 80.

a. 73𝑥

49= 1 b. (8𝑥)−3 = 64 c. (5𝑥)−

1

2 = 20

81. a. 8(2 − 𝑥)3 = 27 b. (5 − 𝑥)−1

2 = 20 c. 2(1 − 4𝑥)3

2 = 16

82. a. log5 (log3 𝑥) = 0 b. log6(log4 (log2 𝑥)) = 0

83. If log8 3 = 𝑟 and log8 5 = 𝑠, express the given logarithm in terms of r and s.

a. log8 75 b. log8 225 c. log8 0.12 d. log83

64

84. When a certain drug enters the blood stream, it gradually dilutes, decreasing exponentially with

a half-life of 3 days. If the initial amount of the drug in the blood stream is A0, what will the

amount be 30 days later?

85. A certain bond investment will double your money every 8 years. If you invest $1000 now, what

will it be worth in 20 years?

Solve. 86. 𝑒2𝑥 − 5𝑒𝑥 + 6 = 0

87. 𝑒2𝑥+2 + 𝑒𝑥+2 = 6𝑒2

88. a. log3(𝑥2 − 7) = 2 b. log5 √2𝑥2 − 3 =3

2

89. What are the rectangular coordinates of the points P and Q where the line y=1/2 intersects the

unit circle? (*Calculator available)

Evaluate without using a calculator.

90. a. log2 sin (𝜋

6) b. log2 sin (

3𝜋

4)

91. log3

√243√81 √33

log2 √644

+log𝑒 𝑒−10

92. A phonograph record turns at 45 RPM (revolutions per minute).

a. Through how many degrees does it turn in a minute?

b. Through how many radians does it turn in a minute?

c. If the diameter of the record is 18cm, find the distance that a point on the rim travels in 1

minute.

93. In ∆ABC, a = 5, b = 8, and c = 7. (Given SSS) (*You may use a calculator)

a. Solve ∆ABC.

b. Find the area of the triangle. (Hint: Check out Hero’s Formula)

c. Find the length of the altitude to AC.

d. Find the radius R of the circumscribed circle.

e. Find the radius r of the inscribed circle.

94. √𝑥 + 15 + √𝑥 = 15