AP Calculus AB Summer Packet 2015 - fallriverschools.org Calculus AB Summer Assignment 2015.pdf · AP Calculus AB Summer Packet 2015 ... not rely on a calculator to do all of the

B.M.C. Durfee High School Mathematics Department Page 1

AP CALCULUS AB

Summer Assignment 2015

AP Calculus AB Summer Packet 2015


Welcome to AP Calculus Part AB. This will be the toughest class yet in your mathematical careers, but the benefit you will receive by having this experience in high school is immense. Because of the unique nature of this class, it is very important that you are ready to start working on the first day. We will NOT spending time to review the material in this packet. The reason for this is to make sure we have adequate time complete all of the required material for this course well before the AP exam. In AP Calculus, there are extremely high expectations of students taking the course. We expect a certain level of independence to be demonstrated by anyone taking AP Calculus. Your first opportunity to demonstrate your capabilities and resourcefulness to us is through this summer work packet which will help you maintain/improve your skills. This packet is a requirement for those entering AP Calculus AB and is due on the first day of class and will be counted as an assessment grade

There are certain math skills that have been taught to you over the previous years that are necessary to be successful in calculus. If you do not fully understand the topics in this packet, it is possible that you will get calculus problems wrong in the future not because you do not understand the calculus concept, but because you do not understand the algebra or trigonometry behind it. Don’t fake your way through any of these problems because you will need to understand the underlying concepts to be successful in this course. You may work with someone else while you do this, but copying will not be tolerated. Also, don’t wait until the last minute to do everything in the packet because you may run out of time and rush through them. Likewise, do not do all the problems right at the beginning of summer and completely forget how to do all of them by the time school starts again. For this packet you must show all of your work for the multiple choice questions on a separate sheet of paper and fill in your answers on the attached answer sheet. Also, do not rely on a calculator to do all of the work for you. Half of the AP exam does not allow any calculator at all. This packet will be worth a test grade. Enjoy your summer and don’t forget about the packet. September will be here before you know it! If you lose your packet, you will be able to access the packets on-line at the school website or a copy can be picked up in the Registrar’s office. Best of luck with the assignment! I look forward to seeing you in September! Mrs. Brogan



Directions: Identify the choice that best completes the statement or answers the question.

Write your answers on the answer sheet that has been provided in this packet.

1. Simplify:

a. c.

b.

d.

2. Simplify:

a.

c.

b.

d.

3. Simplify: +

a.

c.

b.

d.

4. Simplify:

a.

c.

b.

d.



5. Simplify: Assume that no denominator is equal to 0.

a.

c.

b.

d.

6. Use your graphing calculator to determine consecutive values of x between which each real zero is located.

a. There is a zero between x = 1 and x = 2. b. There are zeros between x = 2 and x = 3, x = 1 and x = 0, x = –2 and x = –3. c. There are zeros between x = 1 and x = 2, x = –1 and x = –2. d. There is a zero between x = –1 and x = –2.

7. Use your graphing calculator to find the x-coordinares at which the relative maxima and relative minima occur for the function.

a. The relative maximum is at , and the relative minimum is at . b. The relative maximum is at , and the relative minimum is at . c. The relative maximum is at , and the relative minimum is at . d. The relative maximum is at , and the relative minimum is at .

8. Find the inverse of the given function:

a.

c.

b.

d.

9. Simplify .

a.

c.

b. d.



10. Simplify .

a.

c.

b.

d.

11. Find the domain of the function:

a. All real numbers

b. All real numbers

c. All real numbers

d. All real numbers

12. Find the vertical asymptote(s), if any, for

.

[A]

[B]

[C]

[D] No vertical asymptotes

13. Find the horizontal asymptotes, if any, of the graph of

a.

b.

c.

d. No horizontal asymptotes

x 3 1,

x – , –5 4

x – , –3 1

x 5 4,

x x 7 2,

x x x 2 3 7, ,

x x 2 3,

y 2

3

y 8

y 0



14. Evaluate the expression: .

a. e

c.

b. d. 2

15. Evaluate the expression .

a.

c.

b. 14 d. e

16. Solve the given equation. Round to the nearest ten-thousandth, if necessary.

a. 1.4

c. 0.5666

b. 0.6592 d. 0.1682 17. Solve the exponential equation algebraically:

a. b.

c. d.

18. Find the value of x.

a. 19.049 b. 1.083

c. 17.333 d. 0.367

5 67 970 04e x– .

– .0557

– .44 794

– .150000

– .19 454



19. What is written as a single logarithm? a. b.

c.

d.

20. What is the solution of ? a. b.

c. d.

21. Simplify:

a. 2

b.

c.

d.

22. Factor the expression and use the fundamental identities to simplify.

a.

b.

c. 1

d.

3

3

2

3

3

cos sec cos2 2 2x x x

cos cot2 2x x

cos2x sin2x



23. Find the expression that completes the identity:

a.

b. 0

c.

d.

24. Which equation represents a line through (-1, 1) with a slope of

?

a.

b.

c.

d.

25. Which of the following equations is shown in the graph below?

a.

b.

c.

d.

1 cos

sin

sin

1 cos

u

u

u

u

2csc u 2sin u

2 cos u



26. Which expression equals

a.

b.

c.

d.

27. What is

in simplest form?

a.

b.

c.

d.


Factoring Short Answer Section: You must show necessary work to receive credit.

28. Factor completely:




32. Factor completely and simplify:

Trigonometry Short Answer Section: You must show necessary work to receive credit.

33 If si θ < 0 a d ta θ > 0, the i which quadra t does θ ie?

34. Determine the exact value of csc

.

35. Determine the exact value of si

.

36 Determi e two va ues of θ betwee 0˚ a d 0˚ that satisfy the equatio 3

sin2

37 Determi e two va ues of θ betwee 0 a d 2 radians that satisfy the equation tan 3



38. Solve the fo owi g trigo ometric equatio s o the i terva [0, π

a. cos

b. si cos si 0

c. si

Misc Topics:

39. Write the absolute value expression as a piecewise expression:

40. If , describe in words what the following would do to the graph of :

a.

b.

c. –

d.

e.



41. Solve each equation:

a. 0

b.

c.

d. 0 0

42. Find the vertical and horizontal asymptotes of

, if they exist.

43. Simplify the following complex fraction:



Use the graph below to answer questions 46-61.

44.

)(lim3

xfx

45.

)(lim4

xfx

46.

)(lim3

xfx

47.

)(lim4

xfx

48.

)(lim3

xfx

49.

)(lim6

xfx

50.

)(lim6

xfx

51.

)(lim2

xfx

52.

)(lim6

xfx

53.

)(lim2

xfx

54.

)(lim2

xfx

55.

)(lim xfx

56.

)(lim4

xfx

57.

)(lim xfx

58. )2(f 59. )3(f



60. Find the 1

1lim

21

x

x

x 61. Find the )432(lim 2

5

xx

x

62. Find the x

xx

x 35

12lim

23

2

63. Find the

9

6lim

2

2

3

x

xx

x

64. Find the 2

8lim

3

2

x

x

x



Free Response Section

Part I:

Let f be a function defined on the closed interval [0,7]. The graph of f, consisting of four line segments, is shown

above.

65. Write the piecewise function for f(x).

66. Find the values of x for which . Justify your answer.

67. For what interval(s) if x is the rate of change of positive? Justify your answer.

68. For what interval(s) if x is the rate of change of negative? Justify your answer.

69. For what interval(s) if x does f have a rate of change of 2? Justify your answer.

70. On which interval of x is the rate of change of f the greatest? Justify your answer.

71. For what x value does reach a maximum value? Justify your answer.



Part II

The graph of f(x) is shown above.

72. Write an equation of the secant line connecting the points (0, -1) and (2, -1). Draw all of the possible tangent lines of

f(x) that is parallel to the secant line.

73. Find any local extrema and justify why these points are local extrema.

74. Find any absolute extrema.

75. Estimate the intervals for which f(x) is concave up and concave down.

76. Estimate the inflection point(s) on f(x)

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AP Calculus AB Summer Packet 2015 - fallriverschools.org Calculus AB Summer Assignment 2015.pdf · AP Calculus AB Summer Packet 2015 ... not rely on a calculator to do all of the