AP CALCULUS BC

SUMMER ASSIGNMENT

Dear BC Calculus Student,

Congratulations on your wisdom in taking the BC course! We know you will find it rewarding and a great way to spend your junior/senior year. This course is primarily concerned with developing your understanding of the concepts of

calculus and providing experience with its methods and applications. The course emphasizes a multi-representational

approach to calculus, with concepts, results, and problems being expressed graphically, numerically, analytically, and

verbally. In order to be successful in this course you need the proper foundation (i.e. knowledge of algebra, geometry,

trigonometry, analytic geometry, and elementary functions). You will have to be very familiar with the basic families of

functions, and all of their representations, in order to be successful in your study of calculus. The concept of functions

underlies everything that calculus considers.

You will also need to be able to carry out certain computational tasks (i.e. algebra skills) with efficiency and accuracy if

you are going to be successful in calculus. These include manipulations of functional symbolism, solving algebraic

equations involving the functions mentioned above, interpreting numerical values given by formulas, graphs, and tables,

using and manipulating data, and knowing how and when to use your calculator.

This is a rigorous college course. The curriculum and pace of the course is intense and is designed to prepare students

for the AP exam. Since this is a college class you can expect to spend approximately 1-2 hours completing homework or

studying for every hour that you are in class learning. Each test and quiz that is given is cumulative and will be graded as

per the College Board guidelines. Therefore, this course will be challenging and demanding. Next year, the AP exam will

be on Tuesday, May 14th. Circle this day on your calendar. The College Board requires the use of a graphing calculator on the exam. If you do not own a graphing calculator, we strongly encourage you to purchase one this summer. Spend some time familiarizing yourself with the capabilities of your calculator. The school does have class sets of TI-84’s that

will be available in class but not outside of class.

Complete this packet over the summer. Please be advised that on the third day of class you will be required to turn in the packet (50 pts) and take a quiz (50 pts) on the information within the first two weeks of school. This quiz will cover the following concepts and skills: limits, all differentiation methods and application of derivatives. These were covered in the first three chapters of Calculus AB/Math Analysis. Moreover, you will be assessed on this material throughout the entire year, so it is in your best interest to review the sample questions provided on the subsequent

pages, and prepare yourself prior to the first day of school. Be advised that these are the skills that we expect you to

possess prior to the first day of school. Therefore, if you are unable to answer some of these questions, we suggest that you start studying!

Feel free to contact your teachers with any questions or concerns that you or your parents may have. Our email addresses are included below. Have a restful summer and be ready to talk math again in August.

Sincerely,

Mrs. Maria Quinn James

maria.james@lcps.org

1. Find an equation for the line that contains the points (2, -3) and (6, 9).

2. Find the value of y for which the line through A and B has the given slope m: A(-2, 3), B(4, y),

.3

2m

3. Find an equation for the line that contains the coordinate (5, 1) and is perpendicular to the line

.236 yx

For questions 4-9, let 3)( xxf and .5)( 2xxg

4. )3)(( gf 5. )6)(( fg 6. )(1 xg

7. ))(( xfg 8. )(

1

xf9. )3)(( gf

10. Algebraically find the inverse of .12

3

xy

For questions 11-19, simplify each expression completely.

11. x

x12. 3lne 13. ln 1

14 7ln e 15. 8log2

116. xe ln3

17. 53

1

2

12

4

yx

xy18. 3

2

27 19. 2)1(

)12(2)1(3

x

xxx

20. Rewrite xxx ln6)2ln()3ln(2

1as a single logarithmic expression.

21. Solve for t: 2)045.1( t

22. Solve for x: 1)4(loglog 55 xx

23. Solve for x: 32 927 xx

24. Solve for x: 16)3ln( 2x

25. Evaluate 5log 2 to the nearest thousandth.

26. Solve: 01553 23 xxx

27. Solve: 089 24 xx

28. Without using a calculator, find the exact value of 5

17coscos 1 . Justify your answer.

For questions 29-34, find the exact values of each trigonometric function.

29. 6

7sin 30. 31.

32. 3

2sec 33.

2tan 34. )135cot(

csc cos34

35

35.

Simplify )cos()sin())tan()(csc( xxxx

36. List the three Pythagorean Identities.

37. List the double angle formulas

a) sin 2x =

b) cos 2x =

38. List the sum and difference formulas.

a) )cos(

b) )sin(

39. Prove that csc2sin

cos1

cos1

sin

40. Prove that xxx 2sin1)cos(sin 2

41. For the solution of the equation sin1sin2 2 for .20 x

42. Find the domain for 145)( 2 xxxk .

43. Determine all points of intersection for 432 xxy and 115xy .

44. Find the points of intersection in the graphs of 1xy and .622 xy

45. Use a graphing calculator to approximate all of the function’s real zeros. Round your results to 3

decimal places. 1453)( 2356 xxxxxxf

46. Algebraically determine whether the function is even, odd, or neither:

a. 72)( 2xxf b. xxxf 24)( 3 c. 444)( 2 xxxf

47. If ,1)( 2xxf describe in words what the following would do the graph of ).(xf

a. 4)(xf b. )4(xf c. )2(xf

d. 3)(5 xf

48. Let 3 2)( xxf and .2)( 3xxg Which of the following are true?

I. )()( 1 xfxg for all values of x.

II. 1))(( xgf for all values of x.

III. The function is one-to-one

49. For the function below, give the zeros (if non exist, write none), domain, range, VA’s, HA’s,

and/or points of discontinuity (holes-as ordered pairs)if any exist. Also sketch the function’s

graph.

352

3)(

2 xx

xxf

For questions 50-55, graph each function on the attached graph paper. Give its domain and

range.

50. xey 51. y = 23x

52. xy sin23 53. 21 xy

54. ,1

,1)(xf

0

0

x

x55.

,3

,2

,2

)( 2

x

x

x

xf

),2(

)2,1(

)1,(

63. )(lim3

xfx

64. )(lim4

xfx

65. )(lim3

xfx

66. )(lim4

xfx

67. )(lim3

xfx

68. )(lim6

xfx

69. )(lim6

xfx

70. )(lim2

xfx

71. )(lim6

xfx

72. )(lim2

xfx

73. )(lim2

xfx

74. )(lim xfx

75. )(lim4

xfx

76. )(lim xfx

77. )2(f 78. )3(f

Use the graph below to answer questions 63-78

79. Find the 1

1lim

21 x

x

x 80. Find the )432(lim 2

5xx

x

81. Find the x

xx

x 35

12lim

23

2 82. Find the

9

6lim

2

2

3 x

xx

x

83. Find the 25

5lim

25 x

x

x 84. Find the

)32)(1(

56lim

2

tt

tt

t

85. Find the xx

coslim 86. Find the x

x

x

24lim

0

87. Find the 295

2lim

2

2

xx

x

x 88. Find the

x

xx

x 3lim

2

89. Find the 2

8lim

3

2 x

x

x

90. Find the slope of the tangent line to the graph of 4)( 2xxg at the point (1, -3).

91. Find an equation of the tangent line to the hyperbola x

y3

at the point (3, 1).

92. Find an equation of the tangent line to the graph of 98)( 2 xxxf a the point (3,-6).

For questions 93-95, find the x-values (if any) at which f is not continuous. Which of the

discontinuities are removable.

93. xx

xxf

2)( 94. xxxf cos3)( 95.

103

2)(

2 xx

xxf

For questions 96-116, find the derivative of each given function.

96. 5

2

)( xxf 97. .2)( 23 xxxxf

98. 35 xey 99. 3)( xxf

100. 7

10)(

xxg 101.

x

xxxf

34)(

2

102. xexxf 2)( 103. 2

)(x

exf

x

104. )38(

)(x

xxf 105. x1tan

106. xxy cossin 107. xxy cos2

108. xxxf tan2cos)( 109. xxxxf seccot2)(

110. 1003 )1()( xxg 111. xxy 72

112. 435 )1()12( xxxy 113. )1ln( 3xy

114. Find y of .log10 xy

115. If xexxf 2)( 2, find the value of )1('f

116. If 322)( xxxf , find ),(),( xfxf and )(xf

For questions 117-118, find dx

dyby implicit differentiation.

117. 2522 yx 118. 832 yxyx

119. If a billiard ball is dropped from a height of 100 feet, its height s at time t is given by the position

function: 10016 2ts where s is measured in feet and t is measured in seconds. Find the

average velocity of each the following time intervals.

a) [1, 2] b) [1, 1.5] c) [1, 1.1]

120. A particle moves according to a law of motion ttttfs 3612)( 23 , 0t , where t is

measured in seconds and s in meters.

a) Find the velocity at time t.

b) What is the velocity after 3 s?

c) When is the particle at rest?

d) Find the total distance traveled during the first 8 s.

e) Find the acceleration at time t.

f) Find the acceleration after 3 s.

122. A rectangle has its vertices on the x-axis, the y-axis, the origin, and the graph of 24y x

in the first quadrant. Find the maximum possible area for such a rectangle. Justify your answer

121. The length of a rectangle is decreasing by 2 inches per second and the width is increasing by 3 inches

per second. When the length is 10 inches and the width is 6 inches, how fast is the (a) perimeter and (b)

area changing?