AP Calculus Summer Packet - southampton.k12.va.us



TO: SHS Calculus Students – 2016-2017

FROM: Col. Bradshaw

SUBJECT: Calculus-DC Summer Assignment

DATE: July 14, 2016

DUE DATE: Tuesday, September 6, 2016

Welcome to Calculus-Dual Enrollment! You are expected to complete the attached homework assignment before the beginning of the course. This is because of class time constraints and the amount of material that must be covered during the accelerated schedule this school year. I will grade your assignment packet, and after a short review, we will have a test on the material to determine if you will be successful in this course.

SUMMER HELP: This assignment is challenging. Expect to work a minimum of about 12 hours.

You are being asked to review all of the algebra, geometry, and trigonometry skills that you have ever learned – and that’s a lot! Please don’t get discouraged if you have some trouble; it is to be expected. Refer to your precalculus notes when working these problems. I will be glad to help you with questions during the summer, anytime except the second week in August. You may either call or text me at 703-209-3030(C), or e-mail me at jbradshaw@pdc.edu.

TEXTBOOK: Our textbook for this class is “Calculus”, 10e, by Ron Larson and Bruce Edwards. We will use the book extensively in class. You are expected to study and learn from this textbook. If you want your own book at home, you can buy the 8th or 9th edition for less than $40 from Amazon or Ebay. These two editions are very similar to the 10th; however, the exercise problems are somewhat different. You may also check out an 8th edition book from the school to use at home.

ONLINE RESOURCES: Your main online resource is calc10. The videos are great. While working this packet, you can click on the “Algebra Help” tab. Another good online resource is MIT Open Courseware, . While doing your homework assignments, you may use CalcChat, , to help you solve problems. Do not copy the solutions, and do not use software such as for your homework problems that you turn in for grades. That is cheating!

GRAPHING CALCULATORS: We will use the color TI-84 Plus CE in class. If you don’t have one, you can order online from either , , or ebay. You can also find them at local book stores, including Paul D. Camp, and retail stores such as Wal-Mart.

NOTEBOOK and HOMEWORK: You are required to use a loose leaf three-ring binder for taking notes and completing assignments for this course as there are several handouts, and I collect your assignments for grading. All homework sections should be labeled separately, and the original problems should be written out, except for word problems. Do your work neatly in pencil.

This assignment is worth 200 points. I will randomly pick ten problems from the worksheet to grade for correctness and effort; I will also grade your assignment on overall neatness and thoroughness.

Complete all your work on separate paper. If you need more space for a problem, put it on separate paper stapled to the back, labelled clearly, and put a note for that problem that there is an attachment.

All answers, except graphs and tables, should be boxed. I expect high quality work!

Colonel Bradshaw

[pic]

Simplify using only positive exponents.

1. [pic] 2. [pic] 3. [pic]

4. [pic] 5. [pic] 6. [pic]

7. [pic] 8. [pic]

Find the domain and range of the following functions without a calculator if possible.

Make sure to use interval notation (ex: [0, 3)).

9. [pic] 10. [pic] 11. [pic] 12. [pic]

13. [pic] 14. [pic] 15. [pic] 16. [pic]

17. [pic] 18. [pic] 19. [pic] 20. [pic]

Factor completely.

21. [pic] 22. [pic] 23. [pic]

Solve the following inequalities by factoring and making sign charts.

24. [pic] 25. [pic] 26. [pic]

27. [pic] 28. [pic] 29. [pic]

Describe, in words, the transformations that would take place to [pic] in each of the following.

30. [pic] 31. [pic] 32. [pic]

33. [pic] 34. [pic] 35. [pic]

Determine if each function is even, odd, or neither. Show all work.

36. [pic] 37. [pic]

38. [pic] 39. [pic]

Solve each equation by factoring, graphing, or using the quadratic formula.

40. [pic] 41. [pic] 42. [pic]

43. [pic] 44. [pic] 45. [pic]

46. [pic] 47. [pic] 48. [pic]

Find the equations of all vertical (x = ?) and horizontal (y = ?) asymptotes (if they exist).

49. [pic] 50. [pic] 51. [pic]

52. [pic] 53. [pic] 54. [pic]

Simplify the following.

55. [pic] 56. [pic] 57. [pic]

58. [pic] 59. [pic] 60. [pic]

If [pic], find the following.

61. [pic] 62. [pic] 63. [pic] 64. [pic]

Solve each equation.

65. [pic] 66. [pic] 67. [pic]

68. [pic] 69. [pic] 70. [pic]

TRIGONOMETRY: Trig functions are used extensively in calculus. You must know the special triangles, 30-60-90 and 45-45-90, and you must memorize exact values of sin(θ), cos(θ), and tan(θ) for θ = 0, π/6, π/4, π/3, π/2, π and 3π/2 radians, and be able to use these to find values in

Quadrants II, III, and IV, as well as for evaluating secant, cosecant, and cotangent functions. We will work primarily with radians in calculus. You must know the trig definitions and the Pythagorean identity, cos2 θ + sin2 θ = 1. You should also be able to derive variations of the identity by dividing with cos2 θ or sin2 θ and rearranging terms.

.

Radian and Degree Measure

Use [pic] to convert back and forth between degrees and radians.

71. Convert to degrees: a. [pic] b. [pic] c. 2.63 radians

72. Convert to radians: a. [pic] b. [pic] c. 237[pic]

Angles in Standard Position

73. Sketch the angle in standard position.

a. [pic] b. [pic] c. [pic] d. 1.8 radians

74. Evaluate the following without using a calculator.

a) [pic] b) [pic] c) [pic]

d) [pic] e) [pic] f) [pic]

Solve each equation on the interval [pic]. Give exact values [pic] if possible.

75. [pic] 76. [pic] 77. [pic]

78. [pic] 79. [pic] 80. [pic]

81. [pic] 82. [pic] 83. [pic]

Equation of a line

Slope intercept form: [pic] Vertical line: x = c (slope is undefined)

Point-slope form: [pic] Horizontal line: y = c (slope is 0)

84. Use slope-intercept form to find the equation of the line having a slope of 3 and a y-intercept of 5.

85. Determine the equation of a line passing through the point (5, -3) with an undefined slope.

86. Determine the equation of a line passing through the point (-4, 2) with a slope of 0.

87. Use point-slope form to find the equation of the line passing through the point (0, 5) with a slope of 2/3.

88. Find the equation of a line passing through the point (2, 8) and parallel to the line [pic].

89. Find the equation of a line perpendicular to the y- axis passing through the point (4, 7).

90. Find the equation of a line passing through the points (-3, 6) and (1, 2).

91. Find the equation of a line with an x-intercept (2, 0) and a y-intercept (0, 3).

Answer the following questions over a variety of topics.

92. Let f be a linear function where [pic] and [pic]. Find [pic].

93. Find an equation for the line, in point-slope form, that contains [pic] and is perpendicular to [pic].

94. Find the distance between the points [pic] and [pic].

95. Use the table to calculate the average rate of change from t = 1 to t = 4.

|t |0 |1 |2 |3 |4 |

|[pic] |8 |7 |5 |1 |2 |

96. Order the points A, B, and C, from least to greatest, by their rates of change.

97. If [pic], find [pic] (the inverse of g).

98. Find the points of intersection in the graphs of [pic] and [pic].

99. Rewrite [pic] as a single logarithmic expression.

100. Sketch a graph of the piecewise function [pic].

101. Describe the left and right end-behavior of the function [pic].

102. Find the domain and range of each function (without a calculator if possible).

a) [pic] b) [pic] c) [pic]

d) [pic] e) [pic] f) [pic]

103. The circle below has a radius of 6 ft. Find the area and circumference of the circle, then find s.

104. Find the area of the trapezoid.

105. Find the missing sides and angles of the triangle. Then find its area.

106. Find the volume of a washer with outer radius of 18 ft., inner radius of 15 ft., and height of 3 ft.

107. Rewrite [pic] into an equivalent expression using only natural logarithms.

108. Three sides of a fence and an existing wall form a rectangular enclosure. The total length of fence used for the three sides is 240 ft. Find x if the area enclosed is 5500 ft2.

109. The number of elk after t years in a state park is modeled by the function [pic].

a) What was the initial population?

b) When will the number of elk be 750?

c) What is the maximum number of elk possible in the park?

110. Simplify [pic].

111. Use long division, or synthetic division, to rewrite the expression [pic].

112. Rewrite [pic] in vertex form [pic] by completing the square.

113. Sketch a graph of the piecewise function [pic].

114. Use a graphing calculator to solve [pic].

115. Do the lines [pic] and [pic] intersect?

116. The function [pic] is graphed below. Find the following.

a) [pic] b) [pic] c) [pic]

d) [pic] e) [pic] f) [pic]

-----------------------

Exponent Rules:

| |Rule |Example |

|1 |x1 = x |51 = 5 |

|2 |x0 = 1 |50 = 1 |

|3 |x-1 = 1/x1 |5-1 = 1/5 |

|4 |(xm)(xn) = xm+n |(x2)(x3) = x2+3 = x5 |

|5 |xm/xn = xm-n |x3x2 = x3-2 = x1 |

|6 |(xm)n = x(m)(n) |(x3)2 = x(3)(2) = x6 |

|7 |(xy)n = xnyn |(xy)3 = x3y3 |

|8 |(xy)-n = x-ny-n |(xy)-3 =x-3y-3 |

|9 |x-n = 1/xn |x-2 = 1/x2 |

|Transformations of Function Graphs |

|-f (x) |reflect f (x) over the x-axis |

|f (-x) |reflect f (x) over the y-axis |

|f (x) + k |shift f (x) up k units |

|f (x) - k |shift f (x) down k units |

|f (x + h) |shift f (x) left h units |

|f (x - h) |shift f (x) right h units |

|A•f (x) |multiply y-values by A |

|f (Bx) |divide x-values by B |

Vertical Asymptotes

Determine the vertical asymptotes for the function. Set the denominator equal to zero to find the x-value for which the function is undefined. If the numerator is not equal to zero at that x-value, that is the vertical asymptote.

Horizontal Asymptotes

Determine the horizontal asymptotes using the three cases below.

Case I. Degree of the numerator is less than the degree of the denominator. The asymptote is y = 0.

Case II. Degree of the numerator is the same as the degree of the denominator. The asymptote is the ratio of the leading coefficients.

Case III. Degree of the numerator is greater than the degree of the denominator. There is no horizontal asymptote. The function increases without bound. (If the degree of the numerator is exactly 1 more than the degree of the denominator, then there exists a slant asymptote, which is determined by long division.)

Complex Fractions

When simplifying complex fractions, multiply by a fraction equal to 1 which has a numerator and denominator composed of the common denominator of all the denominators in the complex fraction.

Examples:

[pic]

[pic]

Inverse Functions

To find the inverse of a function, simply switch the x and the y and solve for the new “y” value.

Example:

[pic]

Logarithms

We will work mostly with natural logarithms in Calculus, but we will use some common logarithms. The rules are the same for both, because the natural logarithm is just a logarithm to the base e.

logb (x) = n   means   bn = x

ln x = n   means   en = x

Log Rules:

1) logb (mn) = logb (m) + logb (n)

2) logb (m/n) = logb (m) – logb (n)

3) logb (mn) = n · logb (m)

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