AP Calculus Summer Project: Pre-Requisites for AP Calculus

AP Calculus Summer Project: Pre-Requisites for AP Calculus

AP Calculus Summer Project: Pre-Requisites for AP Calculus

Show all work on separate paper. Write your name in pen at the top of each page. For multi-step problems, work vertically. Use graph paper for all graphs. Prepare to be tested on all topics on the first day of class. If you have any questions during the summer, email the instructor at . Allow at least 72 hours for a response. The completed problem set is due at the beginning of the first day of class.

Topic: Slope, Lines, and Linear Equations

  1. Find slope from two given points: (4 , –7) and (–5 , 8)
  2. Write the slope and an equation for a horizontal line through (4 , 7).
  3. Write the slope and an equation for a vertical line through (–3 , 5).

Point-Slope Form: y – y1 = m(x – x1)

4. Use point-slope form to write an equation for the line through (4 , –7) and (–5 , 8).

5. Define and illustrate a tangent line.

6. Define and illustrate a secant line.

Topic: Semi-Circles

7. Write the equation for a semi-circle with radius r.

8. Give examples of 3 semi-circle functions with their graphs.

Do the following AP released free response questions found online at

9. 2010 FRQ #2(a)

10. 2008 FRQ #2(a)

11. 2011 FRQ #2(a)

12. 2005 FRQ #3(a)

Topic: Functions

Graph the parent functions listed below. Use graph paper with a separate graph for each.

  1. f (x) = = x2 17. f (x) = sin(x)

14. f (x) = x318. f (x) = cos(x)

15. f (x) = 19. f (x) = [[x]]

16. f (x) = | x |20. f (x) =

Topic: Trigonometric Ratios

21. Know sine, cosine, and tangent of the following angles to automaticity.

θ =

If necessary, make flashcards. Practice until you can identify all 48 trig ratios to 100% accuracy in five minutes or less.

Topic: Sinusoidal Waves

Know the amplitude, period, horizontal shift, and vertical shift for trigonometric

functions in the forms:  f (x) = a sin b(x – c) + d and f (x) = a cos b(x – c) + d

Graph and label these sinusoidal waves. Use a separate graph for each.

22. f (x) = 3 sin 2 + 1 23. f (x) = –2 cos 3 – 3

Topic: Basic Trigonometric Identities

24. Know the Pythagorean identities.

(a) sin2θ + cos2θ = 1

(b) tan2θ + 1 = sec2θ

(c) 1 + cot2θ = csc2θ

25. Know the Addition and Subtraction Identities for Sine and Cosine:

(a) sin(α ± β) = sin α cos β ± cos α sin β

(b) cos(α ± β) = cos α cos β ± sin α sin β

26. Know the Double Angle Identities for Sine and Cosine:

(a) sin 2θ = 2sinθ cosθ

(b) cos 2θ = cos2θ – sin2θ

(c) cos 2θ = 2cos2θ – 1

(d) cos 2θ = 1 – 2sin2θ

Topic: Absolute Value Functions

27. Define absolute value, f (x) = | x |, as a piecewise function.

Graph and label the following absolute value functions on four separate graphs.

28. f (x) = | x + 3 | – 5 30. f (x) = –| x – 2 | + 1

29. f (x) = 2| x | – 4 31. f (x) = | 3 – x |

Topic: Piecewise Functions Graph the following piecewise functions.

32. f (x) =34. f (x) =

33. f (x) =35. f (x) = 2[[x]]

Topic: Area Formulas for Basic Geometric Shapes

36. Area of a Square with side s: ______

37. Area of a Semi-Circle with diameter D: ______

38. Area of an Isosceles Right Triangle with leg x: ______

39. Area of an Isosceles Right Triangle with Hypotenuse h: ______

40. Area of a Trapezoid: ______

41. Area of a Rectangle with width x and length equal to 3 times the width: ___

42. Area of an Equilateral Triangle with side s: ______

Topic: Exponential and Logarithmic Functions

Graph with precision.

43. f (x) = 2x 47. f (x) = –5x

44. f (x) = ex48. f (x) = 4x + 3

45. f (x) = ex+2 – 449. f (x) = log(x)

46. f (x) = ln(x) 50. f (x) = ln(x – 1) + 2

Topic: Solving Exponential and Logarithmic Equations

Solve for x.

51. ln(x – 2) + 4 = ln x 53. log5(x–2) = log5x + log57

52. 3(x+1) = 7(x–2) 54. ex(ex–4 ) = 1

Topic: Polynomials and Rational Expressions

Analyze polynomials and rational expressions for x-intercepts, horizontal asymptotes, vertical asymptotes, holes, and end behavior.

63. X-Intercepts occur when ______.

64. Vertical Asymptotes occur when ______.

65. Horizontal Asymptotes occur when ______.

66. Holes occur when ______.

Let n = the degree of the numerator and d = the degree of the denominator.

67. If n < d, then the horizontal asymptote is ______.

68. If n = d, then the horizontal asymptote is ______.

69. If n > d, and n = d + 1, then the non-vertical asymptote is a ______.

70. If n > d, and n = d + 2, then the non-vertical asymptote is a ______.

71. If n > d, and n = d + 3, then the non-vertical asymptote is a ______.

Name the x-intercepts, vertical asymptotes, horizontal asymptotes, and holes.

72. f (x) = 74. f (x) =

73. f (x) = 75. f (x) =