Download AP Calculus Summer Project: Pre-Requisites for AP Calculus Show

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 multistep 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 [email protected]. 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
http://apcentral.collegeboard.com/apc/members/exam/exam_information/1997.html
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.
13. f (x) = = x2
17. f (x) = sin(x)
14. f (x) = x3
18. f (x) = cos(x)
15. f (x) = x
19. f (x) = [[x]]
1
16. f (x) = | x |
20. f (x) =
x
Topic: Trigonometric Ratios
21. Know sine, cosine, and tangent of the following angles to automaticity.
    2 3 5
7 5 4 3 5 7 11
θ = , , , , , , , , , , , , , ,
,2
6 4 3 2 3 4 6
6 4 3 2 3 4 6
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  x   + 1
23. f (x) = –2 cos 3  x   – 3
4
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.
3  x
x0
2

x
x 1
2
0 x3
32. f (x) = 
34. f (x) =  9  x
 3 x  4 x  1
x  3
x3

8  x 2  2  x  2
33. f (x) =  2
35. f (x) = 2[[x]]
 x
elsewhere
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
x
44. f (x) = e
48. f (x) = 4x + 3
45. f (x) = ex+2 – 4
49. 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.
x5  7 x 2
x3  7 x 2
72. f (x) = 4
74. f (x) = 3
x  4x2  x
x  3x 2  5
6x4  2x2
x 2  x  12
73. f (x) = 2
75.
f
(x)
=
x  3x 4  4 x
x2  2x  3  x