# 20 Pre-Requisites for Calculus

*AP Calculus Summer Project Solutions*

- Show all work on separate paper.
- Write your name in pen at the top of each page.
- Use graph paper for all graphs. Do all examples.
- 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 w at least 48 hours for a response.

Topic: Slope, Lines, and Linear Equations

1. Find slope from two given points. Example: (4 , –7) and (–5 , 8)

Slope (m) of the line through (x1,y1) and (x2,y2) is m =

Slope of the line through (4 , –7) and (–5 , 8) is m = = =

2. Write the slope and an equation for a horizontal line through (4, 7). m = 0

y = 7

3. Write the slope and an equation for a vertical line through (-3 , 5).

no slope or undefined slope ; equation: x = –3

Point-Slope Form: y – y1 = m(x – x1)

4. Use point-slope form to write an equation for the line through (6 , –5) with

slope 2.

y – (–5) = 2(x – 6)

y + 5 = 2x – 12

y = 2x – 17

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

First find the slope: m =

Then choose either point and use point-slope form: y – y1 = m(x – x1)

y – 8 = ( x – (–5) )

y – 8 = ( x + 5)

y – 8 =

y = or 5x + 3y = –1

6. Convert from point slope form to slope-intercept form: y – 7 =

Distribute and simplify: y – 7 =

y =

7. Define and illustrate a tangent line.

A tangent line is a line that touches a curve at exactly one point.

8. Define and illustrate a secant line.

A secant line is a line that crosses a curve at two points.

9. Define “average rate of change” algebraically and graphically.

The ARC on [ a , b ] is the slope of the secant line through (a , f (a)) and (b , f (b)).

Algebraically: ARC = or

Geometrically: Average rate of change is slope of a secant line.

10. Define “instantaneous rate of change” algebraically and graphically.

Algebraically: IRC is the derivative or the limit as h approaches zero of the DQ.

Geometrically: IRC is the instantaneous slope of a tangent line

Do the following AP released free response questions found online at ap central.

- 2010 FRQ #2(a)ARC = hundred entries per hour
- 2011 FRQ #2(a)ARC@ 3.5 = ° Celsius/minute

13. 2008 FRQ #2(a)ARC@(5.5) =

14. 2005 FRQ #3(a)ARC@(7) = ° Celsius/minute

Topic: Functions

Graph the 15 parent functions listed below. Use graph paper. State the domain and

range for each.

DomainRange

15. f (x) = x2all Ry ≥ 0

16. f (x) = x3all Rall R

17. f (x) =x ≥ 0y ≥ 0

18. f (x) = all Rall R

19. f (x) = |x|all Ry ≥ 0

20. f (x) = [x]all Rall integers

21. f (x) = sin(x)all R[ –1 , 1 ]

22. f (x) = cos(x)all R[ –1 , 1 ]

23.f (x) = tan(x)all R except odd multiples of all R

24. f (x) =all R except x = 0all R except y = 0

Topic: Semi-Circles

25. Write the general function for a semi-circle: f (x) = , where r is the radius.

26. Give examples of three semi-circle functions and graphs. (Answers may vary.)

y = ; y = ; y = ; y =

Be able to instantly recognize the equation and graph of a semi-circle!

Topic: Special Right Triangles

- Describe and illustrate the ratios for the sides of a 30º–60º–90ºtriangle.

In a 30º-60º-90º triangle, the hypotenuse is double the length of the short side

and the longer leg is times the length of the short side.

The ratio of the sides is 1 : 2 :(short leg : hypotenuse : long leg)

- Describe and illustrate the ratios for the sides of a 45º–45º–90º right triangle.

In a 45º-45º-90º triangle, the hypotenuse is times the length of either leg.

The ratio of the sides is 1 : 1 :(leg : leg : hypotenuse)

- Label all three sides for 30º–60º–90º triangles with hypotenuse of length

1 , 2 , 5 , 8 , 10, and x.

Short Leg / Long Leg / Hypotenuse/ / 1

1 / / 2

/ / 5

4 / 4 / 8

5 / 5 / 10

/ / x

30. Label all three sides for 45º–45º–90º triangles with hypotenuse of length

1 , 2 , 5 , 8 , 10, and x.

Leg / Hypotenuseor / 1

/ 2

or / 5

4 / 8

10 / 10

or / x

Topic: Trigonometric Ratios

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

θ =

If necessary, make 48 flashcards. Identify all 48 trigonometric ratios from randomly-ordered flashcards to 99% accuracy in four minutes or less.

Topic: Seven Deadly Trig Identities

Prove these seven identities and their corollaries and memorize the results.

32. Pythagorean Identity #1: sin2θ + cos2θ = 1

Proof: From a right triangle diagram with acute angle θ, adjacent leg labeled x,

opposite leg labeled y, and hypotenuse h:

x2 + y2 = h2 by the Pythagorean Theorem

Divide through by h2:

cos2θ + sin2θ = 1 or sin2θ+ cos2θ = 1

Remember: “This sign costs one dollar.”

33. Pythagorean Identity #2: tan2θ + 1 = sec2θ

sin2θ + cos2θ= 1

Divide through by sin2θ:

Remember: “One vampire got caught in a casket.” or

You cannot buy just “one cotton ball at Costco”.

34. Pythagorean Identity #3: 1 + cot2θ = csc2θ

sin2θ +cos2θ = 1

Divide through by cos2θ:

Remember: You should only “tan for one second”.

Note: You need to prove the addition and subtraction identities before the double angle identities. I use a geometric proof starting with a rectangle ABCD. Right triangle ∆AED is inscribed in the rectangle with right angle AEF and hypotenuse AF = 1.

35. Sine Addition and Sine Subtraction: sin(α ± β) = sin α cos β ± cos α sin β

36. Cosine Addition and Cosine Subtraction: cos(α ± β) = cos α cos β ± sin α sin β

Next we can prove these:

37. Sine Double Angle Identity: sin 2θ = 2sinθ cosθ

Start with sin 2Ө= sin(Ө + Ө)

= sin Ө cos Ө + cos Ө sin Ө

= 2 sin Ө cos Ө

38. Cosine Double Angle Identities

(a) cos 2θ = cos2θ – sin2θ

Start with cos 2Ө= cos(Ө + Ө)

= cosӨ cos Ө – sin Ө sin Ө

= cos2Ө – sin2Ө

(b) cos 2θ = 2cos2θ – 1

Start with cos 2Ө = cos2Ө – sin2Ө

Use the first Pythagorean Identity: sin2Ө = 1 – cos2Ө

Substitute: cos 2Ө = cos2Ө – (1 – cos2Ө)

cos 2Ө = cos2Ө – 1 + cos2Ө

cos 2Ө = 2cos2Ө – 1

(c): cos 2θ = 1 – 2sin2θ

Start with cos 2Ө = cos2Ө – sin2Ө

Use the second Pythagorean Identity: cos2Ө = 1 – sin2Ө

Substitute: cos 2Ө = (1 – sin2Ө) – sin2Ө

cos 2Ө = 1 – 2sin2Ө

Topic: Sinusoidal Waves

Know the amplitude, period, horizontal shift, and vertical shift for trigonometric functions in the following 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.

39. f (x) = 3 sin 2 + 1 40.f (x) = –2 cos 3 – 3

Topic: Absolute Value Functions

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

Algebraically: | x | = x when x ≥ 0 and | x | = –x when x < 0.

Graphically: | x | is the distance from 0 to x on the number line.

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

42.f (x) = | x + 3 | – 5 V-shape with vertex at ( –3 , –5 )

43. f (x) = 2| x | – 4 Skinny V-shape with vertex at ( 0 , –4 )

44. f (x) = –| x – 2 | + 1 A-shape with vertex at ( 2 , 1 )

45. f (x) = | 3 – x |V-shape with vertex at ( 3 , 0 )

Topic: Greatest Integer Function

46. Define the greatest integer function.

[[ x ]] is the greatest integer that is less than or equal to x. It is a rounding DOWN function.

Graph and label the following on four separate graphs. These are all “stair step” functions with an open circle on the left and a closed circle on the right.

47. f (x) = 2[x]49. f (x) = [x + 3]

48. f (x) = –[x]50. f (x) = [0.5x]

Topic: Piecewise Functions Graph the following piecewise functions.

51. f (x) =52. f (x) =

53. f (x) =

Topic: Exponential Functions

Graph. Label the y-intercept, the horizontal asymptote, and one anchor point.

Function GraphHADomainRangeAnchor Point

54. f (x) = 2xincreasing for all Ry = 0all Ry > 0(0 , 1)

55. f (x) = exincreasing for all Ry = 0all Ry > 0(0 , 1)

56. f (x) = 10xincreasing for all R y = 0all Ry > 0(0 , 1)

57. f (x) = ex+2– 4 same as ex shifted y = –4all Ry > –4(0, –3)

left 2 and down 4

58. f (x) = –5xdecreasing for all Ry = 0all Ry < 0(0 , –1)

59. f (x) = 4x + 3increasing for all Ry = 3all Ry > 3(0 , 4)

60. f (x) = 10x+2 + 3same as 10x shifted

left 2 and up 3y = 3all Ry > 3(0 , 4)

31. f (x) = log(x – 3) same as log(x) shifted right 3x > 3all R

32. f (x) = log(x + 5) – 1 same as ln(x) shifted left 5 and down 1x > –5all

Topic: Logarithmic Functions

61. Graph f (x) = ln(x) and g(x) = ex on one graph.

Label the x- and y-intercepts and one anchor point on each graph.

*( 1 , 0 ) and (e , 1) are on f (x) = ln(x)*

( 0 , 1 ) and (1 , e) are on g(x) = ex

The y-intercept for g(x) = exis y = 1.

The x-intercept for f (x) = ln(x) is ( 1 , 0 ).

62. Graph f (x) = log(x) and g(x) = 10x on one graph.

Label the x- and y-intercepts and one anchor point on each graph.

The y-intercept for g(x) = 10xis y = 1.

( 0 , 1 ) and (1 , 10) are on g(x) = 10x

The x-intercept for f (x) = log(x) is ( 1 , 0 ).

*( 1 , 0 ) and (10 , 1) are on f (x) = log(x)*

Topic: Transformations of Logarithmic Functions

Graph. Label the x-intercept, the vertical asymptote (VA) and one anchor point. Use four separate graphs.

- f (x) = ln(x – 1) + 2VA: x = 1Anchor Point: ( 2 , –3)

- f (x) = ln(x – 3) + 5VA: x = 3Anchor Point: ( 4 , 5 )
- f (x) = ln(x + 2) + 4VA: x = –2Anchor Point: (–1 , 4)

- f (x) = ln(x + 4) – 2VA: x = –4Anchor Point: ( –3 , –2)

Topic: Solving Logarithmic Equations

67. ln(x – 2) + 4 = ln x

4 = ln x – ln(x – 2)

4 = ln

e4 =

e4 (x – 2) = x

xe4 – 2e4 = x

xe4 – x = 2e4

x( e4 – 1 ) = 2e4

x =

68. 3(x+1) = 7(x–2)

ln 3(x+1) = ln 7(x–2)

( x + 1) ln 3 = ( x – 2 ) ln 7

x ln 3 + ln 3 = x ln 7 – 2 ln 7

ln 3 + 2 ln 7 = x ln 7 – x ln 3

ln 3 + 2 ln 7 = x ( ln 7 – ln 3 )

= x

= x

= x

69. log5(x–2) = log5x + log57

log5(x–2) – log5x = log57

log5 = log57

= 7

x – 2 = 7x

–2 = 6x

= x

70. ex(ex–4 ) = 1

ex–4 =

ex–4= e-x

x – 4 = –x

–4 = –2x

2 = x

Topic: Inverse Functions

- What is inverse function notation for f (x)? f -1(x)
- How do you find the inverse of f (x) algebraically? Switch x and y, then solve for x.

73. How do you find the inverse of f (x) graphically?

Reflect the graph across the diagonal line y = x.

74. Give 3 examples of points on a function and points on the inverse.

( 5 , –2 ) (–2 , 5 )

( 3 , 4 ) ( 4 , 3 )

(–7 , –10 ) (–10 , –7 )

75. If f (x) and g(x) are inverses, what is f (g(x))? What is g(f (x))?

f (g(x)) = x

g(f (x)) = x

76. Give 3 examples of slopes of functions and their inverses at corresponding points.

Slope of Function at ( x , y ) / Slope of Inverse at ( y , x )5 / 1/5

–3 / –⅓

½ / 2

77. Inverse functions are reflections across what line? y = x

78. The slopes of inverse functions are reciprocals of each other at corresponding points reflected across the line y = x.

Topic: Polynomials and Rational Expressions

79. X-Intercepts occur when the numerator of a rational expression equals zero.

80. Vertical Asymptotes occur when the denominator of a rational expression equals zero.

81. Horizontal Asymptotes occur when:

the degree of the numerator ≤ the degree of the denominator.

82. Holes occur when there is a:

zero of both the numerator and denominator with multiplicity ≥ in the numerator.

Non-Vertical Asymptotes:In a rational expression, let n = the degree of the numerator and

d = the degree of the denominator. Describe the end behavior.

83. If n < d, then the horizontal asymptote is y = 0 (the x-axis).

84. If n = d, then the horizontal asymptote is y =

where LCON is the leading coefficient of the numerator and

LCOD is the leading coefficient of the denominator.

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

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

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

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

Graph each function separately on graph paper. Describe the end behavior.

Optional: Find and .

88.

HA: y = 0

VA: none

Holes: none

x-intercepts: x = 0 and x = –7

= 0 and = 0

89. =

HA: none, but there is a parabolic asymptote at approx. y = –2

VA: none

Holes: x = 0

x-intercepts: none

= -2and = -2

90. =

HA: none, but there is a parabolic asymptote at approx. y = x2

VA: none

Holes: x = 0

x-intercepts: x =

= ∞and = ∞

91. =

HA: y = 1

VA: none

Holes: none

x-intercepts: x = 4 and x = –3

= 1 and = 1

Topic: Area Formulas for Basic Geometric Shapes

92. Area of a Square with side s.A = s2

93. Area of a Semi-Circle with diameter DA = or A =

94. Area of an Isosceles Right Triangle with leg x.A =

95. Area of an Isosceles Right Triangle with Hypotenuse h. A =

96. Area of a TrapezoidA =

97. Area of a Rectangle with width x and length equal to three times the width. A = 3x2

98. Area of an Equilateral Triangle with side s. A =

Do the following AP released free response questions at

99. 2011B #6: Find the area of the triangle on [–2π , 4π].A = 6π2

100. 2011 #4: Find the area between the graph and the x-axis on the intervals

(a) [–4 , –3]:A = units2

(b) [–3 , 0]:A = units2

(c) [0 , 1.5]:A = units2

101. 2010 #5: Find the area between the graph and the x-axis on the intervals

[–7, –2]:A = units2

[–2 , 2]: A = units2

[2 , 4.5]:A = units2

[4.5, 5]:A = units2

102. 2010B #4: Find the area of the three trapezoids on [0 , 18].

1st Trapezoid on [0 , 9]:A = = 140 units2

2nd Trapezoid on [9 , 15]:A = = 50 units2

3rd Trapezoid on [15 , 18]:A = = 25 units2

Topic: Difference Quotient and Derivative

- Find the difference quotient.
- Find the derivative, f '(x). Use the limit of the difference quotient as h approaches 0.

103. f (x) = –x – 4

DQ =

DQ =

DQ =

DQ = = –1f '(x) =

104. f (x) = x2

DQ =

DQ =

DQ =

DQ =

DQ = 2x + hf '(x) =

105. f (x) = x3 + x

DQ =

DQ =

DQ =

DQ =

DQ = f '(x) =

DQ = 3x2 + 3xh + h2 + 1f '(x) = 3x2 + 1

106. f (x) = 4x2 – 3x – 2

DQ =

DQ =

DQ =

DQ =

DQ = f '(x) =

DQ = 8x + 4h – 3 f '(x) = 8x – 3

107. f (x) =

DQ =

DQ =

DQ = .

DQ = f '(x) =

DQ = f '(x) =

DQ = f '(x) =

108. f (x) =

DQ =

DQ =

DQ =

DQ =

DQ = f '(x) =

DQ = f '(x) =

109. f (x) = sin(x)

DQ =

DQ =

DQ = You can STOP here.

We will find the derivative, f '(x) = cos(x) soon.

110. f (x) = cos(x)

DQ =

DQ =

DQ = You can STOP here.

We will find the derivative, f '(x) = –sin(x) soon.