1. When you see…
Find the zeros
You think…

2. To find the zeros...

3. Find equation of the line tangent to f(x) at (a, b)
When you see…
Find equation of the line tangent to f(x) at (a, b)
You think…

4. Equation of the tangent line

5. Find equation of the line normal to f(x) at (a, b)
When you see…
Find equation of the line normal to f(x) at (a, b)
You think…

6. Equation of the normal line

7. When you see…
Show that f(x) is even
You think…

8. Even function

9. When you see…
Show that f(x) is odd
You think…

10. Odd function

11. Find the interval where f(x) is increasing
When you see…
Find the interval where f(x) is increasing
You think…

12. f(x) increasing

13. Find the interval where the slope of f (x) is increasing
When you see…
Find the interval where the slope of f (x) is increasing
You think…

14. Slope of f (x) is increasing

15. Find the minimum value of a function
When you see…
Find the minimum value of a function
You think…

16. Local Minimum value of a function

17. When you see…
Find critical numbers
You think…

18. Find critical numbers

19. Find inflection points
When you see…
Find inflection points
You think…

20. Find inflection points

21. When you see…
Show that exists
You think…

22. Show exists
Show that

23. Show that f(x) is continuous
When you see…
Show that f(x) is continuous
You think…

24. .f(x) is continuous

25. Show that f(x) is differentiable at x = a
When you see…
Show that f(x) is differentiable at x = a
You think…

26. f(x) is differentiable

27. When you see…
Find vertical
asymptotes of f(x)
You think…

28. Find vertical asymptotes of f(x)
Factor/cancel f(x)
Set denominator = 0

29. Find horizontal asymptotes of f(x)
When you see…
Find horizontal asymptotes of f(x)
You think…

30. Find horizontal asymptotes of f(x)

31. Find the average rate of change of f(x) at [a, b]
When you see…
Find the average rate of change of f(x) at [a, b]
You think…

32. Average rate of change of f(x)
Find
f (b) - f ( a)
b - a

33. Find the instantaneous rate of change of f(x)
When you see…
Find the instantaneous rate of change of f(x)
at x = a
You think…

34. Instantaneous rate of change of f(x)
Find f ‘ ( a)

35. When you see…
You think…

36. Average value of the function

37. Find the absolute maximum of f(x) on [a, b]
When you see…
Find the absolute maximum of f(x) on [a, b]
You think…

38. Find the absolute maximum of f(x)

39. When you see…
Show that a piecewise function is differentiable at the point a where the function rule splits
You think…

40. Show a piecewise function is differentiable at x=a

41. Given s(t) (position function),
When you see…
Given s(t) (position function),
find v(t)
You think…

42. Given position s(t), find v(t)

43. Given v(t), find how far a particle travels on [a, b]
When you see…
Given v(t), find how far a particle travels on [a, b]
You think…

44. Given v(t), find how far a particle travels on [a,b]

45. Find the average velocity of a particle
When you see…
Find the average velocity of a particle
on [a, b]
You think…

46. Find the average rate of change on [a,b]

47. Given v(t), determine if a particle is speeding up at
When you see…
Given v(t), determine if a particle is speeding up at
t = k
You think…

48. Given v(t), determine if the particle is speeding up at t=k

49. When you see…
Given v(t) and s(0),
find s(t)
You think…

50. Given v(t) and s(0), find s(t)

51. Show that the Mean Value Theorem holds
When you see…
Show that the Mean Value Theorem holds
on [a, b]
You think…

52. Show that the MVT holds on [a,b]

53. Show that Rolle’s Theorem holds on [a, b]
When you see…
Show that Rolle’s Theorem holds on [a, b]
You think…

54. Show that Rolle’s Theorem holds on [a,b]

55. When you see…
Find the domain
of f(x)
You think…

56. Find the domain of f(x)

57. When you see…
Find the range
of f(x) on [a, b]
You think…

58. Find the range of f(x) on [a,b]

59. When you see…
Find the range
of f(x) on
You think…

60. Find the range of f(x) on

61. Find f ’(x) by definition
When you see…
Find f ’(x) by definition
You think…

62. Find f ‘( x) by definition

63. Find the derivative of the inverse of f(x) at x = a
When you see…
Find the derivative of the inverse of f(x) at x = a
You think…

64. Derivative of the inverse of f(x) at x=a

65. y is increasing proportionally to y
When you see…
y is increasing proportionally to y
You think…

66. .y is increasing proportionally to y

67. When you see… You think…
Find the line x = c that divides the area under f(x) on [a, b] into two equal areas
You think…

68. Find the x=c so the area under f(x) is divided equally

69. When you see…
You think…

70. Fundamental Theorem

71. When you see…
You think…

72. Fundamental Theorem, again
Given:

73. The rate of change of population is …
When you see…
The rate of change of population is …
You think…

74. Rate of change of a population

75. The line y = mx + b is tangent to f(x) at (a, b)
When you see…
The line y = mx + b is tangent to f(x) at (a, b)
You think…

76. .y = mx+b is tangent to f(x) at (a,b)

77. When you see…
Integrate
You think…

78. Methods for Integration
1. Estimation:
LRAM
RRAM (Riemann Sums)
MRAM
Trapezoid
2. Geometry
Antiderivative
Straight Forward
Substitution
Rewrite (Simplify)

79. Find area using Left Riemann sums
When you see…
Find area using Left Riemann sums
You think…

80. Area using Left Riemann sums

81. Find area using Right Riemann sums
When you see…
Find area using Right Riemann sums
You think…

82. Area using Right Riemann sums

83. Find area using Midpoint rectangles
When you see…
Find area using Midpoint rectangles
You think…

84. Area using midpoint rectangles

85. Find area using trapezoids
When you see…
Find area using trapezoids
You think…

86. Area using trapezoids

87. Solve the differential equation …
When you see…
Solve the differential equation …
You think…

88. Solve the differential equation...

89. When you see…
Meaning of
You think…

90. Meaning of the integral of f(t) from a to x

91. When you see…
Given a base, cross sections perpendicular to the x-axis that are squares
You think…

92. Semi-circular cross sections perpendicular to the x-axis

93. Find where the tangent line to f(x) is horizontal
When you see…
Find where the tangent line to f(x) is horizontal
You think…

94. Horizontal tangent line

95. Find where the tangent line to f(x) is vertical
When you see…
Find where the tangent line to f(x) is vertical
You think…

96. Vertical tangent line to f(x)

97. Find the minimum acceleration given v(t)
When you see…
Find the minimum acceleration given v(t)
You think…

98. Given v(t), find minimum acceleration

99. Approximate the value f(0.1) of by using the
When you see…
Approximate the value f(0.1) of by using the
tangent line to f at x = a
You think…

100. Approximate f(0.1) using tangent line to f(x) at x = 0

101. anti-derivative of f is F, find F(b)
When you see…
Given the value of F(a) and the fact that the
anti-derivative of f is F, find F(b)
You think…

102. Given F(a) and the that the anti-derivative of f is F, find F(b)

103. When you see…
Find the derivative of
f(g(x))
You think…

104. Find the derivative of f(g(x))
Think Chain Rule

105. When you see…
Given , find
You think…

106. Given area under a curve and vertical shift, find the new area under the curve

107. Given a graph of find where f(x) is increasing When you see…
You think…

108. Given a graph of f ‘(x) , find where f(x) is increasing

109. When you see…
Given v(t) and s(0), find the greatest distance from the origin of a particle on [a, b]
You think…

110. Given v(t) and s(0), find the greatest distance from the origin of a particle on [a, b]

111. When you see…
Given a water tank with g gallons initially being filled at the rate of F(t) gallons/min and emptied at the rate of E(t) gallons/min on
, find

112. the amount of water in
the tank at m minutes
You think…

114. Amount of water in the tank at t minutes
initial gallons

115. b) the rate the water
amount is changing
at m
You think…

116. Rate the amount of water is changing at t = m

117. c) the time when the
water is at a minimum
You think…

118. The time when the water is at a minimum

119. When you see…
Given a chart of x and f(x) on selected values between a and b, estimate where c is between a and b.
You think…

121. When you see…
Given , draw a
slope field
You think…

122. Draw a slope field of dy/dx

123. Find the area between curves f(x) and g(x) on [a,b]
When you see…
Find the area between curves f(x) and g(x) on [a,b]
You think…

124. Area between f(x) and g(x) on [a,b]

125. Find the volume if the area between the curves f(x) and g(x)
When you see…
Find the volume if the area between the curves f(x) and g(x)
with a representative rectangle perpendicular to the axis of rotation
You think…

126. Volume generated by rotating area between f(x) and g(x) with a representative rectangle perpendicular to the axis of rotation

127. Find the volume if the area between the curves f(x) and g(x)
When you see…
Find the volume if the area between the curves f(x) and g(x)
with a representative rectangle parallel to the axis of rotation
You think…

128. Volume generated by rotating area between f(x) and g(x) with a representative rectangle parallel to the axis of rotation
Remember: Always Big - small