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1 When you see… Find the zeros You think…

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2 To find the zeros...

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

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4 Equation of the tangent line

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

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6 Equation of the normal line

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

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8 Even function

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

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10 Odd function

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

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12 f(x) increasing

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

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14 Slope of f (x) is increasing

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

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16 Local Minimum value of a function

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17 You think… When you see… Find critical numbers

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18 Find critical numbers

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19 You think… When you see… Find inflection points

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20 Find inflection points

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21 You think… When you see… Show that exists

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22 Show exists Show that

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

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24. f(x) is continuous

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

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26 f(x) is differentiable

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

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

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

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30 Find horizontal asymptotes of f(x)

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

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

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

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

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35 You think… When you see…

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36 Average value of the function

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

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38 Find the absolute maximum of f(x)

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

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40 Show a piecewise function is differentiable at x=a

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

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

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

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

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

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

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

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

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

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

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

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

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53 You think… When you see… Show that Rolles Theorem holds on [a, b]

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54 Show that Rolles Theorem holds on [a,b]

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

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56 Find the domain of f(x)

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

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

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

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60 Find the range of f(x) on

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

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62 Find f ( x) by definition

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

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

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

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66 y is increasing proportionally to y. y is increasing proportionally to y

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

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

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69 You think… When you see…

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70 Fundamental Theorem

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71 You think… When you see…

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72 Fundamental Theorem, again Given:

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

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74 Rate of change of a population

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

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

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77 You think… When you see… Integrate

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78 1. Estimation: LRAM RRAM (Riemann Sums) MRAM Trapezoid 2. Geometry 3.Antiderivative Straight Forward Substitution Rewrite (Simplify) Methods for Integration

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

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80 Area using Left Riemann sums

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

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82 Area using Right Riemann sums

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

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84 Area using midpoint rectangles

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

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86 Area using trapezoids

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

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88 Solve the differential equation...

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89 You think… When you see… Meaning of

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

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

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

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

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94 Horizontal tangent line

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

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96 Vertical tangent line to f(x)

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

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98 Given v(t), find minimum acceleration

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

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

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

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

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

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

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105 You think… When you see… Given, find

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

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

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

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

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

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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

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

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113

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

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

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

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

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118 The time when the water is at a minimum

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119 You think… 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.

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

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122 Draw a slope field of dy/dx

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

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

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125 You think… 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

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

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127 You think… 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

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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

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