Calculus AB Golden Notes

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Derivative Power Rule

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Everything you need to know & understand for the AB calculus exam

102 Terms

1

Derivative Power Rule

If f

<p>If f</p>
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2

Derivative exponential rule

<p></p>
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3

Derivative e Rule

<p></p>
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4

Derivative Ln Rule

<p></p>
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5

Derivative Square Root Rule

<p></p>
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6

Derivative Tangent Rule

<p></p>
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7

Derivative Sine Rule

<p></p>
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8

Derivative Cosine Rule

<p></p>
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9

Derivative Inverse Sine Rule

<p></p>
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10

Derivative Inverse cos rule

<p></p>
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11

Derivative Inverse tan rule

<p></p>
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12

Derivative constant Rule

<p></p>
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13

Derivative Chain Rule

<p></p>
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14

Derivative Product Rule

<p></p>
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15

Derivative Quotient Rule

<p></p>
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16

Derivative Addition Rule

<p></p>
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17

Anti-derivative power rule

<p></p>
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18

Anti-derivative expanded power rule

<p></p>
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19

Anti-derivative exponential rule

<p></p>
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20

Anti-derivative expanded exponential rule

<p></p>
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21

Anti-derivative Ln Rule

<p></p>
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22

Anti-derivative Ln expanded rule

<p></p>
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23

Anti-derivative sine rule

<p></p>
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24

Anti-derivative expanded sin rule

<p></p>
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25

Anti-derivative cos rule

<p></p>
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26

Anti-derivative expanded cos rule

<p></p>
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27

Derivative of inverse f(x)

<p></p>
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28

Displacement

<p></p>
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29

Total Distance

<p></p>
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30

Derivative of an integral

<p></p>
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31

Differentiable if

continuous, no corner or vertical tangent

<p>continuous, no corner or vertical tangent</p>
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32

Continuous if

No removable discontinuity, jumps, or vertical asymptotes.

<p>No removable discontinuity, jumps, or vertical asymptotes.</p>
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33

Limits if x->∞ then

  1. compare terms that add

  2. Factor & divide

  3. Left & Right

  4. L'hopital's rule

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34

Place in order of growing fastest as x ->∞: x^99, e^x, lnx

lnx, x^99, e^x

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35

Find the average value of f(x)

<p></p>
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36

Find the average rate of change

<p></p>
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37

v(t) is the

rate at which x is changing; tangent slope; instantaneous rate of change

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38

Average value of f'(x) is the same as

average rate of change

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39

secant slope is the

average rate of change

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40

Find the secant slope

<p></p>
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41

e^(lnA)

A

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42

lne^A

A

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43

e^(A+B)

e^Ae^B

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44

ln12-ln4

ln(12/4)

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45

f(x) has a critical point when

f'(x)=0 or f'(x)=undefined

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46

Min-Max Theorem

The absolute Max/Min of f(x) is at the beginning of f(x) at the end of f(x) or at a critical point on f(x)

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47

f(x) has an inflection point when

f(x) changes concavity, OR f'(x) changes I to D or D to I or when f"(x) changes sign

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48

L'Hopitals Rule

<p></p>
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49

The limit exists if

<p></p>
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50

Area of a semicircle

<p></p>
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51

Solve an Equation

Find value which makes equation true OR graph both halves of equation & find intersection

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52

The particular solution y=B(t) of a differential equation dB/dt=1/5(100-B) with initial condition B(0)=20 what would you use?

Use SACI

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53

SACI

Separate, Anti Differentiate, Constant-tate, Isolate

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54

Speed is increasing when

v(t) and a(t) are the same sign

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55

Approximate the instant rate of change by:

calculating the average rate of change

<p>calculating the average rate of change</p>
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56

Approximate the tangent slope by:

calculating the nearest secant slope

<p>calculating the nearest secant slope</p>
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57

When the in rate is E(t) and the out rate is L(t) what is the equation for the rate?

A'(t)=E(t)-L(t)

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58

Solve an anti-derivative

  1. Rule 2. u substitution 3. Algebra trick

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59

Average rate of change of velocity is the same as

average acceleration

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60

average rate of change of position is the same as

average velocity

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61

secant slope is the same as

average rate of change of f(x)

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62

secant slope or average roc or f(x)

<p></p>
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63

average roc of x(t) or average velocity

<p></p>
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64

average roc of v(t) or average acceleration

<p></p>
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65

speed

<p></p>
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66

F'(x)=

f(x)

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67

anti-derivative of f(x)

F(x)

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68

anti-derivative of f'(x)

f(x)

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69

integral from a to b of a(t) equals

v(b)-v(a)

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70

integral from a to b of v(t) equals

x(b)-x(a)

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71

integral from a to b of f(x) equals

F(b)-F(a)

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72

integral from a to b of f'(x) equals

f(b)-f(a)

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73

integral of a rate equals

change in amount

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74

Mean Value Theorem

If f(x) is continuous and differentiable the "tangent slope at c" = secant slope

<p>If f(x) is continuous and differentiable the &quot;tangent slope at c&quot; = secant slope</p>
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75

Tangent line formula

<p></p>
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76

If f(x) is concave down the tangent line is

an OVER approximation

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77

If f(x) is concave up the tangent line is

an UNDER approximation

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78

Trapezoidal riemann sum formula

<p></p>
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79

f'(x)=dy/dx= Formula to find:

  1. Instantaneous rate of change of f(x)

  2. Slope of line tangent to f(x)

  3. Slope of f(x) at a point

  4. Instant rate at which f(x) is changing

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80

f(x) has relative/local max when

f'(x) changes + to - or when f"(x) changes I to D

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81

lne^2

2

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82

lne

1

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83

lne^0

0

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84

ln1

0

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85

ln(1/e)

-1

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86

lne^(-1)

-1

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87

ln(1/e^-2)

-2

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88

rate of change of position

x'(t) or v(t)

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89

rate of change of velocity

v'(t) or a(t)

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90

Vertical Tangent when

number/0

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91

Jump discontinuity when

the left limit is different from the right limit

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92

Removable discontinuity when

the value is different than the limits on the left and right. Limits must be the same on left and right.

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93

Horizontal asymptote

the value of the limit as x->infinity

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94

When given a rate and then asked to find the amount use

Fundamental Theorem

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95

When given a rate that includes the output variable and then asked to find the amount use

SACI

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96

f has an inflection point when

f changes concavity

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97

f has a relative or local max when

f changes from increasing to decreasing

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98

f has a relative extrema when

f changes from I to D or D to I or when f' changes + to - or - to +

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99

f has a critical point when

the slope of f is 0 or undefined or when f' has a y-coord. of 0 or und

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100

tangent slope means

instantaneous rate of change

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