AP Calc AB exam review

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d/dx (x)

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102 Terms

1

d/dx (x)

1

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2

d/dx sin(x)

cos(x)

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3

d/dx cos(x)

-sin(x)

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4

d/dx tan(x)

sec^2(x)

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5

d/dx sec(x)

sec(x)tan(x)

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6

d/dx csc(x)

-csc(x)cot(x)

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7

d/dx cot(x)

-csc^2(x)

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8

d/dx sin^-1(x)

1/sqrt(1-x^2)

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9

d/dx cos^-1(x)

-1/sqrt(1-x^2)

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10

d/dx tan^-1(x)

1/(1+x^2)

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11

d/dx a^x

a^x*ln(a)

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12

d/dx e^x

e^x

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13

d/dx ln(x)

1/x

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14

d/dx mx

m

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15

d/dx c

0

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16

d/dx x^n

n*x^(n-1)

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17

d/dx sqrt(x)

1/2sqrt(x)

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18

∫ m dx

mx+c

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19

∫ x dx

x^2/2

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20

∫ x^n dx

(1/n+1)x^(n+1)+c

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21

∫ x^-1 dx

ln|x|+c

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22

∫ 1/(ax+b) dx

(1/a)ln|ax+b|+c

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23

∫ e^x dx

e^x+c

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24

∫ cos(x) dx

sin(x)+c

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25

∫ sin(x) dx

-cos(x)+c

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26

∫ sec^2(x) dx

tan(x)+c

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27

∫ sec(x)tan(x) dx

sec(x)+c

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28

∫ csc(x)cot(x) dx

-csc(x)+c

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29

∫ csc^2(x) dx

-cot(x)+c

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30

∫ a^x dx

(a^x/ln(a))+c

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31

∫ sqrt(x) dx

(2/3)x^(3/2)+c

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32

slope of line tangent to y=f(x) at x=a

m = f’(a)

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33

tan line formula at x=a

y=f(a)+f´(a)(x-a)

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34

(cf´(x))

cf´(x)

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35

(f+/- g)´

f´(x)+/-g´(x)

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36

product rule (fg)´

f´g+fg´

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37

quotient rule (f/g)´

lo**dhi-hi**dlow/lolo

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38

power rule (x^n)´

(n)x^(n-1)

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39

chain rule (f(g(x)))

f´(g(x))g´(x)

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40

sin(0)

0

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41

cos(0)

1

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42

sin(π/6)

1/2

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43

cos(π/6)

sqrt(3)/2

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44

sin(π/4)

sqrt(2)/2

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45

cos(π/4)

sqrt(2)/2

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46

sin(π/3)

sqrt(3)/2

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47

cos(π/3)

1/2

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48

sin(π/2)

1

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49

cos(π/2)

0

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50

sin(2π/3)

sqrt(3)/2

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51

cos(2π/3)

-1/2

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52

sin(3π/4)

sqrt(2)/2

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53

cos(3π/4)

-sqrt(2)/2

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54

sin(5π/6)

1/2

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55

cos(5π/6)

-sqrt(3)/2

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56

sin(π)

0

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57

cos(π)

1

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58

sin(7π/6)

-1/2

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59

cos(7π/6)

-sqrt(3)/2

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60

sin(5π/4)

-sqrt(2)/2

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61

cos(5π/4)

-sqrt(2)/2

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62

sin(4π/3)

-sqrt(3)/2

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63

cos(4π/3)

-1/2

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64

sin(3π/2)

-1

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65

cos(3π/2)

0

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66

sin(5π/3)

-sqrt(3)/2

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67

cos(5π/3)

1/2

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68

sin(7π/4)

-sqrt(2)/2

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69

cos(7π/4)

sqrt(2)/2

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70

sin(11π/6)

-1/2

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71

cos(11π/6)

sqrt(3)/2

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72

AROC formula

(f(b)-f(a))/(b-a)

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73

IROC formula

(f(a+h)-f(a))/h

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74

tan line overestimates if

f is concave down, f´ is decreasing, f´´ is negative

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75

tan line underestimates if

f is concave up, f´ is increasing, f´´ is positive

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76

average value formula

1/(b-a)∫ a to b f(x)dx

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77

mean value theorem

if f(x) is continuous and differentiable on the open interval (a,b) then there is a number c so f´(c)=(f(b)-f(a))/(b-a)

the slope of the tan line and sec line are equal

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78

rel min in f

f´ sign change - to +, f´´ is positive

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79

rel max in f

f´ sign change + to -, f´´ is negative

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80

f is increasing

f´ is positive

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81

f is decreasing

f´ is negative

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82

POI (concavity change) in f

f´ has rel extrema, f´´ changes sign

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83

f is concave up

f´ is increasing, f´´ is positive

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84

f is concave down

f´ is decreasing, f´´ is negative

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85

f has saddle point

f´=0, f´´=0 or DNE

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86

f has cusp

f´ DNE, f´´ DNE

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87

LRAM underestimates when

f is increasing, f´ is positive

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88

LRAM overestimates when

f is decreasing, f´ is negative

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89

RRAM overestimates when

f is increasing, f´ is positive

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90

RRAM underestimates when

f is decreasing, f´ is negative

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91

trapezoidal sums underestimate when

f is concave down, f´ is decreasing, f´´ is negative

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92

trapezoidal sums overestimate when

f is concave up, f´ is increasing, f´´ is positive

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93

position

x(t), location of an object at time t

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94

velocity

v(t), rate of change in position

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95

acceleration

a(t), rate of change in velocity

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96

derivative of position

velocity x´(t)=v(t)

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97

derivative of velocity

acceleration v´(t)=a(t)

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98

2nd derivative of position

acceleration x´´(t)=a(t)

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99

antiderivative of acceleration

total change in velocity ∫a(t)dt=v(t)+c

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100

antiderivative of velocity

total change in position ∫v(t)dt=x(t)+c

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