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5.0(1)

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Hint

1

d/dx (x^n)

nx^(n-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 [cot x]

-csc^2(x)

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6

d/dx [sec x]

sec(x)tan(x)

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7

d/dx [csc x]

-csc(x)cot(x)

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8

d/dx [ln u] (use option-8)

1/u•du/dx

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9

d/dx [e^u] use option-8

e^u•du/dx

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10

d/dx [arcsin x]

1/(1-x^2)^0.5

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11

d/dx [arccos x]

-1/(1-x^2)^0.5

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12

d/dx [arctan x]

1/(1+x^2)

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13

d/dx [arccot x]

-1/(1+x^2)

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14

d/dx [arcsec x]

1/[|x|(x^2-1)^0.5]

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15

d/dx [arccsc x]

-1/[|x|(x^2-1)^0.5]

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16

d/dx [a^x] use option-8

a^x•ln(a)

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17

d/dx [log base "a" of (x)]

1/[xln(a)]

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18

Chain rule of d/dx[f(u)] use option-8

f'(u)•du/dx

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19

product rule of d/dx • [uv]

u'v+uv'

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20

quotient rule of d/dx • [f/g]

(gf'-fg')/g^2

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21

Extrema

The biggest/smallest y-values over a certain range

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22

(if you typed something like it, then override quizlet and say you were correct)

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23

Intermediate Value Theorem (IVT)

If f(x) is continuous on [a, b] then and N is a value between f(a) and f(b), then there must be a value "c" in (a, b) such that f(c) = N

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24

Mean Value Theorem (MVT)

If f(x) is continuous on [a, b] and differentiable on (a, b) then there must be "c" in (a, b) such that f'(c) = [f(b)-f(a)]/(b-a)

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25

Rolle's Theorem

If f(x) is continuous on [a, b] and differentiable on (a, b) and f(a) = f(b), then there is at least one critical number in (a, b)

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26

Is an Extrema an x or y value

y-value

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27

critical number

the x-values where extrema occur

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28

critical point

(critical number, extrema) in (x,y) format

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29

Absolute Maximum

Largest y-value in the entire function

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30

Absolute Minimum

smallest y-value in the entire function

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31

Local/Relative Maximum

Largest y-value in a certain area of the function marked by bounds

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32

Local/Relative Minimum

Smallest y-value in a certain interval of the function marked by bounds

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33

Where do critical numbers occur?

endpoints of interval, f'(x)=0, f'(x) does not exist

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34

You forgot something on your indefinite integral. What is it?

+C

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35

You forgot something on your indefinite and definite integral. What was it?

dx

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36

Concave up like a cup is when

f'(x) is increasing

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37

Concave down like a frown is when

f'(x) if decreasing

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38

inflection points occur when

concavity changes

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39

Law of Sines with angles A, B, and C and corresponding sides a, b, and c

[sin (A)]/a = [sin (B)]/b = [sin (C)]/c

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40

Law of Cosines with angles A,B, and C and sides a, b, and c

c^2 = a^2+b^2-2ab•cos(C) *note* you can do this for any angle as long as it corresponds with the opposite side

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41

L'Hospital's Rule (pronounced Lo-Pay-Tall) -- option-5 keyboard shortcut and shift - option +

If the limit as x approaches a of f(x)/g(x) = 0/0, ±∞/±∞, 1^∞, of ∞^0, then it becomes the limit as x approaches a of f'(x)/g'(x)

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42

derivatives are

tangent slopes

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43

definite integrals model for

area under a curve

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44

integral of a rate (ie: integral of v(t)) =

total amount (ie: in the example, it would be the total distance

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45

local minimums occur where f'(x) goes

(-,0,+) or (-, undefined, +)

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46

local minimums also occur where f''(x) is

greater than zero (up like a cup)

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47

local maximums occur where f'(x) goes

(+,0,-) or (+, undefined, -)

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48

local maximums also occur where f''(x) is

less than zero (down like a frown)

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49

points of inflection (concavity changes) occur when f''(x) goes

(+,0,-), (-,0,+), (+, undefined, -), or (-, undefined, +)

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50

interval of continuity for a polynomial

(-∞,∞)

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51

interval of continuity for f(x) = ln(x)

(0,∞)

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52

interval of continuity for f(x) = e^x

(-∞,∞)

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53

interval of continuity for f(x) = arctan(x)

(-∞,∞)

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54

interval of continuity for f(x) = x^0.5

[0,∞)

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55

interval of continuity for f(x) = sin(x)

(-∞,∞)

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56

interval of continuity for f(x) = cos(x)

(-∞,∞)

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57

factor a^2 - b^2 (difference of squares)

(a-b)(a+b)

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58

factor a^3 - b^3 (difference of cubes)

(a-b)(a^2+ab+b^2)

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59

factor a^3 + b^3 (sum of cubes)

(a+b)(a^2-ab+b^2)

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60

range of arcsin (use option - p)

[-π,π]

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61

range of arccos

[0,π]

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62

range of arctan

[-π,π]

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63

It is an even function if

f(x) = f(-x)

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64

It is an odd function if

f(-x) = -f(x)

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65

is sin(x) even or odd

odd

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66

is cos(x) even or odd?

even

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67

Fun fact that you should know on related rates (generally distance problems), the minimums and maximums are the same for

f(x) and [f(x)]^0.5 Now you can differentiate without the evil radical in the denominator.

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68

volume of a sphere

(4πr^3)/3

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69

surface area of a sphere

4πr^2

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70

volume of a cone (bonus hint for related rates problems-- use similar triangles to relate the radius and height)

π•r^2•h/3

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71

volume of a cylinder

π•r^2•h

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72

area of an equilateral triangle in terms of side s

3^0.5•s^2/4 (root 3 times s squared over 4-- useful in integration volume problems)

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73

what is revenue (R)

how much an item sells for

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74

what is cost (C)

how much it takes to make an item

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75

what is profit (P)

net gain from selling an item

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76

Profit formula (Cost C, profit P, and revenue R)

P=R-C

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77

revenue formula (related rates/optimization)

(units sold)(price per unit)

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78

average cost formula (denoted with C with a line above it)

total cost/units sold

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79

marginal revenue

the derivative of the revenue function

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80

marginal profit

the derivative of the profit function

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81

marginal cost

the derivative of the cost function

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82

what does marginal revenue represent

approximately the revenue generated from selling one more item

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83

what does marginal profit represent

approximately the profit of selling one more item

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84

what does marginal cost represent

approximately the cost of making one more item

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85

| a | equals

a or -a

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86

| ab | =

|a|•|b|

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87

|a-b| =

|b-a|

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88

(a^2)^0.5 =

|a|

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89

(f+g)(x) =

f(x) + g(x)

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90

(f-g)(x)

f(x)-g(x)

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91

(fg)(x) =

f(x)-g(x)

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92

(f/g)(x) =

f(x)/g(x)

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93

Arsec(x) in terms of arccos so you can do it on a calculator

arccos(1/x)

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94

Arccsc x in terms of arcsin so you can do it on a calculator

arsin(1/x)

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95

Arccot x in terms of arctan

π/2 - arctan x

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96

Log base a of x in terms of ln

ln(x)/ln(a)

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97

double angle sin(2x) =

2sinxcosx

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98

cos(2x) =

1-2sin^2x = cos^2 (x) - sin^2 (x)

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99

tan(2x) =

2tanx/(1-tan^2x)

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100

half angle sin(x/2) =

± square root of (1-cosx)/2

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