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5.0(1)

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Hint

1

Average value of a function f(x)

1/(b-a) * ∫ f(x) dx from a to b

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2

Average rate of change of f(x) from a to b

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

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3

Polar to rectangular coordinates: x=?

rcosθ

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4

Polar to rectangular coordinates: y=?

rsinθ

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5

Rectangular to polar coordinates: r=?

v(x²+y²)

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6

Rectangular to polar coordinates: θ=?

arctan(y/x)

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7

Area of a sector of a circle

½θr²

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8

Area of a region defined by a polar function

½∫r²dθ from θ1 to θ2

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9

Area between two polar curves

½ ∫r_{outer curve}²-r_{inner curve}²dθ from θ1 to θ2

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10

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11

Displacement of f(x)

∫f’(x)dx from a to b

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12

Distance of f(x)

∫|f’(x)|dx from a to b

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13

Speed of f(x)

|f’(x)|

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14

Arclength of f(x)

∫v(1+[f’(x)]²)dx from a to b

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15

Displacement of ⟨x(t), y(t)⟩

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16

Distance of ⟨x(t), y(t)⟩

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17

Speed of ⟨x(t), y(t)⟩

v((x’(t))²+(y’(t))²)

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18

Arc length of ⟨x(t), y(t)⟩

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19

∫x^a dx

[x^(a+1)] / (a+1) + c

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20

∫e^x dx

e^x + c

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21

∫lnx dx

xlnx - x + c

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22

∫cosx dx

sinx + c

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23

∫secxtanx dx

secx + c

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24

∫csc²x dx

-cotx + c

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25

∫secx dx

ln|secx + tanx| + c

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26

∫cscx dx

ln|cscx - cotx| + c

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27

∫1/(a²+x²) dx

1/a arctan(x/a) + c

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28

∫1/x dx

ln|x| + c

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29

∫a^x dx

a^x/lna + c

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30

∫sinx dx

-cos(x) + c

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31

∫sec²x dx

tanx + c

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32

∫cscxcotx dx

-cscx + c

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33

∫tanx dx

ln|secx| + c

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34

∫cotx dx

ln|sinx| + c

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35

∫1/v(a²-x²) dx

arcsin(x/a) + c

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36

∫1/xv(x²-a²) dx

1/a * arcsec(x/a) + c

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37

In the exponential model y=Ce^(Kx), what is y?

Final amount

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38

In the exponential model y=Ce^(Kx), what is C?

Initial amount

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39

In the exponential model y=Ce^(Kx), what is K?

Growth/decay constant

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40

In the exponential model y=Ce^(kx), what is x?

Time

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41

In the logistic model y = a/(1+be^(-aKt)), what is a?

Carrying capacity

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42

What does a geometric series look like?

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43

When does a geometric series diverge?

When |r| ≥ 1

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44

How do you find the sum that a geometric series converges to?

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45

Using the N-th term test, when does a series diverge?

If the limit as n approaches infinity of a_{n }does not equal 0

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46

What are the criteria for f(x) in the integral test for convergence?

f(x) must be positive, continuous, and decreasing

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47

When does a series converge according to the p series test?

When p>1

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48

When does a series diverge according to the p series test

When 0<p≤1

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49

What is it called when p=1 in a p series?

Harmonic series

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50

What does a p series look like?

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51

What is the first condition for an alternating series to converge?

The limit as n approaches infinity of a_{n} = 0

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52

What is the second condition for an alternation series to converge?

a_{n+1} ≤ a_{n} (a_{n} are non-increasing)

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53

How is the error for an estimate of an alternating series found?

error ≤ |a_{n+1}|

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54

sinx’ = ?

cosx

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55

cosx’ = ?

-sinx

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56

tanx’ = ?

sec²x

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57

secx’ = ?

secxtanx

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58

cotx’ = ?

-csc²x

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59

cscx’ = ?

-cscxcotx

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60

arcsinx’ = ?

1/v(1-x²)

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61

arccosx’ = ?

-1/v(1-x²)

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62

arctan’ = ?

1/(1+x²)

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63

arccotx’ = ?

-1/(1+x²)

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64

arcsecx’ = ?

1/(|x|v(x²-1))

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65

arccscx’

-1/(|x|v(x²-1))

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66

lnx’ = ?

1/x

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67

log_{a}x’ = ?

1/xlna

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68

e^{x}’ = ?

e^{x}

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69

a^{x}’ = ?

a^{x}lna

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