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4.5(6)

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Hint

1

Y=f(x) must be continuous at each:

-Critical Point or undefined and endpoints

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2

Local Minimum

Goes (-,0,+) or (-, und, +)

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3

Local Maximum

Goes (+,0,-) or (+, und, -)

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4

Point of Inflection

Concavity Changes

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

(+,und,-) or (-,und,+)

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5

D/dx(sinx)

Cosx

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6

D/dx(cosx)

-sinx

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7

D/dx(tanx)

Sec²x

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8

D/dx(cotx)

-csc²x

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9

D/dx(secx)

Secxtanx

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10

D/dx(cscx)

-cscxcotx

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11

D/dx(lnx)

1/x

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12

D/dx(ln(n))

1/n

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13

D/dx(eⁿ)

Eⁿ

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14

∫Cosx

Sinx

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15

∫-sinx

Cosx

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16

∫Sec²x

Tanx

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17

∫-csc²x

Cotx

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18

∫Secxtanx

Secx

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19

∫-cscxcotx

Cscx

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20

∫1/n

Ln(n)

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21

∫Eⁿ

Eⁿ

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22

When doing integrals never forget

Constant (+c)

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23

∫Axⁿ

A/n+1(xⁿ⁺¹)+C

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24

∫Tanx

Ln|secx|+c

Ln|cosx|+c

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25

∫Secx

Ln|secx+tanx|+c

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26

D/dx(sin⁻¹u)

1/√1-u²

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27

D/dx(cos⁻¹x)

-1/√1-x²

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28

D/dx(tan⁻¹x)

1/1+x²

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29

D/dx(cot⁻¹x)

-1/1+x²

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30

With derivative inverses

You plug in the number of the trigonometric function into x

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31

D/dx(sec⁻¹x)

1/|x|√x²-1

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32

D/dx(csc⁻¹x)

-1/|x|√x²-1

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33

D/dx(aⁿ)

Aⁿln(a)

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34

D/dx(Logₙx)

1/xln(a)

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35

Chain Rule

Take derivative of outside of parenthesis

Take derivative of inside parenthesis and keep the original of what was in the parenthesis

For example, sin(x²+1)→ 2xcos(x²+1)

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36

Product Rule

d/dx first times second + first times d/dx second

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37

Quotient Rule

LoDHi-HiDLo/LoLo

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38

Fundamental Theorem of Calculus

∫(a to b) f(x) dx = F(b) - F(a)

Basically saying that F’(x)=f(x)

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39

*f* relative max→*f* ‘ goes from

Positive to negative

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40

*f* relative min→*f* ‘ goes from

Negative to positive

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41

Intermediate Value Theorem

If I pick an X value that is included on a continuous function, I will get a Y value, within a certain range, to go with it.

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42

Mean Value Theorem

If function is continuous on [a,b] and first derivative exist on interval (a,b), then there is at least one number (c) such that f’(c)=f(b)-f(a)/b-a

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43

Rolle’s Theorem with Mean Value Theorem

If function is continuous on [a,b] and first derivative exist on interval (a,b), and f(a)=f(b) then there is at least one number x=c such that f’(c)=0

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44

Trapezoidal Rule

The average of left hand point method and right hand point method

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45

Theorem of mean value and average value

1/b-a ∫f(x)dx which is the average value

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46

Disk Method

V=π∫[R(x)]²dx

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47

Washer Method

π∫(R(x))²-(r(x))²dx

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48

General volume equation

V=∫area(x)dx

-If no shape is given, then you just take the integral of the function

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49

Distance, Velocity, and Acceleration

Velocity is d/dx of position

Acceleration is d/dx of velocity

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50

Displacement

∫(Velocity)dt

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51

Distance

∫|Velocity|dt

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52

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53

Speed

The absolute value of velocity

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54

Average Velocity

Final Position-Initial Position/Total time

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55

cos(0)

1

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56

sin(0)

0

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57

Pie circle starts with

π/6

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58

tan(π/3)

√3

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59

tan(π/6)

√3/3

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60

tan(90⁰)

Undefined

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61

Double Argument

sin2x=2sinxcosx

cos2x=cos²x-sin²x=1-2sin²x

cos²x=1/2(1+cos2x)

sin²x=1/2(1-cos2x)

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62

Pythagorean Identity

sin²x+cos²x=1

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63

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64

L’Hopital Rule

If limit equals 0/0 or ∞/∞, then you can take the derivative

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