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AP Calculus AB Chapter 5 Memory Sheet Check (Mr. Wong & Mr. Baker)

1

**Definition of Derivative** - Meaning of Derivative:

instantaneous rate of change

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2

**Definition of Derivative** - Numerical Interpretation

Limit of the average rate of change over the interval from ** c** to

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3

**Definition of Derivative** - Geometrical Interpretation of Derivative

Slope of the tangent line

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4

**Definition of Definite Integral** - Meaning of Definite Integral

Product of (** b**-

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5

**Definition of Definite Integral** - Geometrical Interpretation of Definite Integral

Area under the curve between **a** and **b**

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6

Verbal Definition of Limit

** L** is the limit of

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7

**The Limit Theorems** (provided lim *x→c* *f*(x) and the lim *x→c* *g*(x) exists) - Limit of a Product of Functions

lim x→c [** f**(x) * g(x)] = lim

The limit of a product equals the product of its limits.

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8

**The Limit Theorems** (provided lim *x→c* ** f**(x) and the lim

**lim** ** x→c** [

The limit of a sum equals the sum of its limits.

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9

**The Limit Theorems** (provided lim *x→c* ** f**(x) and the lim

**lim** **x →c** [

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10

**The Limit Theorems** (provided lim *x→c* ** f**(x) and the lim

**lim** ** x→c** [k *

The limit of a constant times a function equals the constant times its limit.

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11

**The Limit Theorems** (provided lim *x→c* ** f**(x) and the lim

**lim** *x→c***x=c** The limit of x as x approaches c is c.

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12

**The Limit Theorems** (provided lim *x→c* ** f**(x) and the lim

If **k** is a constant, then **lim** *x→***k = k** The limit of the constant is the constant.

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13

Property of Equal Left and Right Limits

**lim** ** x→c f**(x) exists if and only if

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14

Definition of Continuity at a Point

(c)*f***exists****lim**(x) exists, and*x→c f***lim**(x) =*x→>c f*(c)*f*

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15

Horizontal Asymptote

If **lim** ** x→∞ f**(x) = L or

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16

Vertical Asymptote

If **lim** ** x→c f**(x) = ∞ or

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17

Intermediate Value Theorem (IVT)

If ** f** is continuous for all

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18

Definition of a Derivative at a Point (**x=c form**)

*f***‘**(c) = lim ** x→c** [

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19

Definition of Derivative at a Point (**Δx** or **h form**)

*f***'**(x) = lim *Δx→0***Δy/Δx = lim** ** Δx→0** [

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20

Power Rule

If ** f** (x) = xⁿ, where

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21

**Properties of Differentiation** - Derivative of a Sum of Functions

If ** f**(x) =

The derivative of the Sum equals the Sum of the derivatives.

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22

**Properties of Differentiation** - Derivative of a Constant Times a Function

If ** f**(x) = k *

The derivative of a constant times a function equals the constant times the derivative

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23

**Properties of Differentiation -** Derivative of a Constant Function

If ** f**(x) =

The derivative of a constant is 0.

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24

**Chain Rule (dy/dx form)**

dy/dx = dy/du * du/dx

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25

Chain Rule (** f**(x) form)

[** f**(

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26

**Limit of (sin x) / x**

lim *x→0* **sin x/x** = **1**

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27

**Relationship between a Graph and its Derivatives Graph** - Increasing/Decreasing

** f** is increasing when

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28

**Relationship between a Graph and its Derivatives Graph** - Local maximum( or relative maximum)

occurs when *f***’**(x) changes from positive to negative at *x***=** ** c**.

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29

**Relationship between a Graph and its Derivatives Graph** - Local minimum (or relative minimum)

occurs when *f***’**(x) changes from negative to positive at *x***=** ** c**.

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30

**The Calculus of Motion** - Velocity

dx/dt, where x is the displacement

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31

**The Calculus of Motion** - Acceleration

dv/dt = dx²/dt², where v is the velocity.

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32

**The Calculus of Motion** - Distance

[*displacement*]

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33

**The Calculus of Motion** - Speed

[*velocity*]

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34

**The Calculus of Motion** - Speeding Up

occurs when velocity and acceleration are the same sign

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35

**The Calculus of Motion** - Slowing Down

occurs when velocity and acceleration are the opposite signs

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36

**Equation of a Tangent Line**

The equation of the line tangent to the graph of ** f** at x = c is given by y =

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37

**Product Rule**

if y = *uv*, then y’ = *u’v* + *uv’*

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38

**Quotient Rule**

if y = u/v, then y’ = [*u’v* - *uv’*]/*v*², (v ≠ 0)

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39

**Relationship between Differentiability and Continuity**

If ** f** is differentiable at

__Contrapositive__: If ** f** is not continuous at

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40

**Derivative of an Inverse Function**

The derivative of *f***⁻¹**(x) is 1/*f* **’**(y)

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41

**Derivative of Trig Functions** - d/dx(sin x)

*cos* x

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42

**Derivative of Trig Functions** - d/dx(cos x)

-*sin* x

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43

**Derivative of Trig Functions** - d/dx(tan x)

*sec*² x

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44

**Derivative of Trig Functions** - d/dx(sec x)

sec x tan x

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45

**Derivative of Trig Functions** - d/dx(cot x)

-*csc*² x

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46

**Derivative of Trig Functions** - d/dx(csc x)

-csc x cot x

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47

**Derivative of Inverse Trig Functions** - d/dx(sin⁻¹ x)

1/*sqrt*(1-x²)

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48

**Derivative of Inverse Trig Functions** - d/dx(cos⁻¹ x)

-1/*sqrt*(1-x²)

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49

**Derivative of Inverse Trig Functions** - d/dx(tan⁻¹x)

1/(1+x²)

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50

**Derivative of Inverse Trig Functions** - d/dx(cot⁻¹ x)

-1/(1+x²)

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51

**Derivative of Inverse Trig Functions** - d/dx(sec⁻¹ x)

1/[|x|*sqrt*(x²-1)]

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52

**Derivative of Inverse Trig Functions** - d/dx(csc⁻¹ x)

-1/[|x|*sqrt*(x²-1)]

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53

__Integral of a Constant times a Function__

If k is a constant, then **∫** k * *f* (x)**dx** = k * **∫** *f*(x)**dx**

The integral of a constant times a function equals the constant times the integral.

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54

**Integral of a Sum of Functions**

∫[f(x) + g(x)]dx = ∫f(x)dx + ∫g(x)dx.

The integral of a sum equals the sum of the integrals.

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55

**Derivative and Integrals of Logarithmic and Exponential Graphics** - d/dx(ln(x))

1/x

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56

**Derivative and Integrals of Logarithmic and Exponential Graphics** - ∫ 1/x dx

ln|x| + C

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57

**Derivative and Integrals of Logarithmic and Exponential Graphics** - d/dx(e^x)

e^x

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58

**Derivative and Integrals of Logarithmic and Exponential Graphics** - ∫ e^x dx

e^x + C

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59

**Derivative and Integrals of Logarithmic and Exponential Graphics** - d/dx(log b X)

1/ln(b) * 1/x

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60

**Derivative and Integrals of Logarithmic and Exponential Graphics** - d/dx(b^x)

b^x * ln(b)

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61

**Derivative and Integrals of Logarithmic and Exponential Graphics** - ∫ b^x dx

( 1/ln(b) * b^x ) + C

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62

**Integrals that Yield Trig Functions** - ∫ cos x dx

sin x + C

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63

**Integrals that Yield Trig Functions** - ∫ sin x dx

-cos x + C

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64

**Integrals that Yield Trig Functions** - ∫ sec² x dx

tan x + C

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65

**Integrals that Yield Trig Functions** - ∫ csc² x dx

-cot x + C

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66

**Integrals that Yield Trig Functions** - ∫ sec x tan x dx

sec x + C

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67

**Integrals that Yield Trig Functions** - ∫ csc x cot x dx

-csc x + C

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68

**Integrals that Yield Inverse Trig Functions** - ∫ 1/sqrt(1-x²) dx

sin⁻¹ x + C

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69

**Integrals that Yield Inverse Trig Functions** - ∫ -1/sqrt(1-x²) dx

cos⁻¹ x + C

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70

**Integrals that Yield Inverse Trig Functions** - ∫ 1/(1+x²) dx

tan⁻¹ + C

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71

**Integrals that Yield Inverse Trig Functions** - ∫ -1/(1+x²) dx

cot⁻¹ x + C

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72

**Integrals that Yield Inverse Trig Functions** - ∫ 1/[|x|sqrt(x²-1)] dx

sec⁻¹ x + C

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73

**Integrals that Yield Inverse Trig Functions** - ∫ -1/[|x|sqrt(x²-1)] dx

csc⁻¹ x + C

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74

**Definition of Definite Integral**: a ^b∫ f(x)dx = lim Δx → 0 i=1 ^nΣ f(ci)Δx

__Verbally__

Limit of a Riemann Sum

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75

**Definition of Definite Integral**: a ^b∫ f(x)dx = lim Δx → 0 i=1 ^nΣ f(ci)Δx

__Meaning of Definite Integral__

Product of (*b*-*a*) and *f* (x)

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76

**Definition of Definite Integral**: a ^b∫ f(x)dx = lim Δx → 0 i=1 ^nΣ f(ci)Δx

__Geometrical Interpretation of Definite Integral__

Area under the curve between *a* and *b*

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77

**Definition of Integrability**

*f* (x) is integrable on an interval if and only if *f* (x) is continuous on that interval

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78

**Mean Value Theorem**

If

*f*is differentiable on the open interval (a,b), and*f*is continuous on the close interval [a,b].then there is at least one number x = c (a,b) such that f’(c) = [f(b) - f(a)]/(b-a)

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79

**Rolle’s Theorem**

If

*f*is differentiable on the open interval (a, b), and*f*is continuous on the closed interval [a, b], andf(a) = f(b) = 0

then there is at least one number x = c (a, b) such that f’(c) = 0

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80

**Fundamental Theorem of Calculus**

If g(x) = ∫ f(x) dx, then a ^b∫ f(x) dx = g(b) - g(a)

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81

**Second Fundamental Theorem of Calculus**

If g(x) = a ^x∫ f(t) d(x), where a is a constant, then g’(x) = f(x)

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82

**Properties of Definite Integrals** - __Reversal Of Limits__

a ^b∫ f(x) dx = -a ^b∫ f(x) dx

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83

**Properties of Definite Integrals** - __Sum of Integrals with Same Integrand__

a ^b∫ f(x) dx = a ^c∫ f(x) dx + c ^b∫f(x) dx

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84

**Properties of Definite Integrals** - __Symmetric Limits__

If *f* is odd , -a ^a∫ f(x) dx = 0. If *f* is even, then -a ^a∫ f(x) dx = 2 [0 ^a ∫f(x) dx]

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85

**Properties of Definite Integrals** - __Integral of a Sum of Functions__

a ^b∫[f(x) + g(x)] dx = a^b∫ f(x) dx + a^b∫ g(x) dx

The integral of a sum equals the sum of the integrals.

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86

**Properties of Definite Integrals** - __Integral of a Constant times a Function__

a ^b∫ k *f* (x) dx = k [a ^b∫ f(x) dx]

The integral of a constant times a function is the constant times the integral.

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