THS Step By Step Calculus Chapter 5

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Name: ___________________________________

Class Period: _____________________________

Throughout this packet – there will be blanks you are expected to fill in prior to coming to class. This packet follows your

Larson Textbook. Do NOT throw away! Keep in 3 ring-binder until the end of the course.

Chapter 5.1 The Natural Logarithmic Function: Differentiation

Mathematician: ________________

Definition of Natural Logarithm Function

Properties of Logarithms

The Definition of e

THS Step By Step Calculus Chapter 5

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The Derivative of Natural Logarithmic Function

Why Logarithmic Differentiation???? Log Properties simplify derivative, have function in exponent Steps Logarithmic Differentiation

( )

√

Step 1: Write original equation ( )

√

Step 2: Take natural log of each side (( )

√ )

Step 3: Use logarithmic properties to expand ( )

( )

Step 4: Implicit Differentiate

( )

Step 5: Solve for y’ (

)

Step 6: If possible, substitute for y (( )

√ ) (

)

Step 7: Simplify ( )( )

( )

THS Step By Step Calculus Chapter 5

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5.2 The Natural Logarithmic Function: Integration

Log Rule for Integration

Alternate form of Log Rule: Integrals of Trig Functions:

∫

∫

∫ ∫

| |

∫ ∫

| |

∫ | |

∫ | |

THS Step By Step Calculus Chapter 5

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5.3 Inverse Functions Definition of Inverse:

Reflexive Properties

Existence of Inverse

Steps for finding inverse

Step 1: Verify an inverse exists

Step 2: Solve for ( ) ( )

Step 3: Exchange x and y ( ( ))

Step 4: Verify ( ( )) ( ( ))

Continuity and Differentiability of Inverse:

THS Step By Step Calculus Chapter 5

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Derivative of Inverse

( ) ( )

( ( ))

Steps for finding inverse value at a point

Find ( ) ( )

Step 1: Set ( )

Step 2: Solve for x x=2 Step 3: Identify Function (x, f(x)) Function: (2, 3) Step 4: Use Inverse Relationship to identify inverse Inverse point: (3, 2)

( ) Steps for finding derivative of an inverse function at a point x=c

Find ( ) ( ) ( )

Step 1: Calculate ( ) ( ) (Above)

Step 2: Calculate ( ) ( )

Step 3: Evaluate ( ) at ( ) ( ( )) ( )

( )

Step 4: Evaluate ( )

( ( )) ( ) ( )

THS Step By Step Calculus Chapter 5

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5.4 Exponential Function: Differentiation and Integration

Definition of the Natural Exponential Function

Inverse: ( ) Properties:

Derivative of exponential

Integral of exponential

THS Step By Step Calculus Chapter 5

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5.5 Bases Other than e

Definition of Exponential Function to Base a

Definition of Logarithmic Function to Base a

Properties of Inverse Functions

Derivatives

Integral

∫ (

) ∫ (

)

Limits involving e

Compound Interest:

P = amount of deposit T=number of years A=balance after t years R=annual interest rate (decimal form) N=number of compoundings per year

Compounded n times per year: (

)

Compounded continuously:

THS Step By Step Calculus Chapter 5

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5.6 Inverse Trigonometric Functions: Differentiation Definition of Inverse Trig Functions

THS Step By Step Calculus Chapter 5

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Properties of Inverse Trig Functions

Derivatives of Inverse Trig Functions

THS Step By Step Calculus Chapter 5

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5.7 Inverse Trig Functions: Integration

Integrals involving inverse trigonometric functions

Techniques needed: