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CalculusChapter 5 Day 1

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The Natural Logarithmic Function and DifferentiationThe Natural Logarithmic Function- The number e- The Derivative of the Natural Logarithmic Function

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The Natural Logarithmic Function

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Definition of the Natural Logarithmic Function

▪ The Natural Logarithmic function is defined by

The domain of the natural logarithmic function is the set of all positive real numbers

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Properties of the Natural Logarithmic Function

▪ The natural logarithmic function has the following properties:1. The domain is and the range is 2. The function is continuous, increasing,

and one-to-one3. The graph is concave downward

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Logarithmic Properties

▪ If and are positive numbers and is rational, then the following properties are true

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The number

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Definition of

▪ The letter denotes the positive real number such that

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Derivative of the Natural Logarithmic Function

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Derivative of the Natural Logarithmic Function

▪ Let be a differentiable function of .

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Derivative Involving Absolute Value

▪ If is a differentiable function of such that then

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The Natural Logarithmic Function and IntegrationLog Rule for Integration- Integrals of Trigonometric Functions

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Log Rule for Integration

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Log Rule for Integration

▪ Let be a differentiable function of .

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Practice

▪Click Here for Assignment 5-1:

Assignment 5-1

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Integrals of Trigonometric Functions

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Integrals of the Six Basic Trigonometric Functions

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Inverse FunctionsInverse Functions- Existence of an Inverse Function- Derivative of an Inverse Function

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Inverse Functions

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Definition of an Inverse Function

▪ A function is the inverse of the function if

for each in the domain of and

for each in the domain of .

The function is denoted by (read as “”)

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Reflective Property of Inverse Functions

▪The graph of contains the point if and only if the graph of contains the point .

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Existence of an Inverse Function

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The Existence of an Inverse Function

1.A function has an inverse if and only if it is one-to- one

2.If is strictly monotonic on its entire domain, then it is one-to-one and therefore has an inverse

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Guidelines for Finding the Inverse of a Function

1.Use the Existence of an Inverse to determine whether the function given by has an inverse

2.Solve for as a function of

3.Interchange and . The resulting equation is

4.Define the domain of to be the range of

5.Verify that and

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Derivative of an Inverse Function

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Continuity and Differentiability of Inverse Functions

▪ Let be a function whose domain is an interval . If has an inverse, then the following statements are true.

1. If is continuous on its domain, then is continuous on its domain

2. If is increasing on its domain, then is increasing on its domain

3. If is decreasing on its domain, then is decreasing on its domain

4. If is differentiable at and , then is differentiable at .

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The Derivative of an Inverse Function

▪ Let be a function that is differentiable on an interval . If has an inverse function , then is differentiable at any for which . Moreover,

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Exponential Functions: Differentiation and IntegrationThe Natural Exponential Function- Derivatives of Exponential Functions- Integrals of Exponential Functions

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The Natural Exponential Function

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Definition of the Natural Exponential Function

▪ The Inverse of the natural logarithmic function is called the natural exponential function and is denoted by

That is,

if and only if

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Inverse Relationships

and

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Operations with Exponential Functions

▪ Let and be any real numbers

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Properties of the Natural Exponential Function

1. The domain of is , and the range is .

2. The function is continuous, increasing, and one-to-one on its entire domain

3. The graph of is concave upward on its entire domain

4. and

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Derivatives of Exponential Functions

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The Derivative of the Natural Exponential Function

▪ Let be a differentiable function of .

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Integrals of Exponential Functions

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Integration Rules for Exponential Functions

▪ Let be a differentiable function of .

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Practice

▪Click Here for Assignment 5-2

Assignment 5-2

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Bases Other Than and ApplicationsBases Other Than - Differentiation and Integration- Applications of Exponential Functions

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Bases Other Than

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Definition of Exponential Function to Base

▪ If is a positive real number and is any real number, then the exponential function to the base is denoted by and is defined by

If , then is a constant function.

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Definition of Logarithmic Function to Base

▪ If is a positive real number and is any positive real number, then the logarithmic function to the base is denoted by and is defined as

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Properties of Inverse Functions

1. if and only if

2. 3.

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Differentiation and Integration

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Derivatives for Bases Other than

▪ Let be a positive real number and let be a differentiable function of .

1.

2.

3.

4.

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Integration

∫𝑎𝑥𝑑𝑥=( 1ln𝑎 )𝑎𝑥+𝐶

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The Power Rule for Real Exponents

▪ Let be any real number and let be a differentiable function of .

1. 2.

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Applications of Exponential Functions

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A Limit Involving

lim𝑥→∞ (1+ 1𝑥 )

𝑥

= lim𝑥→∞ ( 𝑥+1

𝑥 )𝑥

=𝑒

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Summary of Compound Interest Formulas

▪ Let amount of deposit, number of years, balance after years, annual interest rate (decimal form), and the number of compoundings per year.

1. Compoundings times per year:

2.Compounded Continuously:

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Practice

▪Click Here for Assignment 5-3

Assignment 5-3