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CHAPTER 5 SECTION 5.4 EXPONENTIAL FUNCTIONS: DIFFERENTIATION AND INTEGRATION PowerPoint PPT Presentation


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CHAPTER 5 SECTION 5.4 EXPONENTIAL FUNCTIONS: DIFFERENTIATION AND INTEGRATION. Definition of the Natural Exponential Function. Recall: . This means…. and…. Exponential and log functions are interchangeable. Start with the base. Change of Base Theorem. Solve. Solve.

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CHAPTER 5 SECTION 5.4 EXPONENTIAL FUNCTIONS: DIFFERENTIATION AND INTEGRATION

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Chapter 5 section 5 4 exponential functions differentiation and integration l.jpg

CHAPTER 5SECTION 5.4EXPONENTIAL FUNCTIONS:DIFFERENTIATION AND INTEGRATION


Definition of the natural exponential function l.jpg

Definition of the Natural Exponential Function


Slide3 l.jpg

Recall:

This means…

and…

Exponential and log functions are interchangeable.

Start with the base.

Change of Base Theorem


Solve l.jpg

Solve.


Solve5 l.jpg

Solve.

We can’t take a log of -1.


Theorem 5 10 operations with exponential functions l.jpg

Theorem 5.10 Operations with Exponential Functions


Properties of the natural exponential function l.jpg

Properties of the Natural Exponential Function


Theorem 5 11 derivative of the natural exponential function l.jpg

Theorem 5.11 Derivative of the Natural Exponential Function


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5.4 Exponential Functions

  • Example 3: Find dy/dx:


Slide10 l.jpg

5.4 Exponential Functions

  • Example 3 (concluded):


Find each derivative l.jpg

Find each derivative:


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5.4 Exponential Functions

  • THEOREM 2

  • or

  • The derivative of e to some power is the product of e

  • to that power and the derivative of the power.


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5.4 Exponential Functions

  • Example 4: Differentiate each of the following with

  • respect to x:


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5.4 Exponential Functions

  • Example 4 (concluded):


Find each derivative15 l.jpg

Find each derivative


Theorem l.jpg

Theorem:

1.Find the slope of the line tangent to f (x) at x= 3.


Theorem17 l.jpg

Theorem:

1.Find the slope of the line tangent to f (x) at x= 3.


4 find extrema and inflection points for l.jpg

4.Find extrema and inflection points for


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4.Find extrema and inflection points for

Crit #’s:

Crit #’s:

Can’t ever work.

none


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Intervals:

Test values:

f ’’(test pt)

f(x)

f ’(test pt)

f(x)

rel max

rel min

Inf pt

Inf pt


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5.4 Exponential Functions

  • Example 7: Graph with x≥ 0. Analyze the graph using calculus.

  • First, we find some values, plot the points, and sketch

  • the graph.


Slide22 l.jpg

  • Example 4 (continued):

  • a) Derivatives. Since

  • b) Critical values. Since the derivative

    for all real numbers x. Thus, the

  • derivative exists for all real numbers, and the equation

  • h(x) = 0 has no solution. There are no critical values.


Slide23 l.jpg

  • Example 4 (continued):

  • c) Increasing. Since the derivative for all real numbers x, we know that h is increasing over the entire real number line.

  • d) Inflection Points.Since we know that the equation h(x) = 0 has no solution. Thus there are no points of inflection.


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5.4 Exponential Functions

  • Example 4 (concluded):

  • e) Concavity. Since for all real numbers x, h’ is decreasing and the graph is concave down over the entire real number line.


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  • Example 4 (continued):


Theorem 5 12 integration rules for exponential functions l.jpg

Theorem 5.12 Integration Rules for Exponential Functions


Theorem27 l.jpg

Theorem:


Theorem28 l.jpg

Theorem:


Ap question l.jpg

AP QUESTION


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Why is x = -1/2 the only critical number???????


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AP QUESTION


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