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

CHAPTER 5SECTION 5.4EXPONENTIAL FUNCTIONS:DIFFERENTIATION AND INTEGRATION



Slide3 l.jpg

Recall:

This means…

and…

Exponential and log functions are interchangeable.

Start with the base.

Change of Base Theorem



Solve5 l.jpg
Solve.

We can’t take a log of -1.





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5.4 Exponential Functions

  • Example 3: Find dy/dx:


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5.4 Exponential Functions

  • Example 3 (concluded):



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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):



Theorem l.jpg
Theorem:

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


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Theorem:

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



4 find extrema and inflection points for19 l.jpg
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.


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  • 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.


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  • 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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