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CHAPTER 5 SECTION 5.4 EXPONENTIAL FUNCTIONS: DIFFERENTIATION AND INTEGRATIONPowerPoint Presentation

CHAPTER 5 SECTION 5.4 EXPONENTIAL FUNCTIONS: DIFFERENTIATION AND INTEGRATION

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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

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CHAPTER 5SECTION 5.4EXPONENTIAL FUNCTIONS:DIFFERENTIATION AND INTEGRATION

Recall:

This means…

and…

Exponential and log functions are interchangeable.

Start with the base.

Change of Base Theorem

We can’t take a log of -1.

5.4 Exponential Functions

- Example 3: Find dy/dx:

5.4 Exponential Functions

- Example 3 (concluded):

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.

5.4 Exponential Functions

- Example 4: Differentiate each of the following with
- respect to x:

5.4 Exponential Functions

- Example 4 (concluded):

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

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

Crit #’s:

Crit #’s:

Can’t ever work.

none

Intervals:

Test values:

f ’’(test pt)

f(x)

f ’(test pt)

f(x)

rel max

rel min

Inf pt

Inf pt

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.

- 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.

- 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.

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.

- Example 4 (continued):

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