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

1 / 54

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.

I am the owner, or an agent authorized to act on behalf of the owner, of the copyrighted work described.

CHAPTER 5 SECTION 5.4 EXPONENTIAL FUNCTIONS: DIFFERENTIATION AND INTEGRATION

Download Policy: Content on the Website is provided to you AS IS for your information and personal use and may not be sold / licensed / shared on other websites without getting consent from its author.While downloading, if for some reason you are not able to download a presentation, the publisher may have deleted the file from their server.

- - - - - - - - - - - - - - - - - - - - - - - - - - E N D - - - - - - - - - - - - - - - - - - - - - - - - - -

## CHAPTER 5SECTION 5.4EXPONENTIAL FUNCTIONS:DIFFERENTIATION AND INTEGRATION

### Definition of the Natural Exponential Function

Recall:

This means…

and…

Exponential and log functions are interchangeable.

Change of Base Theorem

### Solve.

We can’t take a log of -1.

### Theorem 5.11 Derivative of the Natural Exponential Function

5.4 Exponential Functions

• Example 3: Find dy/dx:

5.4 Exponential Functions

• Example 3 (concluded):

### Find each derivative:

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):

### Theorem:

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

### Theorem:

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

### 4.Find extrema and inflection points for

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):

### AP QUESTION

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