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CHAPTER 5SECTION 5.4EXPONENTIAL

FUNCTIONSDIFFERENTIATION 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.

We cant take a log of -1.

Theorem 5.10 Operations with Exponential Functions

Properties of the Natural Exponential Function

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)

Find each derivative

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

4. Find extrema and inflection points for

Crit s

Crit s

Cant ever work.

none

Intervals

Test values

f (test pt)

f(x)

f (test pt)

f(x)

rel min

rel max

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)

Theorem 5.12 Integration Rules for Exponential

Functions

Theorem

Theorem

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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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