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

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 5SECTION 5.4EXPONENTIAL FUNCTIONS:DIFFERENTIATION AND INTEGRATION

This means…

and…

Exponential and log functions are interchangeable.

Change of Base Theorem

We can’t take a log of -1.

• Example 3: Find dy/dx:

• Example 3 (concluded):

• THEOREM 2

• or

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

• to that power and the derivative of the power.

• Example 4: Differentiate each of the following with

• respect to x:

• 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

Test values:

f ’’(test pt)

f(x)

f ’(test pt)

f(x)

rel max

rel min

Inf pt

Inf pt

• 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.

• 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.