CHAPTER 5 SECTION 5 4 EXPONENTIAL FUNCTIONS DIFFERENTIATION

  • Slides: 54
Download presentation
CHAPTER 5 SECTION 5. 4 EXPONENTIAL FUNCTIONS: DIFFERENTIATION AND INTEGRATION

CHAPTER 5 SECTION 5. 4 EXPONENTIAL FUNCTIONS: DIFFERENTIATION AND INTEGRATION

Definition of the Natural Exponential Function

Definition of the Natural Exponential Function

Recall: This means… and… Exponential and log functions are interchangeable. Start with the base.

Recall: This means… and… Exponential and log functions are interchangeable. Start with the base. Change of Base Theorem

Solve.

Solve.

Solve. We can’t take a log of -1.

Solve. We can’t take a log of -1.

Theorem 5. 10 Operations with Exponential Functions

Theorem 5. 10 Operations with Exponential Functions

Properties of the Natural Exponential Function

Properties of the Natural Exponential Function

Theorem 5. 11 Derivative 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: Find dy/dx:

5. 4 Exponential Functions • Example 3 (concluded):

5. 4 Exponential Functions • Example 3 (concluded):

Find each derivative:

Find each derivative:

5. 4 Exponential Functions • THEOREM 2 • • or • The derivative of

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 •

5. 4 Exponential Functions • Example 4: Differentiate each of the following with • respect to x:

5. 4 Exponential Functions • Example 4 (concluded):

5. 4 Exponential Functions • Example 4 (concluded):

Find each derivative

Find each derivative

Theorem: 1. Find the slope of the line tangent to f (x) at x

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

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

4. Find extrema and inflection points for Crit #’s: Can’t ever work. Crit #’s:

4. Find extrema and inflection points for Crit #’s: Can’t ever work. Crit #’s: none

Intervals: Test values: f ’’(test pt) f(x) f ’(test pt) f(x) rel max rel

Intervals: Test values: f ’’(test pt) f(x) f ’(test pt) f(x) rel max rel min Inf pt

5. 4 Exponential Functions • Example 7: Graph with x ≥ 0. Analyze the

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.

• 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

• 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

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):

• Example 4 (continued):

Theorem 5. 12 Integration Rules for Exponential Functions

Theorem 5. 12 Integration Rules for Exponential Functions

Theorem:

Theorem:

Theorem:

Theorem:

AP QUESTION

AP QUESTION

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

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

AP QUESTION

AP QUESTION