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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. 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 5 SECTION 5.4 EXPONENTIAL FUNCTIONS: DIFFERENTIATION AND INTEGRATION

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  1. CHAPTER 5SECTION 5.4EXPONENTIAL FUNCTIONS:DIFFERENTIATION AND INTEGRATION

  2. Definition of the Natural Exponential Function

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

  4. Solve.

  5. Solve. We can’t take a log of -1.

  6. Theorem 5.10 Operations with Exponential Functions

  7. Properties of the Natural Exponential Function

  8. Theorem 5.11 Derivative of the Natural Exponential Function

  9. 5.4 Exponential Functions • Example 3: Find dy/dx:

  10. 5.4 Exponential Functions • Example 3 (concluded):

  11. Find each derivative:

  12. 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.

  13. 5.4 Exponential Functions • Example 4: Differentiate each of the following with • respect to x:

  14. 5.4 Exponential Functions • Example 4 (concluded):

  15. Find each derivative

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

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

  18. 4. Find extrema and inflection points for

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

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

  21. 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.

  22. 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.

  23. 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.

  24. 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.

  25. Example 4 (continued):

  26. Theorem 5.12 Integration Rules for Exponential Functions

  27. Theorem:

  28. Theorem:

  29. AP QUESTION

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

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