2. 5-4: Differentiation and Integration of Exponentials Objectives: Solve logarithmic and exponential equations Differentiate and integrate exponential functions ©2002 Roy L. Gover (www.mrgover.com)

3. “Wrong, Williamson! Wrong Wrong Wrong! You explained the way the ball bounces, when I specifically asked you to show me the way the cookie crumbles!”

4. Definition Natural Exponential Function:the inverse of the natural log function. y=ln x y=exy=ex A working definition:

5. Example Write as a natural logarithm:

6. Example Write as a natural exponential :

7. Example Solve for x :

8. Warm-Up Solve for x :

9. Definition The derivative of the natural exponential function Chain Rule Version e x is its own derivative

10. Example Find the derivative:

11. Try This Find the derivative:

12. Definition Chain Rule Version

13. Example

14. Assignment 1. 357/35-55 odd 85-93 odd, 103, 105

15. Try This Find the derivative:

16. Try This Find the derivative:

17. Try This The first permanent colony in America was established in Jamestown, Virginia, in 1607. For 1610 through 1780, the population, p ( in thousands), in colonial America can be modeled by...

18. where t =10 represents the year 1610. At what rate was the population of colonial America changing in 1650? Include the correct units. Hint:Find p ’ (t) and evaluate at t =50 2369 people per year

19. Definition Chain Rule Version

20. Example

21. Try This

22. Example Find the area bounded by the x -axis and the lines x =-1 and x =0, 0

23. Lesson Close For, what is: The domain? The range? Over what part of the domain is f ( x ) concave up? Do not graph.

24. Assignment 1. 357/35-55 odd 85-93 odd, 103, 105