1. 5 Copyright © Cengage Learning. All rights reserved. Logarithmic, Exponential, and Other Transcendental Functions

2. Exponential Functions: Differentiation and Integration Copyright © Cengage Learning. All rights reserved. 5.4

3. 3 Develop properties of the natural exponential function. Differentiate natural exponential functions. Integrate natural exponential functions. Objectives

4. 4 The Natural Exponential Function

5. 5 The function f(x) = ln x is increasing on its entire domain, and therefore it has an inverse function f –1. The domain of f –1 is the set of all reals, and the range is the set of positive reals, as shown in Figure 5.19. Figure 5.19

6. 6 f(x) = ln x, so, for any real number x, If x happens to be rational, then Because the natural logarithmic function is one-to-one, you can conclude that f –1 (x) and e x agree for rational values of x. The Natural Exponential Function

7. 7 The inverse relationship between the natural logarithmic function and the natural exponential function can be summarized as follows. The Natural Exponential Function

8. 8 Example 1 – Solving Exponential Equations Solve 7 = e x + 1. Solution: You can convert from exponential form to logarithmic form by taking the natural logarithm of each side of the equation.

9. 9 The Natural Exponential Function

10. 10

11. 11 The slope at x=0 appears to be 1. The only function that is its own derivative!

12. 12 The Natural Exponential Function

13. 13 Derivatives of Exponential Functions

14. 14 One of the most intriguing (and useful) characteristics of the natural exponential function is that it is its own derivative. In other words, it is a solution to the differential equation y' = y. This result is stated in the next theorem. Derivatives of Exponential Functions http://www.youtube.com/watch?v=bZivCw3bB6w

15. 15 The inverse function of a natural logarithm, ln x, is the natural exponential function, e x. The following rules are important to remember: Once more: Derivatives of the natural exponential functions: or http://www.youtube.com/watch?v=bZivCw3bB6w

16. 16 Example 3 – Differentiating Exponential Functions

17. 17 Examples  Differentiate:

18. 18 More Practice  Differentiate:

19. 19 More Practice  Differentiate:

20. 20  Differentiate:

21. 21 Homework Day 1. Pg. 356: 1 – 15 odd, 35 – 61 odd

22. 22 Integrals of Exponential Functions

23. 23 HWQ 1/15  Find an equation of the line tangent to at the point (1,0).

24. 24 Warm Up (for day 2)  Differentiate:

25. 25 Integrals of Exponential Functions

26. 26 Find Solution: If you let u = 3x + 1, then du = 3dx Example – Integrating Exponential Functions

27. 27 Practice Problems  Integrate:

28. 28 More Practice:  Integrate:

29. 29 Try this one:  Integrate:

30. 30  Find the particular solution that satisfies the initial conditions:

31. 31 Homework Day 2. Pg. 356: 65, 71, 85 – 105 (odd)