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

2. Definition of the Natural Exponential Function

3. Exponential and log functions are interchangeable.
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: f ’’(test pt) f(x) f ’(test pt) f(x) Test values: rel min
rel max
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. 29

30. 30

31. 31

32. 32

33. AP QUESTION

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

51. AP QUESTION