1. Exponential Functions
OBJECTIVE
Graph exponential functions.
Differentiate exponential functions.

2. 3.1 Exponential Functions
DEFINITION:
An exponential function f is given by
where x is any real number, a > 0, and a ≠ 1. The
number a is called the base.
Examples:

3. 3.1 Exponential Functions
f (x) = a0 · ax, a > 1
positive, increasing, continuous function
as x gets smaller, a0 · ax approaches 0
concave up
x-axis is the horizontal
asymptote

4. 3.1 Exponential Functions
f (x) = a0 · ax, 0 < a < 1
positive, decreasing, continuous function
as x gets larger, a0 · ax approaches 0
concave up
x-axis is the horizontal
asymptote

5. 3.1 Exponential Functions
Example 1: Graph First, we find some function values.

6. 3.1 Exponential Functions
Quick Check 1
For , complete this table of function values. Graph

7. 3.1 Exponential Functions
DEFINITION:
We call e the natural base.

8. 3.1 Exponential Functions
THEOREM 1
The derivative of the function f given by f (x) = ex is itself:

9. 3.1 Exponential Functions
Example 2: Find dy/dx:

10. 3.1 Exponential Functions
Example 2 (concluded):

11. 3.1 Exponential Functions
Quick Check 2
Differentiate:
a.) ,
b.) ,
c.) ,

12. 3.1 Exponential Functions
THEOREM 2
The derivative of e to some power is the product of e
to that power and the derivative of the power. or

13. 3.1 Exponential Functions
Example 3: Differentiate each of the following with
respect to x:

14. 3.1 Exponential Functions
Example 3 (concluded):

15. 3.1 Exponential Functions
Quick Check 3
Differentiate:
a.) ,
b.) ,
c.) ,

16. 3.1 Exponential Functions
Example 4: Graph with x ≥ 0. Analyze the graph using calculus.
First, we find some values, plot the points, and sketch
the graph.

17. 3.1 Exponential Functions
Example 4 (continued):

18. 3.1 Exponential Functions
Example 4 (continued):
a) Derivatives. Since
b) Critical values. Since the derivative
for all real numbers x. The derivative exists for all real numbers, and the equation
has no solution. There are no critical values.

19. 3.1 Exponential Functions
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 and the graph is concave down over the entire real line.

20. 3.1 Exponential Functions
Example 4 (concluded):
e) Concavity. Since for all real
numbers x, we know that h is decreasing and the
graph is concave down over the entire real number
line.

21. 3.1 Exponential Functions
Section Summary
An exponential function is given by f (x) = a0 · ax, where x is any real umber, a > 0 and a ≠ 1. The number a is the base, and the y-intercept is (0, a0).
If a > 1, then f (x) = a0 · ax is increasing, concave up, increasing without bound as x increases, and has a horizontal asymptote y = 0 as x approaches –∞.
If 0 < a < 1, then f (x) = a0 · ax is decreasing, concave up, increasing without bound as x approaches –∞, and has a horizontal asymptote y = 0 as x approaches infinity.

22. 3.1 Exponential Functions
Section Summary
The exponential function , where e ≈ has the derivative That is, the slope of a tangent line to the graph of y = ex is the same as the function value at x.
In general,
The graph of is an increasing function with no critical values, no maximum or minimum values, and no points of inflection. The graph is concave up, with
and