Exponential Functions 3 1 OBJECTIVE Graph exponential functions

Exponential Functions 3. 1 OBJECTIVE • • Graph exponential functions. Differentiate exponential functions. Copyright © 2014 Pearson Education, Inc.

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: Copyright © 2014 Pearson Education, Inc. Slide 3 - 2

3. 1 Exponential Functions Example 1: Graph First, we find some function values. Copyright © 2014 Pearson Education, Inc. Slide 3 - 3

3. 1 Exponential Functions For Quick Check 1 , complete this table of function values. Graph Copyright © 2014 Pearson Education, Inc. Slide 3 - 4 .

3. 1 Exponential Functions DEFINITION: We call the natural base. Copyright © 2014 Pearson Education, Inc. Slide 3 - 5

3. 1 Exponential Functions THEOREM 1 The derivative of the function f given by itself: Copyright © 2014 Pearson Education, Inc. Slide 3 - 6 is

3. 1 Exponential Functions Example 2: Find dy/dx: Copyright © 2014 Pearson Education, Inc. Slide 3 - 7

3. 1 Exponential Functions Example 2 (concluded): Copyright © 2014 Pearson Education, Inc. Slide 3 - 8

3. 1 Exponential Functions Quick Check 2 Differentiate: a. ) , b. ) , c. ) , Copyright © 2014 Pearson Education, Inc. Slide 3 - 9

3. 1 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. Copyright © 2014 Pearson Education, Inc. Slide 3 - 10

3. 1 Exponential Functions Example 3: Differentiate each of the following with respect to x: Copyright © 2014 Pearson Education, Inc. Slide 3 - 11

3. 1 Exponential Functions Example 3 (concluded): Copyright © 2014 Pearson Education, Inc. Slide 3 - 12

3. 1 Exponential Functions Quick Check 3 Differentiate: a. ) , b. ) , c. ) , Copyright © 2014 Pearson Education, Inc. Slide 3 - 13

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. Copyright © 2014 Pearson Education, Inc. Slide 3 - 14

3. 1 Exponential Functions Example 4 (continued): Copyright © 2014 Pearson Education, Inc. Slide 3 - 15

3. 1 Exponential Functions 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. Copyright © 2014 Pearson Education, Inc. Slide 3 - 16

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. Copyright © 2014 Pearson Education, Inc. Slide 3 - 17

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. Copyright © 2014 Pearson Education, Inc. Slide 3 - 18

3. 1 Exponential Functions Section Summary • The exponential function , where has the derivative. That is, the slope of a tangent line to the graph of is the same as the function value at x. • 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 • Calculus is rich in applications of exponential functions. Copyright © 2014 Pearson Education, Inc. Slide 3 - 19