Exponential Functions 4.4 OBJECTIVES Differentiate exponential functions. Solve application problems with exponential functions. Copyright © 2008 Pearson Education, Inc. Publishing as Pearson Addison-Wesley

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. Copyright © 2008 Pearson Education, Inc. Publishing as Pearson Addison-Wesley

Example 1: Look at the graph First, we find some function values. Copyright © 2008 Pearson Education, Inc. Publishing as Pearson Addison-Wesley

DEFINITION: e is a number, named for the Swiss mathematician Leonhard Euler. Copyright © 2008 Pearson Education, Inc. Publishing as Pearson Addison-Wesley

THEOREM 1 Copyright © 2008 Pearson Education, Inc. Publishing as Pearson Addison-Wesley

Example 1: Find dy/dx: Copyright © 2008 Pearson Education, Inc. Publishing as Pearson Addison-Wesley

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Example 1 (concluded): Copyright © 2008 Pearson Education, Inc. Publishing as Pearson Addison-Wesley

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 © 2008 Pearson Education, Inc. Publishing as Pearson Addison-Wesley

Example 2: Differentiate each of the following with respect to x: Copyright © 2008 Pearson Education, Inc. Publishing as Pearson Addison-Wesley

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Example 2 (concluded): Copyright © 2008 Pearson Education, Inc. Publishing as Pearson Addison-Wesley

The sales of a new computer ( in thousands) are given by: t represents time in years. Find the rate of change of sales at each time. after 1 year b) after 5 years c) What is happening to the rate of change of sales? Answers: a) 20 b) 6 c) decreasing Copyright © 2008 Pearson Education, Inc. Publishing as Pearson Addison-Wesley

Write an equation of the tangent line to at x = 0. Answer: y = -6x+2 Copyright © 2008 Pearson Education, Inc. Publishing as Pearson Addison-Wesley