Sec 3.1 – Exponential Functions 2012 Pearson Education, Inc. All rights reserved 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: Examples of problems involving the use of exponential functions: growth or decay, compound interest, the statistical "bell curve,“ the shape of a hanging cable (or the Gateway Arch in St. Louis), some problems of probability, some counting problems, the study of the distribution of prime numbers.

Sec 3.1 – Exponential Functions 2012 Pearson Education, Inc. All rights reserved

Sec 3.1 – Exponential Functions 2012 Pearson Education, Inc. All rights reserved DEFINITION: e is called the natural base. THEOREM 1 The derivative of the function f given by is:

Sec 3.1 – Exponential Functions 2012 Pearson Education, Inc. All rights reserved THEOREM 2 The derivative of e to some power is the product of e to that power and the derivative of the power.

Sec 3.1 – Exponential Functions 2012 Pearson Education, Inc. All rights reserved Find the derivatives:

Sec 3.1 – Exponential Functions 2012 Pearson Education, Inc. All rights reserved Find the derivatives:

Sec 3.1 – Exponential Functions 2012 Pearson Education, Inc. All rights reserved Find the critical values, intervals of inc./dec., points of inflection, and the intervals of concavity.

Sec 3.1 – Exponential Functions 2012 Pearson Education, Inc. All rights reserved

Sec 3.1 – Exponential Functions 2012 Pearson Education, Inc. All rights reserved

Sec 3.1 – Exponential Functions 2012 Pearson Education, Inc. All rights reserved

Sec 3.1 – Exponential Functions 2012 Pearson Education, Inc. All rights reserved