5-4 Exponential Functions: Differentiation and Integration

Objective: Differentiate and integrate natural exponential functions.

Ms. BattagliaAP Calculus

The inverse function of the natural logarithmic function f(x)=lnx is called the natural exponential function and is denoted by

f -1(x) = ex

That is,

y = ex if and only if x = lny

Definition of the Natural Exponential Function

Inverse Relationshipln(ex) = x and elnx = x

Solve 7 = ex+1

Solving Exponential Equations

Solve ln(2x – 3) = 5

Solving a Logarithmic Equation

Let a and b be any real numbers.

1.

2.

Operations with Exponential Functions

1. The domain of f(x)=ex is (-∞, ∞), and the range is (0, ∞)

2. The function f(x)=ex is continuous, and one-to-one on its entire domain.

3. The graph of f(x)=ex is concave upward on its entire domain.

4. and

Properties of the Natural Exponential Function

Let u be a differentiable function of x.

1.

2.

Derivatives of the Natural Exponential Function

a. b.

Differentiating Exponential Functions

Find the relative extrema of f(x)=xex.

Locating Relative Extrema

Show that the standard normal probability density function

has points of inflection when x=1, -1

The Standard Normal Probability Density Function

The numbers y of shares traded (in millions) on the NY Stock Exchange from 1990 through 2005 can be modeled by

where t represents the year, with t=0 corresponding to 1990. At what rate was the number of shares traded changing in 2000?

Shares Traded

Let u be a differentiable function of x.

1. 2.

Integration Rules for Exponential Functions

Find

Integrating Exponential Functions

Find

Integrating Exponential Functions

a. b.

Integrating Exponential Functions

a. b. c.

Finding Areas Bounded by Exponential Functions

AB: Read 5.4 Page 358 #3, 4, 11-14, 37, 38, 39-57 odd

BC: HW: Read 5.4 Page 358 #1-11 odd, 39-59 odd,63,69,79,99, 103,105,111,121,125

Classwork/Homework