# Exponential Functions - SCHOOLinSITES

Exponential Functions Differentiation and Integration Definition of the Natural Exponential Function The inverse function of the natural logarithmic function f(x) = ln x is called the natural exponential

function and is denoted by f-1(x) = ex. That is, y = ex if and only if x = ln y Inverse Relationship ln (ex) = x and eln x = x Solving Exponential Equations

eln 2x = 12 ln (x 2)2 = 12 200e-4x = 15 Operations with Exponential Functions

Let a and b be any real numbers. eaeb = ea + b a e a b e b

e Properties of the Natural Exponential Function 1. The domain of f(x) = ex is (-, ), and the

range is (0, ). 2. The function f(x) = ex is continuous, increasing, and one-to-one on its entire domain. 3. The graph of f(x) = ex is concave upward on its entire domain. x x 4. lim e 0 and lim e x x

Properties of the Natural Exponential Function Properties of the Natural Exponential Function Sketch the graph of y = e-x using transformations. p. 347 problems 25 26

The Derivative of the Natural Exponential Function Let u be a differentiable function of x. d x x 1. e e dx

d u u du e e 2. du dx Integrals of Exponential Functions Let u be a differentiable function of x.

x x 1.e dx e C u u 2.e du e C

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