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3.2 Inverse Functions and Logarithms

3.3 Derivatives of Logarithmic and Exponential

functions

One-to-one functions

- Definition A function f is called a one-to-one

function if it never takes on the same value

twice that is - f(x1) ? f(x2) whenever x1 ? x2.
- Horizontal line test A function f is one-to-one

if and only if no horizontal line intersects its

graph more than once. - Examples f(x) x3 is one-to-one
- but f(x) x2 is not.

Inverse functions

- Definition Let f be a one-to-one function with

domain A and range B. Then the inverse function f

-1 has domain B and range A and is defined by - for any y in B.
- Note f -1(x) does not mean 1 / f(x) .
- Example The inverse of f(x) x3 is f -1(x)x1/3

- Cancellation equations

How to find the inverse function of a one-to-one

function f

- Step 1 Write yf(x)
- Step 2 Solve this equation for x in terms of y

(if possible) - Step 3 To express f -1 as a function of x,

interchange x and y. The resulting equation is y

f -1(x) - Example Find the inverse of f(x) 5 - x3

Another example

Solve for x

Inverse functions are reflections about y x.

Switch x and y

Derivative of inverse function

First consider an example

At x 2

At x 4

Calculus of inverse functions

- Theorem If f is a one-to-one continuous function

defined on an interval then its inverse function

f -1 is also continuous. - Theorem If f is a one-to-one differentiable

function with inverse function f -1 and f ' (f -1

(a)) ? 0, then the inverse function is

differentiable and - Example Find (f -1 )' (1) for f(x) x3 x 1

- Solution By inspection f(0)1, thus f -1(1)

0 - Then

Logarithmic Functions

Consider where agt0 and a?1

This is a one-to-one function, therefore it has

an inverse.

The inverse is called the logarithmic function

with base a.

Example

The most commonly used bases for logs are 10

and e

Properties of Logarithms

Since logs and exponentiation are inverse

functions, they un-do each other.

Product rule

Quotient rule

Power rule

Change of base formula

Derivatives of Logarithmic and Exponential

functions

Examples on the board.

Logarithmic Differentiation

- The calculation of derivatives of complicated

functions involving products, quotients, or

powers can often be simplified by taking

logarithms. - Step 1 Take natural logarithms of both sides of

an equation y f (x) and use the properties of

logarithms to simplify. - Step 2 Differentiate implicitly with respect to

x - Step 3 Solve the resulting equation for y'
- Examples on the board

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