MAT 1235Calculus II

Section 6.1

Inverse Functions

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Next Week Scheduling Changes

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Tuesday: no class Thursday: Lab 01

Homework and …

WebAssign HW 6.1 Quiz: 5.3, 6.1 First Exam: Monday Bring your tutoring record tomorrow. Please, Please study for the exam! Please read the quiz solutions Please read the grader’s comment

Preview

Where are we going?• Unfinished business:

• 6.2*: Define the Natural log. function as the antiderivative of

• 6.3*: Define the exponential function as the inverse function of the natural log. function

• 6.4*: General log. and exponential functions

1 ,1

1

nCn

xdxx

nn

?1 dxx

Preview

Part I: Review of the Inverse Functions• Quick review (Read the text carefully if you

do not remember the details) Part II: The relation between the

derivatives of a function and its inverse function

The Quest

Given , we want to find the “undo” function , such that

If exist, it is called the inverse function of

x x)(xff g

A AB

The Quest

f

The Quest

f

g

Existence of Inverse Functions

2?4

2)( xxf ""g

2

A AB

Not all functions have corresponding inverse functions

Existence of Inverse Functions

f

g

f

Existence of Inverse Functions

Not all functions have corresponding inverse functions

In order for an inverse function to exist, this situation cannot happen: Two distinct points have the same function value

2?4

2)( xxf ""g

2

A AB

Properties

and The graph of and are symmetric about

the line Read the text to review other properties

Derivatives of Inverse Functions

Let . How to find ?1 fg )(ag

Derivatives of Inverse Functions

)(xf

1 fg

)(xg

)(xg

easy

easy

????

)(ag

Derivatives of Inverse Functions

)(xf

1 fg

)(xg

)(xg

)(xf easy

easy

easy

????

))((

1)(

agfag

)(ag

easy

Derivatives of Inverse Functions

)(xf

1 fg

)(xg

)(xg

)(xf easy

easy

easy

????

))((

1)(

agfag

)(ag

easy Why?

Derivatives of Inverse Functions

( ( ))f g x x)

(

(

)f xy

y g x

))((

1)(

agfag

Remarks

The formula can be written equivalently as

which we are going to use in later sections (and it is easy to remember due to the wonderful design of the notations)

dydxdx

dy 11

( )( ( ))

g xf g x

)

(

(

)f xy

y g x

Example 1

)1( Find

,12)(Let 13

g

fgxxxf

1( )

( ( ))g a

f g a

Example 1: Step 1

)1( Find

,12)(Let 13

g

fgxxxf

1( )

( ( ))g a

f g a

By inspection, what is the value of such that ?

( ) 1

(1)

f

g

Example 1: Step 2

)1( Find

,12)(Let 13

g

fgxxxf

1( )

( ( ))g a

f g a

( )f x

Example 1: Step 3

)1( Find

,12)(Let 13

g

fgxxxf

1( )

( ( ))g a

f g a

1

2

1(1)

( (1))g

f g

( )f x

Step 1 Use inspection to find such that . Then find .

Step 2 Find .

Step 3 Must state the formula

before using it.

Expectations

1( )

( ( ))g a

f g a

Expectations

The steps are designed to • conform with standard presentation, and• minimize the chance of making mistakes.

In the exam, I will look for the expected steps.