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MAT 1235Calculus II
Section 6.1
Inverse Functions
http://myhome.spu.edu/lauw
Next Week Scheduling Changes
I will be helping with recruiting off campus next Tuesday and Thursday.
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.