(a)
To show:
(a)
Answer to Problem 73E
The function is conformal except
Explanation of Solution
Given information:
Calculation:
In order to show that
By the definition of absolute change in
Differentiate
Product rule of differentiation is used.
Differentiation of
Now substitute
Now substitute
By (1) and (2) it is proved that
Now differentiate
Since
This proves the second result.
(b)
To find: Use intermediate value theorem to show that for any real number
(b)
Explanation of Solution
Given information:
Calculation:
Intermediate theorem states that for any function
Now let
Given that
Hence proved.
(c)
To Prove:
(c)
Explanation of Solution
Given information:
By the proof in part (b)
Calculation:
Given that
Since
Similarly,
(d)
To prove:
(d)
Explanation of Solution
Given information:
Given that
Calculation:
Let
Using the given fact that
Can be written as
Since
The function is increasing on
Since
Similarly,
Since
Hence proved.
(e)
To Prove:
(e)
Explanation of Solution
Given information:
Given that
Calculation:
Let
Using the given fact that
Can be written as
Since
The function is increasing on
Since
Similarly,
Since
Hence proved.
(e)
To Prove:
(e)
Explanation of Solution
Given information:
From part (e)
Calculation:
It is given
From part (b)
Now let
Using (1)
Therefore, from part (b) there is some value
Substitute the value of
Given,
Subtract equation (2) from (3) gives:
Substitute the value of
Hence proved.
Chapter 4 Solutions
Advanced Placement Calculus Graphical Numerical Algebraic Sixth Edition High School Binding Copyright 2020
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