Ma 113 Functions And Inverse Functions, The Exponential Function And The Logarithm Worksheet Page 29
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41Worksheet # 22: Newton’s Method and Antiderivatives
1. Use Newton’s method to find an approximation to
2. You may do this by finding a solution of
3
2 = 0.
5
2
2. Use Newton’s method to approximate the critical points of the function ( ) =
7
+ .
3. Let ( ) =
.
2
1 +
(a) Solve ( ) = 0 without using Newton’s method.
(b) Use Newton’s method to solve ( ) = 0 beginning with the starting point
= 2. Does something
0
interesting happen?
(c) Make a sketch of the graph of
and explain what you observed in part b).
3
4. (a) Let ( ) =
+ 1. Calculate the derivative
( ). Give an anti-derivative of
( ). Give an
3
anti-derivative that is different from ( ).
2
(b) Let ( ) =
+ 1. Let
( ) be any anti-derivative of . What is
( )?
5. Find
given that
( ) = sin( )
sec( ) tan( )
( ) = 1
6. Find
given that
( ) =
9 8
(0) = 1
(0) = 2
2
On the surface of the earth, the acceleration of an object due to gravity is approximately
9 8 m s
.
What situation could we describe using the function ? Be sure to specify what
and its first two
derivatives represent.
7. A small rock is dropped from a bridge and the splash is heard 3 seconds later. How high is the bridge?
2
8. Let
be a function on the domain (
) that satisfies ( )
= 1. This is an example of a differential
equation. Suppose also that we are given an initial value condition (0) = 1.
(a) Show that this does not have a unique solution by finding two different functions that satisfy both
conditions.
(b) What does the fact that there are multiple solutions say about this as a model for physical
phenomena?
9. Find a function ( ) such that
( ) = ( ). Find the solution, given initial condition (0) = .
10. Let ( ) = 1
,
( ) = ln(
), and
ln( )
0
( ) =
ln(
) + 8
0
(a) Is
an anti-derivative of ? Is
an anti-derivative of ? Is
equal to a constant?
(b) Does Theorem 1 on page 275 imply that
is constant? Is the theorem wrong?
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