Homework



Homework Assignment #24
Read Section 4.4
Page 236, Exercises: 1 – 61 (EOO)
Rogawski Calculus
Copyright © 2008 W. H. Freeman and Company
Example, Page236
Find a point c satisfying the conclusion of the MVT for the given
function and the interval.
1
y

x
, 1, 4
1.
y 1  1  1, y  4    4   0.25
1
1
0.25  1
m
 0.25  y   1 x 2
4 1
1
1
 2    x 2  4  x  2  x  2
4
x
Rogawski Calculus
Copyright © 2008 W. H. Freeman and Company
Example, Page236
Find a point c satisfying the conclusion of the MVT for the given
function and the interval.
x
, 3, 6
5. y 
x 1
3
6


3
6
y  3 
 , y 6 

 3  1 4
6 1 7
6 3
3

x  11  x 1

1
1
7
4
28
m


 y 

2
2
63
3 28
x

1
x

1
 
 
 x  1
2
 28  x  1  5.2915  x  4.292
Rogawski Calculus
Copyright © 2008 W. H. Freeman and Company
Example, Page236
Find a point c satisfying the conclusion of the MVT for the given
function and the interval.
9. y  cosh x,  1,1
1
 1
1
1
e   e  
e   e  
y  1 
 1.543, y 1 
 1.543
2
2
1.543  1.543
e x  e x
m
 0  y  sinh x 
0
1   1
2
e x  e x  x  0
Rogawski Calculus
Copyright © 2008 W. H. Freeman and Company
Example, Page236
13.
Determine the intervals on which f (x) is increasing or
decreasing, assuming that Figure 12 is the graph of the derivative,
f ′(x).
The function f (x) is increasing on [0, 2] and [4, 6] since f ′(x) is
positive on these intervals and decreasing on [2,4] since f ′(x) is
negative on this interval.
Rogawski Calculus
Copyright © 2008 W. H. Freeman and Company
Example, Page236
Sketch the graph of the function f (x) whose derivative, f ′(x), has
the given description.
17.
f ′(x) is negative on (1, 3) and positive everywhere else
y


y = f '(x)


x











y = f(x)


Rogawski Calculus
Copyright © 2008 W. H. Freeman and Company
Jon Rogawski
Calculus, ET
First Edition
Chapter 4: Applications of the Derivative
Section 4.4: The Shape of a Graph
Rogawski Calculus
Copyright © 2008 W. H. Freeman and Company
Concavity is used to describe the manner in which a graph is curving
as we proceed from left to right.
Observe in Figure 1, that the slope of the segment of the graph is
decreasing as we go from left to right on the concave down segments.
Similarly, the slope is increasing as we go from left to right on the
concave up segments.
Rogawski Calculus
Copyright © 2008 W. H. Freeman and Company
Figure 2 offers another way to look at concavity. If the curve is
above or up from the tangent line, the curve is concave up. Similarly,
if the curve is below or down from the tangent line, the curve is
concave down.
Rogawski Calculus
Copyright © 2008 W. H. Freeman and Company
A more formal definition of concavity is given below.
Rogawski Calculus
Copyright © 2008 W. H. Freeman and Company
Figure 3 shows the increase in stock price of two companies over
the same time interval. Both companies’ stock currently sells for
$75. If the value (stock price) trends continue, which is the better
investment?
Rogawski Calculus
Copyright © 2008 W. H. Freeman and Company
As noted in Theorem 1, the sign of the second derivative on an
interval indicates the concavity of the graph on that interval.
Rogawski Calculus
Copyright © 2008 W. H. Freeman and Company
As illustrated in Figure 4, a point of inflection is the point on a
curve where the concavity changes from concave up to concave
down or concave down to concave up. The second derivative
equals zero at a point of inflection.
This is formalized in Theorem 2.
Rogawski Calculus
Copyright © 2008 W. H. Freeman and Company
Example, Page 243
Determine the intervals on which the function is
concave up or down and find the points of inflection.
12. y   x  2  1  x 3 
Rogawski Calculus
Copyright © 2008 W. H. Freeman and Company