Download RT Section 4_5 L_Hopitals Rule

Homework



Homework Assignment #25
Read Section 4.5
Page 243, Exercises: 1 – 57 (EOO)
Rogawski Calculus
Copyright © 2008 W. H. Freeman and Company
Homework, Page 243
1. Match the graphs in Figure 12 with the description.
 a .
 c .
f   x   0 for all x
f   x   0 for all x
(b)
 b .
 d .
(c)
(a)
f   x  goes from + to 
f   x  goes from  to +
(d)
Rogawski Calculus
Copyright © 2008 W. H. Freeman and Company
Homework, Page 243
5. If Figure 14 is the graph of the derivative of f ′(x),
where do the points of inflection of f (x) occur, and on
which interval is f (x) concave down?
Points of inflection
occur at a. and b.
the graph of f (x) is
concave down on
[d, f].
Rogawski Calculus
Copyright © 2008 W. H. Freeman and Company
Homework, Page 243
Determine the intervals on which the function is
concave up or down and find the points of inflection.
9. y  x  2cos x
y  x  2cos x  y  1  2sin x  y  sin x
Graph is concave up on  2n ,  2n  1  
Graph is concave down on  2n  1  , 2n 
Points of inflection at x  n 





y " = sin x





Rogawski Calculus
Copyright © 2008 W. H. Freeman and Company
Homework, Page 243
Determine the intervals on which the function is
concave up or down and find the points of inflection.
1
13. y  2
x 3
 1 2 x 
1
2 x

y 2
y 

2
2
2
2
x 3
 x  3  x  3
 x  3  2    2 x  2  x  3  2 x 
2
2
y 

2
x
2
2  x 2  3  8 x 2
x
2
 3
3
 3

2
6x  6
2
x
2
 3
3

2  x  3  8 x 2  x 2  3 

x
6  x 2  1
x
2
2
2
 3
3

2
 3
4
6  x  1 x  1
x
2
 3
3
Rogawski Calculus
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Homework, Page 243
13.
y 
Continued
6  x  1 x  1
x
2
 3
3
x
1
1

y 


The function is concave up on (–∞, –1) and (1, ∞) and
concave down on (–1, 1). Points of inflection are at
x = {–1, 1}.
Rogawski Calculus
Copyright © 2008 W. H. Freeman and Company
Homework, Page 243
Determine the intervals on which the function is
concave up or down and find the points of inflection.
17. y   x 2  3 e x
y  x 2  3 e x  y  x 2  3 e x  e x  2 x   x 2  2 x  3 e x






y   x 2  2 x  3 e x  e x  2 x  2    x 2  4 x  1 e x
x
4.236
0.236
y   x  4.236  x  0.236  e 
y 


x
The function is concave up on (–∞, –4.236) and
(0.236, ∞) and concave down on (–4.236, 0.236).
Points of inflection are at x = {–4.236, 0.236}.
Rogawski Calculus
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Homework, Page 243
21. The growth of a sunflower during its first 100 days
is modeled well by the logistic curve y = h (t) in Figure
15. Estimate the growth rate at the point of inflection
and explain its significance. Then make a rough sketch
of the first and second derivatives of h (t).
The growth rate at the point of inflection appears to be
about 7 cm/day. It is the greatest rate of growth as the
second derivative goes from + to – at that point.Rogawski Calculus
Copyright © 2008 W. H. Freeman and Company
Homework, Page 243
y
y'
20
y'
40
60
80
100
x
Rogawski Calculus
Copyright © 2008 W. H. Freeman and Company
Homework, Page 243
Find the critical points of f (x) and use the Second Derivative Test to
determine whether each corresponds to a local minimum or maximum.
25. f  x   3x 4  8x3  6 x 2
f   x   12 x 3  24 x 2  12 x  12 x  x 2  2 x  1
f   x   36 x 2  48 x  12  12  3 x 2  4 x  1
f   x   0  12 x  x  2 x  1  12 x  x  1  x  0,1
2
2

f  1  12  3 1

f   0   12 3  0   4  0   1  12  Local minimum
2
2

 4 1  1  0  Neither min nor max
Rogawski Calculus
Copyright © 2008 W. H. Freeman and Company
Homework, Page 243
Find the critical points of f (x) and use the Second Derivative Test to
determine whether each corresponds to a local minimum or maximum.
1
29. f  x  
cos x  2
f  x 
1  sin x 

sin x
 cos x  2   cos x  2 
2
cos
x

2

  cos x   sin x  2  cos x  2     sin x 
f   x  
4
 cos x  2 
2
cos x  cos x  2   2sin 2 x  cos x  2 

4
 cos x  2 
cos x  cos x  2   2sin 2 x

3
cos
x

2


2
2
Rogawski Calculus
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Homework, Page 243
29. Continued
f   x   0  x  n
f   0  
cos 0  cos 0  2   2sin 2 0

11  2   2  0 
0
 cos 0  2 
1  2 
cos   cos   2   2sin 2  1 1  2   2  0 
f    

0
3
3
 cos   2 
 1  2 
Maximum at x   2n  1  , minimum at x  2n
3
3
Rogawski Calculus
Copyright © 2008 W. H. Freeman and Company
Homework, Page 243
Find the critical points of f (x) and use the Second Derivative Test to
determine whether each corresponds to a local minimum or maximum.
33. f  x   xe
f   x   xe
 x2
 x2
 2 x   e 1  1  2 x
 x2
f   x   1  2 x  e
2
 x2
2
e
 x2
 2 x   e  4 x    4 x
 x2
f   x   0  1  2 x 2   0  x  
3
 6x  e
 x2
1
2
3

 1 
 1 
 1    x2
f   
   4  
  6 
  e  0  Minimum
2  
2
2 


3

 1 
 1 
 1    x2
f  
   4 
  6
  e  0  Maximum
 2   2
 2 
Rogawski Calculus
Copyright © 2008 W. H. Freeman and Company
Homework, Page 243
Find the intervals on which f is concave up or down, the points of
inflection, and the critical points and determine whether each critical
point corresponds to a local maximum or minimum (or neither).
37. f  x   x3  2 x 2  x
f   x   3 x 2  4 x  1  f   x   6 x  4
1 
3 x  4 x  1  0  x   ,1 Critical points
3 
2
x
2
3
6x  4  0  x    
3
f  

2
Rogawski Calculus
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37.
Homework, Page 243
Continued
1

f    0  Local Maximum
3
f  1  0  Local Minimum
2

2 
Concave down on  ,  ,Concave up on  ,  
3

3 
2
Inflection point at x 
3
Rogawski Calculus
Copyright © 2008 W. H. Freeman and Company
Homework, Page 243
Find the intervals on which f is concave up or down, the points of
inflection, and the critical points and determine whether each critical
point corresponds to a local maximum or minimum (or neither).
41. f  x   x  x
2
1
2
1  12
1 32
f  x  2x  x  f  x   2  x
2
4
1  12
2 x  x  0  x  0.397
2
1 32
2 x  0
4
f is concave up on its domain, there are no
points of inflection and the critical point is a
local minimum
Rogawski Calculus
Copyright © 2008 W. H. Freeman and Company
Homework, Page 243
Find the intervals on which f is concave up or down, the points of
inflection, and the critical points and determine whether each critical
point corresponds to a local maximum or minimum (or neither).
45. f      sin  for 0    2
f     1  cos   f      sin 
1  cos   0      f      sin   0

f  


 f is concave down on  0,  
and concave up on  , 2  with inflection point
at   n . There are no extrema.
Rogawski Calculus
Copyright © 2008 W. H. Freeman and Company
Homework, Page 243
Find the intervals on which f is concave up or down, the points of
inflection, and the critical points and determine whether each critical
point corresponds to a local maximum or minimum (or neither).
3
49. f  x   e cos x for    
2
2
x

f   x   e  x   sin x   cos x  e  x   e  x   sin x  cos x 
f   x   e  x   cos x  sin x     sin x  cos x   e  x 
 2sin xe  x
  3 
x

f  x   0  e   sin x  cos x   cos x   sin x  x   , 
 4 4 
f   x   0  2sin xe  x  x  0,    Inflection points at x  0,  
Rogawski Calculus
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Homework, Page 243
49. Continued
x
0

f   x  


 Concave up on  0,  
  
 3 
Concave down on   ,0  and  , 
 2 
 2 
Relative maximum at x  

4
3
Relative minimum at x 
4
Rogawski Calculus
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Homework, Page 243
52. Water is pumped into a sphere at a variable rate in such a way that
the water level rises at a constant rate c. Let V (t) be the volume at time
t. Sketch the graph of V (t). Where does the point of inflection occur?
y
x
The point of inflection occurs at the time when the
height of the water equals the radius of the tank.Rogawski Calculus
Copyright © 2008 W. H. Freeman and Company
Homework, Page 243
Sketch the graph of a function satisfying the given condition.
57. i.
f   x   0 for all x, and
ii. f   x   0 for x  0 and f   x   0 for x  0
y
x
Rogawski Calculus
Copyright © 2008 W. H. Freeman and Company
Jon Rogawski
Calculus, ET
First Edition
Chapter 4: Applications of the Derivative
Section 4.5: Graph Sketching and
Asymptotes
Rogawski Calculus
Copyright © 2008 W. H. Freeman and Company
Graphs of functions that are at least twice differentiable
are made up of segments shown in Figure 1.
The keys to hand sketching are finding the transition points
and selecting the correct curve shape.
Rogawski Calculus
Copyright © 2008 W. H. Freeman and Company
The text uses solid dots to indicate local extrema and solid
squares to indicate transition points, as shown in Figure 2.
Rogawski Calculus
Copyright © 2008 W. H. Freeman and Company
1 3 1 2
In sketching the graph of y  x  x  2 x  3,it is
3
2
necessary first to find the critical points by setting f   0.
The table below shows finding the sign of f  in the
intervals between the critical points.
Rogawski Calculus
Copyright © 2008 W. H. Freeman and Company
1 3 1 2
In sketching the graph of y  x  x  2 x  3,it is
3
2
next necessary to find where f   0. The table below
shows finding the sign of f  in the intervals between
the zeroes.
Rogawski Calculus
Copyright © 2008 W. H. Freeman and Company
Figure 4 shows the sign combinations of f ′ and f ″ for
1 3 1 2
y  x  x  2 x  3.
3
2
Another way to represent this information is:
1 3 1 2
y  x  x  2 x  3  y  x 2  x  2   x  2  x  1
3
2
y  2 x  1  y  0, x  1,2; y  0, x  1
2
1
x
1
2
2
y 



y 



Rogawski Calculus
 
Copyright © 2008 W. H. Freeman and Company
Calculating the values of f at the critical points and point of
inflection gives us sufficient information to sketch the graph.
Rogawski Calculus
Copyright © 2008 W. H. Freeman and Company
Rogawski Calculus
Copyright © 2008 W. H. Freeman and Company


f   x   12 x  24 x  12 x  12 x x  2 x  1  12 x  x  1
3
2

2

2
f   x   36 x 2  48 x  12  12 3 x 2  4 x  1  12  3 x  1 x  1
x
f  x 
f   x  
1
0
1
3






Rogawski Calculus
Copyright © 2008 W. H. Freeman and Company
f   x    sin x 
1
1
 5
 0  sin x   x 
2
2
6, 6
f   x    cos x  0  x  
x
f  x 
f   x  


6
2
2
5
6






Rogawski Calculus
Copyright © 2008 W. H. Freeman and Company
Example, Page 256
Sketch the graph of the function. Indicate the transition points.

14. y  x  3x  5
3
y









  









x



















Rogawski Calculus
Copyright © 2008 W. H. Freeman and Company

Example, Page 256
Sketch the graph of the function. Indicate the transition points.
34. y  x  4  x   3ln x
y





x




















Rogawski Calculus
Copyright © 2008 W. H. Freeman and Company
Asymptotic behavior refers to the behavior of a function as either x or
f (x) approaches ±∞. A horizontal line y = L is called a horizontal
asymptote if either of the following exists:
lim f  x   L or
x 
lim f  x   L
x 
Similarly, a vertical line y = L is called a vertical asymptote if either
of the following exists:
lim f  x    or
xL
lim f  x   
xL
Rogawski Calculus
Copyright © 2008 W. H. Freeman and Company
Rogawski Calculus
Copyright © 2008 W. H. Freeman and Company
Rogawski Calculus
Copyright © 2008 W. H. Freeman and Company
Example, Page 256
Sketch the graph over the given interval. Indicate the transition
points.
42. y  x  sin x,
0,2 
y






x









Rogawski Calculus
Copyright © 2008 W. H. Freeman and Company

Example, Page 256
Sketch the graph over the given interval. Indicate the transition
points.
0,2 
42. y  2sin x  cos 2 x,
y






x










Rogawski Calculus
Copyright © 2008 W. H. Freeman and Company
Example, Page 256
Calculate the following limits.
3x  20 x
52. lim
x  4 x 2  9
2
Rogawski Calculus
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Rogawski Calculus
Copyright © 2008 W. H. Freeman and Company
Rogawski Calculus
Copyright © 2008 W. H. Freeman and Company
Example, Page 256
Calculate the limit.
66. lim
x 
4x  9
 3x  2 
4
1
3
Rogawski Calculus
Copyright © 2008 W. H. Freeman and Company
Example, Page 256
Sketch the graph of the function. Indicate the asymptotes, local
extrema, and points of inflection.
1
1
86. y  2 
2
x  x  2

y





x


















Rogawski Calculus
Copyright © 2008 W. H. Freeman and Company
Homework



Homework Assignment #26
Read Section 4.6
Page 256, Exercises: 1 – 89 (EOO)
Rogawski Calculus
Copyright © 2008 W. H. Freeman and Company