homework homework assignment #23 read section 4.3 page 227, exercises: 1 – 77 (eoo), skip 57, 69...
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Homework
Homework Assignment #23 Read Section 4.3 Page 227, Exercises: 1 – 77 (EOO), skip 57,
69
Rogawski CalculusCopyright © 2008 W. H. Freeman and Company
Homework, Page 2271. The following questions refer to Figure 15.
(a) How many critical points does
f (x) have?
Three, x = {3, 5, 7}
(b) What is the maximum value of
f (x) on [0, 8]?
6 is the maximum value of f (x) on [0, 8]
(c) What are the local maximum values
of f (x)?
6 at x = 0, 5 at x = 5, and 4 at x = 8
Rogawski CalculusCopyright © 2008 W. H. Freeman and Company
Homework, Page 2271. (d) Find a closed interval on which both the minimum and
maximum occur at critical points.
[2, 6] or [4, 8]
(e) Find an interval on which the
minimum occurs at an endpoint.
[0, 3], [3, 4], [4, 7], or [7, 8]
Rogawski CalculusCopyright © 2008 W. H. Freeman and Company
Homework, Page 227Find all critical points of the function.
5.
Rogawski CalculusCopyright © 2008 W. H. Freeman and Company
3 2954 2
2f x x x x
3 2 2
2
954 2 3 9 54
2
3 18 0 6 3 0 3,6
f x x x x f x x x
x x x x x
Homework, Page 227Find all critical points of the function.
9.
Rogawski CalculusCopyright © 2008 W. H. Freeman and Company
1
3f x x
1 23 3
23
1
31
0 undefined 03
f x x f x x
x
Homework, Page 227Find all critical points of the function.
13.
Rogawski CalculusCopyright © 2008 W. H. Freeman and Company
1sin 2f x x x
2 2
2 2 2
1 12 0 2
1 1
1 1 3 31 1
2 4 4 2
f xx x
x x x x
Homework, Page 22717. Find the minimum value of .
Rogawski CalculusCopyright © 2008 W. H. Freeman and Company
1 2tanf x x x
22
21 1
2 1 10 2 1 0 2 1
21
1 1 1 1tan tan 0.245
2 2 2 4
minimum value = 0.245
xf x x x x
x x
f
Homework, Page 22721. Plot f (x) = ln x – 5 sin x on [0, 2π] and approximate both the critical points and the extreme values.
The critical points are x = {0.204, 1.431, 4.754}
Relative maximum y(0.204) = –2.603
Absolute minimum y(1.430) = –4.593
Absolute maximum y(4.754) = 6.555
Rogawski CalculusCopyright © 2008 W. H. Freeman and Company
Homework, Page 227Find the minimum and maximum values of the function on the given interval.
25.
Rogawski CalculusCopyright © 2008 W. H. Freeman and Company
2 6 1, 2, 2y x x
2 2
2 6 0 Critical point: 3
2 2 6 2 1 15; 2 2 6 2 1 9
Absolute maximum 2 15
Absolute minimum 2 9
y x x
y y
y
y
Homework, Page 227Find the minimum and maximum values of the function on the given interval.
29.
Rogawski CalculusCopyright © 2008 W. H. Freeman and Company
3 6 1, 2, 0y x x
23 6 0 Critical point: 2
2 8 12 1 3
2 2 2 6 2 1 12.314
0 0 0 1 1
Absolute maximum 2 3
Absolute minimum 2 12.314
y x x
y
y
y
y
y
Homework, Page 227Find the minimum and maximum values of the function on the given interval.
33.
Rogawski CalculusCopyright © 2008 W. H. Freeman and Company
3 23 9 2, 4, 4y x x x
2 23 6 9 0 2 3 0 Critical point: 3,1
4 64 48 36 2 22; 3 27 27 27 2 29
1 1 3 9 2 3; 4 64 48 36 2 78
Absolute maximum 4 78
Absolute minimum 1 3
y x x x x x
y y
y y
y
y
Homework, Page 227Find the minimum and maximum values of the function on the given interval.
37.
Rogawski CalculusCopyright © 2008 W. H. Freeman and Company
2 1
, 5, 64
xy
x
2 2 2 2
2 2 2
2
2 2
4 2 1 1 2 8 1 8 1
4 4 4
8 1 0 Critical point: 0.123,8.123
5 1 6 15 26; 6 18.5
5 4 6 4
Absolute maximum 5 26
Absolute minimum 6 18.5
x x x x x x x xy
x x x
x x x
y y
y
y
Homework, Page 227Find the minimum and maximum values of the function on the given interval.
41.
Rogawski CalculusCopyright © 2008 W. H. Freeman and Company
22 2 2 , 0, 2y x x
22
2 2
2 22 2
2
2 2 2
2 2
2 22 2 12 2 2 1
2 2 2 2 2
2 2 2 2 2 2 0
2 2
4 4 2 4 4 0 2 10 0 Critical points: None
0 2 0 2 2 0 2 6; 2 2 2 2 2 2 4 2
Absolute maximum 0 4 2
Absolute minimum 2 2
xxy x x
x x
x xx x
x
x x x x x
y y
y
y
6
Homework, Page 227Find the minimum and maximum values of the function on the given interval.
45.
Rogawski CalculusCopyright © 2008 W. H. Freeman and Company
sin cos , 0,2
y x x
2 2sin sin cos cos cos sin cos 2
cos 2 0 Critical points: 4
2 2 10 sin 0cos 0 0; sin cos
4 4 4 2 2 2
1sin cos 0 Absolute maximum :
2 2 2 4 2
Absolute minimum : 0 0, 2
y x x x x x x x
x x
y y
y y
y y
0
Homework, Page 227Find the minimum and maximum values of the function on the given interval.
49.
Rogawski CalculusCopyright © 2008 W. H. Freeman and Company
2sin , 0, 2y
11 2cos 0 2cos 1 cos
25
Critical points: ,3 3
0 0 2sin 0 0; 2sin 0.6853 3 3
5 5 52sin 6.968; 2 2 2sin 2 2
3 3 3
5Absolute maximum : 6.968 3
Abso
y
y y
y y
y
lute minimum : 0.685 3y
Homework, Page 227Find the minimum and maximum values of the function on the given interval.
53.
Rogawski CalculusCopyright © 2008 W. H. Freeman and Company
ln, 1,3
xy
x
2 2
1ln 1
1 ln0 1 ln 0 ln 1
Critical point:
ln1 ln 1 ln 31 0; 0.368; 3 0.366
1 3
Absolute maximum : 0.368
Absolute minimum : 1 0
x xxx
y x x x ex x
x e
ey y e y
e e
y e
y
Homework, Page 227Find the critical points and the extreme values on [0, 3]. Refer to Figure 18.
61.
Rogawski CalculusCopyright © 2008 W. H. Freeman and Company
2 4 12y x x
2
2
2
Critical point : 2
0 0 4 0 12 12
2 2 4 2 12 0
3 3 4 3 12 9
Absolute maximum : 0 12
Absolute minimum : 2 0
x
y
y
y
y
y
Homework, Page 227Verify Rolle’s Theorem for the given interval.
65.
Rogawski CalculusCopyright © 2008 W. H. Freeman and Company
3sin , ,4 4f x x
2 23 3sin ; sin4 4 4 42 23cos cos 02 4 2 4
f f
f x x
Homework, Page 22773. Migrating fish tend to swim at a velocity that minimizes the total expenditure of energy E. According to one model, E is proportional to where vr is the velocity of the river water.
(a) Find the critical points of f (v).
Rogawski CalculusCopyright © 2008 W. H. Freeman and Company
3
r
vf v
v v
2 3
2 3 3 2
2
3 10
3 0 2 3 0
32 3 0 ,
2
r
r
r r
rr r
v v v vf v f v
v v
v v v v v v v
vv v v v v
Homework, Page 22773. (b) Choose a value of vr, (say vr = 10) and plot f (v). Confirm that f (v) has a minimum value at the critical point.
Rogawski CalculusCopyright © 2008 W. H. Freeman and Company
Let 10rv
Homework, Page 227Plot the function and find the critical points and extreme values on [–5, 5].
77.
Rogawski CalculusCopyright © 2008 W. H. Freeman and Company
2 21 4
xf x
x x
Critical points for are 2
Minimum value : 0.667
Maximum value : 0.667
f x x
y
y
Rogawski CalculusCopyright © 2008 W. H. Freeman and Company
Chapter 4: Applications of the DerivativeSection 4.3: The Mean Value Theorem and
Monotonicity
Jon Rogawski
Calculus, ETFirst Edition
Rogawski CalculusCopyright © 2008 W. H. Freeman and Company
If a function is defined over a closed interval, then there is some point x = c on the interval such that the slope at point c equals theslope of the secant line joining the end points of the interval. This is known as the Mean Value Theorem.
Theorem 1 is illustrated in Figure 1.
Rogawski CalculusCopyright © 2008 W. H. Freeman and Company
The Mean Value Theorem set forth on the previous slide is a generalization of Rolle’s Theorem from the previous section. A corollary to the Mean Value Theorem is:
Example, Page236Find a point c satisfying the conclusion of the MVT for the given function and the interval.
2.
Rogawski CalculusCopyright © 2008 W. H. Freeman and Company
, 4,9y x
Rogawski CalculusCopyright © 2008 W. H. Freeman and Company
A function f (x) is monotonic if it is strictly increasing or strictlydecreasing on some interval (a, b).
1 2 1 2
1 2
1 2 1 2
on , if for all , ,
such that
on , if for all , ,
a b f x f x x x a b
x x
a b f x f x x x a b
Increasing
Decreasing
1 2 such that x x
Rogawski CalculusCopyright © 2008 W. H. Freeman and Company
The derivative of the function on the left of Figure 3 would be positiveand the derivative of the function on the right of Figure 3 would be negative, as noted in Theorem 2
Rogawski CalculusCopyright © 2008 W. H. Freeman and Company
For which intervals isf (x) as graphed in Figure 5 increasing?
Decreasing?
What happens to the derivative at the point thefunction changes from decreasing to increasing?
Rogawski CalculusCopyright © 2008 W. H. Freeman and Company
Using a sign chart to display the information in Figure 6,
1 3 5we have This indicates
that ( ) changes sign from positive to negative at 1, and
from negative to positive at 3. If w
x
F x
F x x
x F x
ere , then
would have a local maximum at 1 and local minimum
at 3.
f x
f x x
x
Rogawski CalculusCopyright © 2008 W. H. Freeman and Company
Figure 7 further illustrates the connection between the graphs of f (x)and f ′(x).
Rogawski CalculusCopyright © 2008 W. H. Freeman and Company
What we observed in Figures 6 and 7 lead us to Theorem 3.
Not stated, but implied in Theorem 3 is that the sign of f ′(x) may change at a critical point, but it may not change anywhere in the interval between two consecutive critical points.
Rogawski CalculusCopyright © 2008 W. H. Freeman and Company
Figure 8 shows how the critical points of y = f ′(x)correspond to the localminimum and maximumsof y = f (x).
Rogawski CalculusCopyright © 2008 W. H. Freeman and Company
The chart below is a variation of a sign chart used to analyze the behavior of a function on different intervals
The sign chart might also be drawn as below.
56 2 6x
f x
The sign charts on the previous slide both show that the function has:A relative maximum at π/6 because f ′(x) changes from increasing to decreasing at x = π/6.A relative minimum at π/2 because f ′(x) changes from decreasing to increasing at x = π/2.A relative maximum at 5π/6 because f ′(x) changes from increasing to decreasing at x = 5π/6.
Rogawski CalculusCopyright © 2008 W. H. Freeman and Company
If we were asked to justify that a relative maximum occurs at x = π/6,we could say: “ A local maximum occurs at x = π/6 by the First Derivative Test since the sign of f ′(x) changes from positive to negative at x = π/6.”
Rogawski CalculusCopyright © 2008 W. H. Freeman and Company
A critical point without an accompanying change of sign of the derivative is neither a minimum nor a maximum, as shown in Figure 9.
Example, Page23624. Figure 13 shows the graph of the derivative f ′(x) of a function f (x). Find the critical points of f (x) and determine whether they are local minima, maxima, or neither.
Rogawski CalculusCopyright © 2008 W. H. Freeman and Company
Rogawski CalculusCopyright © 2008 W. H. Freeman and Company
Rogawski CalculusCopyright © 2008 W. H. Freeman and Company
The table below summarizes the significance of the sign change of f ′(x) at a critical point. We always evaluate the sign change in the direction of increasing values of x.
Homework
Homework Assignment #24 Read Section 4.4 Page 2236, Exercises: 1 – 61 (EOO)
Rogawski CalculusCopyright © 2008 W. H. Freeman and Company