Homework Homework Assignment #23 Read Section 4.3 Page 227, Exercises: 1 – 77 (EOO), skip 57, 69 Rogawski Calculus Copyright © 2008 W. H. Freeman and Company

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


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]


Homework, Page 227Find all critical points of the function.

5.


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


9.


1

3f x x

1 23 3

23

1

31

0 undefined 03

f x x f x x

x


13.


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 .


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


Homework, Page 227Find the minimum and maximum values of the function on the given interval.

25.


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


29.


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 minimum 2 12.314

y x x

y

y

y

y

y


33.


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



y x x x x x

y y

y y

y

y


37.


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 minimum 6 18.5

x x x x x x x xy

x x x

x x x

y y

y

y


41.


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


xxy x x

x x

x xx x

x

x x x x x

y y

y

y

6


45.


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


49.


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


53.


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.


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.


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).


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.


Let 10rv

Homework, Page 227Plot the function and find the critical points and extreme values on [–5, 5].

77.


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


Chapter 4: Applications of the DerivativeSection 4.3: The Mean Value Theorem and

Monotonicity

Jon Rogawski

Calculus, ETFirst Edition


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.


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.


, 4,9y x


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


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


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?


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


Figure 7 further illustrates the connection between the graphs of f (x)and f ′(x).


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.


Figure 8 shows how the critical points of y = f ′(x)correspond to the localminimum and maximumsof y = f (x).


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.


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.”


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.




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)


Documents

Homework Homework Assignment #23 Read Section 4.3 Page 227, Exercises: 1 – 77 (EOO), skip 57, 69 Rogawski Calculus Copyright © 2008 W. H. Freeman and Company