Contributed by:

In this section, we draw graphs by first considering the checklist that follows-

1. We don’t assume that you have a graphing device.

2. However, if you do have one, you should use it as a check on your work.

1. We don’t assume that you have a graphing device.

2. However, if you do have one, you should use it as a check on your work.

1.
4

APPLICATIONS OF DIFFERENTIATION

APPLICATIONS OF DIFFERENTIATION

2.
APPLICATIONS OF DIFFERENTIATION

So far, we have been concerned with some

particular aspects of curve sketching:

Domain, range, and symmetry (Chapter 1)

Limits, continuity, and asymptotes (Chapter 2)

Derivatives and tangents (Chapters 2 and 3)

Extreme values, intervals of increase and decrease,

concavity, points of inflection, and l’Hospital’s Rule

(This chapter)

So far, we have been concerned with some

particular aspects of curve sketching:

Domain, range, and symmetry (Chapter 1)

Limits, continuity, and asymptotes (Chapter 2)

Derivatives and tangents (Chapters 2 and 3)

Extreme values, intervals of increase and decrease,

concavity, points of inflection, and l’Hospital’s Rule

(This chapter)

3.
APPLICATIONS OF DIFFERENTIATION

It is now time to put all this information

together to sketch graphs that reveal

the important features of functions.

It is now time to put all this information

together to sketch graphs that reveal

the important features of functions.

4.
APPLICATIONS OF DIFFERENTIATION

4.5

Summary of

Curve Sketching

In this section, we will learn:

How to draw graphs of functions

using various guidelines.

4.5

Summary of

Curve Sketching

In this section, we will learn:

How to draw graphs of functions

using various guidelines.

5.
SUMMARY OF CURVE SKETCHING

You might ask:

Why don’t we just use a graphing calculator

or computer to graph a curve?

Why do we need to use calculus?

You might ask:

Why don’t we just use a graphing calculator

or computer to graph a curve?

Why do we need to use calculus?

6.
SUMMARY OF CURVE SKETCHING

It’s true that modern technology

is capable of producing very accurate

However, even the best graphing devices

have to be used intelligently.

It’s true that modern technology

is capable of producing very accurate

However, even the best graphing devices

have to be used intelligently.

7.
SUMMARY OF CURVE SKETCHING

We saw in Section 1.4 that it is extremely

important to choose an appropriate viewing

rectangle to avoid getting a misleading graph.

See especially Examples 1, 3, 4, and 5

in that section.

We saw in Section 1.4 that it is extremely

important to choose an appropriate viewing

rectangle to avoid getting a misleading graph.

See especially Examples 1, 3, 4, and 5

in that section.

8.
SUMMARY OF CURVE SKETCHING

The use of calculus enables us to:

Discover the most interesting aspects of graphs.

In many cases, calculate maximum and

minimum points and inflection points exactly

instead of approximately.

The use of calculus enables us to:

Discover the most interesting aspects of graphs.

In many cases, calculate maximum and

minimum points and inflection points exactly

instead of approximately.

9.
SUMMARY OF CURVE SKETCHING

For instance, the

figure shows

the graph of:

f(x) = 8x3 - 21x2 +

18x + 2

For instance, the

figure shows

the graph of:

f(x) = 8x3 - 21x2 +

18x + 2

10.
SUMMARY OF CURVE SKETCHING

At first glance, it

seems reasonable:

It has the same shape as

cubic curves like y = x3.

It appears to have no

maximum or minimum

point.

At first glance, it

seems reasonable:

It has the same shape as

cubic curves like y = x3.

It appears to have no

maximum or minimum

point.

11.
SUMMARY OF CURVE SKETCHING

However, if you compute

the derivative,

you will see that there is

a maximum when

x = 0.75 and a minimum

when x = 1.

Indeed, if we zoom in

to this portion of the

graph, we see that

behavior exhibited

in the next figure.

However, if you compute

the derivative,

you will see that there is

a maximum when

x = 0.75 and a minimum

when x = 1.

Indeed, if we zoom in

to this portion of the

graph, we see that

behavior exhibited

in the next figure.

12.
SUMMARY OF CURVE SKETCHING

Without calculus,

we could easily

have overlooked it.

Without calculus,

we could easily

have overlooked it.

13.
SUMMARY OF CURVE SKETCHING

In the next section, we will graph

functions by using the interaction

between calculus and graphing devices.

In the next section, we will graph

functions by using the interaction

between calculus and graphing devices.

14.
SUMMARY OF CURVE SKETCHING

In this section, we draw graphs by first

considering the checklist that follows.

We don’t assume that you have a graphing device.

However, if you do have one, you should use it

as a check on your work.

In this section, we draw graphs by first

considering the checklist that follows.

We don’t assume that you have a graphing device.

However, if you do have one, you should use it

as a check on your work.

15.
GUIDELINES FOR SKETCHING A CURVE

The following checklist is intended as a

guide to sketching a curve y = f(x) by hand.

Not every item is relevant to every function.

For instance, a given curve might not have

an asymptote or possess symmetry.

However, the guidelines provide all the information

you need to make a sketch that displays the most

important aspects of the function.

The following checklist is intended as a

guide to sketching a curve y = f(x) by hand.

Not every item is relevant to every function.

For instance, a given curve might not have

an asymptote or possess symmetry.

However, the guidelines provide all the information

you need to make a sketch that displays the most

important aspects of the function.

16.
A. DOMAIN

It’s often useful to start by determining

the domain D of f.

This is the set of values of x for which f(x)

is defined.

It’s often useful to start by determining

the domain D of f.

This is the set of values of x for which f(x)

is defined.

17.
B. INTERCEPTS

The y-intercept is f(0) and this tells us

where the curve intersects the y-axis.

To find the x-intercepts, we set y = 0

and solve for x.

You can omit this step if the equation is difficult

to solve.

The y-intercept is f(0) and this tells us

where the curve intersects the y-axis.

To find the x-intercepts, we set y = 0

and solve for x.

You can omit this step if the equation is difficult

to solve.

18.
C. SYMMETRY—EVEN FUNCTION

If f(-x) = f(x) for all x in D, that is, the equation

of the curve is unchanged when x is replaced

by -x, then f is an even function and the curve

is symmetric about the y-axis.

This means that our work is cut in half.

If f(-x) = f(x) for all x in D, that is, the equation

of the curve is unchanged when x is replaced

by -x, then f is an even function and the curve

is symmetric about the y-axis.

This means that our work is cut in half.

19.
C. SYMMETRY—EVEN FUNCTION

If we know what the

curve looks like for x ≥ 0,

then we need only reflect

about the y-axis to obtain

the complete curve.

If we know what the

curve looks like for x ≥ 0,

then we need only reflect

about the y-axis to obtain

the complete curve.

20.
C. SYMMETRY—EVEN FUNCTION

Here are some

y = x2

y = x4

y = |x|

y = cos x

Here are some

y = x2

y = x4

y = |x|

y = cos x

21.
C. SYMMETRY—ODD FUNCTION

If f(-x) = -f(x) for all x in D, then f

is an odd function and the curve is

symmetric about the origin.

If f(-x) = -f(x) for all x in D, then f

is an odd function and the curve is

symmetric about the origin.

22.
C. SYMMETRY—ODD FUNCTION

Again, we can obtain

the complete curve

if we know what it looks

like for x ≥ 0.

Rotate 180°

about the origin.

Again, we can obtain

the complete curve

if we know what it looks

like for x ≥ 0.

Rotate 180°

about the origin.

23.
C. SYMMETRY—ODD FUNCTION

Some simple

examples of odd

functions are:

y=x

y = x3

y = x5

y = sin x

Some simple

examples of odd

functions are:

y=x

y = x3

y = x5

y = sin x

24.
C. SYMMETRY—PERIODIC FUNCTION

If f(x + p) = f(x) for all x in D, where p

is a positive constant, then f is called

a periodic function.

The smallest such number p is called

the period.

For instance, y = sin x has period 2π and y = tan x

has period π.

If f(x + p) = f(x) for all x in D, where p

is a positive constant, then f is called

a periodic function.

The smallest such number p is called

the period.

For instance, y = sin x has period 2π and y = tan x

has period π.

25.
C. SYMMETRY—PERIODIC

If we know what the

graph looks like in

an interval of length p,

then we can use

translation to sketch the

entire graph.

If we know what the

graph looks like in

an interval of length p,

then we can use

translation to sketch the

entire graph.

26.
D. ASYMPTOTES—HORIZONTAL

Recall from Section 2.6 that, if either

lim f ( x) L or xlim f ( x ) L ,

x

then the line y = L is a horizontal asymptote

of the curve y = f (x).

If it turns out that lim f ( x) (or -∞), then

x

we do not have an asymptote to the right.

Nevertheless, that is still useful information

for sketching the curve.

Recall from Section 2.6 that, if either

lim f ( x) L or xlim f ( x ) L ,

x

then the line y = L is a horizontal asymptote

of the curve y = f (x).

If it turns out that lim f ( x) (or -∞), then

x

we do not have an asymptote to the right.

Nevertheless, that is still useful information

for sketching the curve.

27.
D. ASYMPTOTES—VERTICAL Equation 1

Recall from Section 2.2 that the line x = a

is a vertical asymptote if at least one of

the following statements is true:

lim f ( x) lim f ( x)

x a x a

lim f ( x) lim f ( x)

x a x a

Recall from Section 2.2 that the line x = a

is a vertical asymptote if at least one of

the following statements is true:

lim f ( x) lim f ( x)

x a x a

lim f ( x) lim f ( x)

x a x a

28.
D. ASYMPTOTES—VERTICAL

For rational functions, you can locate

the vertical asymptotes by equating

the denominator to 0 after canceling any

common factors.

However, for other functions, this method

does not apply.

For rational functions, you can locate

the vertical asymptotes by equating

the denominator to 0 after canceling any

common factors.

However, for other functions, this method

does not apply.

29.
D. ASYMPTOTES—VERTICAL

Furthermore, in sketching the curve, it is very

useful to know exactly which of the statements

in Equation 1 is true.

If f(a) is not defined but a is an endpoint of

the domain of f, then you should compute xlim a

f ( x)

or lim f ( x) , whether or not this limit is infinite.

x a

Furthermore, in sketching the curve, it is very

useful to know exactly which of the statements

in Equation 1 is true.

If f(a) is not defined but a is an endpoint of

the domain of f, then you should compute xlim a

f ( x)

or lim f ( x) , whether or not this limit is infinite.

x a

30.
D. ASYMPTOTES—SLANT

Slant asymptotes are discussed

at the end of this section.

Slant asymptotes are discussed

at the end of this section.

31.
E. INTERVALS OF INCREASE OR DECREASE

Use the I /D Test.

Compute f’(x) and find the intervals

on which:

f’(x) is positive (f is increasing).

f’(x) is negative (f is decreasing).

Use the I /D Test.

Compute f’(x) and find the intervals

on which:

f’(x) is positive (f is increasing).

f’(x) is negative (f is decreasing).

32.
F. LOCAL MAXIMUM AND MINIMUM VALUES

Find the critical numbers of f (the numbers c

where f’(c) = 0 or f’(c) does not exist).

Then, use the First Derivative Test.

If f’ changes from positive to negative at

a critical number c, then f(c) is a local maximum.

If f’ changes from negative to positive at c,

then f(c) is a local minimum.

Find the critical numbers of f (the numbers c

where f’(c) = 0 or f’(c) does not exist).

Then, use the First Derivative Test.

If f’ changes from positive to negative at

a critical number c, then f(c) is a local maximum.

If f’ changes from negative to positive at c,

then f(c) is a local minimum.

33.
F. LOCAL MAXIMUM AND MINIMUM VALUES

Although it is usually preferable to use the

First Derivative Test, you can use the Second

Derivative Test if f’(c) = 0 and f’’(c) ≠ 0.

f”(c) > 0 implies that f(c) is a local minimum.

f’’(c) < 0 implies that f(c) is a local maximum.

Although it is usually preferable to use the

First Derivative Test, you can use the Second

Derivative Test if f’(c) = 0 and f’’(c) ≠ 0.

f”(c) > 0 implies that f(c) is a local minimum.

f’’(c) < 0 implies that f(c) is a local maximum.

34.
G. CONCAVITY AND POINTS OF INFLECTION

Compute f’’(x) and use the Concavity Test.

The curve is:

Concave upward where f’’(x) > 0

Concave downward where f’’(x) < 0

Compute f’’(x) and use the Concavity Test.

The curve is:

Concave upward where f’’(x) > 0

Concave downward where f’’(x) < 0

35.
G. CONCAVITY AND POINTS OF INFLECTION

Inflection points occur

where the direction of concavity

Inflection points occur

where the direction of concavity

36.
H. SKETCH AND CURVE

Using the information in items A–G,

draw the graph.

Sketch the asymptotes as dashed lines.

Plot the intercepts, maximum and minimum points,

and inflection points.

Then, make the curve pass through these points,

rising and falling according to E, with concavity

according to G, and approaching the asymptotes

Using the information in items A–G,

draw the graph.

Sketch the asymptotes as dashed lines.

Plot the intercepts, maximum and minimum points,

and inflection points.

Then, make the curve pass through these points,

rising and falling according to E, with concavity

according to G, and approaching the asymptotes

37.
H. SKETCH AND CURVE

If additional accuracy is desired near

any point, you can compute the value of

the derivative there.

The tangent indicates the direction in which

the curve proceeds.

If additional accuracy is desired near

any point, you can compute the value of

the derivative there.

The tangent indicates the direction in which

the curve proceeds.

38.
GUIDELINES Example 1

Use the guidelines to sketch

the curve 2

2x

y 2

x 1

Use the guidelines to sketch

the curve 2

2x

y 2

x 1

39.
GUIDELINES Example 1

A. The domain is:

{x | x2 – 1 ≠ 0} = {x | x ≠ ±1}

= (-∞, -1) U (-1, -1) U (1, ∞)

B. The x- and y-intercepts are both 0.

A. The domain is:

{x | x2 – 1 ≠ 0} = {x | x ≠ ±1}

= (-∞, -1) U (-1, -1) U (1, ∞)

B. The x- and y-intercepts are both 0.

40.
GUIDELINES Example 1

C. Since f(-x) = f(x), the function

is even.

The curve is symmetric about the y-axis.

C. Since f(-x) = f(x), the function

is even.

The curve is symmetric about the y-axis.

41.
GUIDELINES Example 1

2

D. lim

2 x 2

2

lim 2

2

x x 1 x 1 1/ x

Therefore, the line y = 2 is a horizontal

2

D. lim

2 x 2

2

lim 2

2

x x 1 x 1 1/ x

Therefore, the line y = 2 is a horizontal

42.
GUIDELINES Example 1

Since the denominator is 0 when x = ±1,

we compute the following limits:

2 x2 2x2

lim 2 lim 2

x 1 x 1 x 1 x 1

2 2

2x 2x

lim 2 lim 2

x 1 x 1 x 1 x 1

Thus, the lines x = 1 and x = -1 are vertical asymptotes.

Since the denominator is 0 when x = ±1,

we compute the following limits:

2 x2 2x2

lim 2 lim 2

x 1 x 1 x 1 x 1

2 2

2x 2x

lim 2 lim 2

x 1 x 1 x 1 x 1

Thus, the lines x = 1 and x = -1 are vertical asymptotes.

43.
GUIDELINES Example 1

This information about

limits and asymptotes

enables us to draw the

preliminary sketch,

showing the parts of the

curve near the

This information about

limits and asymptotes

enables us to draw the

preliminary sketch,

showing the parts of the

curve near the

44.
GUIDELINES Example 1

2 2

E. f '( x) 4 x ( x 1) 2 x 2 x 4 x

2 2

2 2

( x 1) ( x 1)

Since f’(x) > 0 when x < 0 (x ≠ 1) and f’(x) < 0

when x > 0 (x ≠ 1), f is:

Increasing on (-∞, -1) and (-1, 0)

Decreasing on (0, 1) and (1, ∞)

2 2

E. f '( x) 4 x ( x 1) 2 x 2 x 4 x

2 2

2 2

( x 1) ( x 1)

Since f’(x) > 0 when x < 0 (x ≠ 1) and f’(x) < 0

when x > 0 (x ≠ 1), f is:

Increasing on (-∞, -1) and (-1, 0)

Decreasing on (0, 1) and (1, ∞)

45.
GUIDELINES Example 1

F. The only critical number is x = 0.

Since f’ changes from positive to negative

at 0, f(0) = 0 is a local maximum by the First

Derivative Test.

F. The only critical number is x = 0.

Since f’ changes from positive to negative

at 0, f(0) = 0 is a local maximum by the First

Derivative Test.

46.
GUIDELINES Example 1

2 2 2 2

4( x 1) 4 x 2( x 1)2 x 12 x 4

G. f ''( x ) 2 4

2 3

( x 1) ( x 1)

Since 12x2 + 4 > 0 for all x, we have

f ''( x) 0 x2 1 0 x 1

and f ''( x) 0 x 1

2 2 2 2

4( x 1) 4 x 2( x 1)2 x 12 x 4

G. f ''( x ) 2 4

2 3

( x 1) ( x 1)

Since 12x2 + 4 > 0 for all x, we have

f ''( x) 0 x2 1 0 x 1

and f ''( x) 0 x 1

47.
GUIDELINES Example 1

Thus, the curve is concave upward on

the intervals (-∞, -1) and (1, ∞) and concave

downward on (-1, -1).

It has no point of inflection since 1 and -1

are not in the domain of f.

Thus, the curve is concave upward on

the intervals (-∞, -1) and (1, ∞) and concave

downward on (-1, -1).

It has no point of inflection since 1 and -1

are not in the domain of f.

48.
GUIDELINES Example 1

H. Using the

information in E–G,

we finish the sketch.

H. Using the

information in E–G,

we finish the sketch.

49.
GUIDELINES Example 2

Sketch the graph of:

2

x

f ( x)

x 1

Sketch the graph of:

2

x

f ( x)

x 1

50.
GUIDELINES Example 2

A. Domain = {x | x + 1 > 0}

= {x | x > -1}

= (-1, ∞)

B. The x- and y-intercepts are both 0.

C. Symmetry: None

A. Domain = {x | x + 1 > 0}

= {x | x > -1}

= (-1, ∞)

B. The x- and y-intercepts are both 0.

C. Symmetry: None

51.
GUIDELINES Example 2

x2

D. Since lim , there is no horizontal

x x 1

Since x 1 0 as x → -1+ and 2f(x) is

x

always positive, we havexlim ,

1

x 1

and so the line x = -1 is a vertical asymptote

x2

D. Since lim , there is no horizontal

x x 1

Since x 1 0 as x → -1+ and 2f(x) is

x

always positive, we havexlim ,

1

x 1

and so the line x = -1 is a vertical asymptote

52.
GUIDELINES Example 2

2

E. f '( x) 2 x x 1 x 1/(2 x 1) x(3 x 4)

3/ 2

x 1 2( x 1)

We see that f’(x) = 0 when x = 0 (notice that

-4/3 is not in the domain of f).

So, the only critical number is 0.

2

E. f '( x) 2 x x 1 x 1/(2 x 1) x(3 x 4)

3/ 2

x 1 2( x 1)

We see that f’(x) = 0 when x = 0 (notice that

-4/3 is not in the domain of f).

So, the only critical number is 0.

53.
GUIDELINES Example 2

As f’(x) < 0 when -1 < x < 0 and f’(x) > 0

when x > 0, f is:

Decreasing on (-1, 0)

Increasing on (0, ∞)

As f’(x) < 0 when -1 < x < 0 and f’(x) > 0

when x > 0, f is:

Decreasing on (-1, 0)

Increasing on (0, ∞)

54.
GUIDELINES Example 2

F. Since f’(0) = 0 and f’ changes from

negative to positive at 0, f(0) = 0 is

a local (and absolute) minimum by

the First Derivative Test.

F. Since f’(0) = 0 and f’ changes from

negative to positive at 0, f(0) = 0 is

a local (and absolute) minimum by

the First Derivative Test.

55.
GUIDELINES Example 2

3/ 2 2 1/ 2

G. f ''( x) 2( x 1) (6 x 4) (3 x 4)3( x 1)

3

4( x 1)

2

3x 8 x 8

5/ 2

4( x 1)

Note that the denominator is always positive.

The numerator is the quadratic 3x2 + 8x + 8,

which is always positive because its discriminant

is b2 - 4ac = -32, which is negative, and the coefficient

of x2 is positive.

3/ 2 2 1/ 2

G. f ''( x) 2( x 1) (6 x 4) (3 x 4)3( x 1)

3

4( x 1)

2

3x 8 x 8

5/ 2

4( x 1)

Note that the denominator is always positive.

The numerator is the quadratic 3x2 + 8x + 8,

which is always positive because its discriminant

is b2 - 4ac = -32, which is negative, and the coefficient

of x2 is positive.

56.
GUIDELINES Example 2

So, f”(x) > 0 for all x in the domain of f.

This means that:

f is concave upward on (-1, ∞).

There is no point of inflection.

So, f”(x) > 0 for all x in the domain of f.

This means that:

f is concave upward on (-1, ∞).

There is no point of inflection.

57.
GUIDELINES Example 2

H. The curve is

sketched here.

H. The curve is

sketched here.

58.
GUIDELINES Example 3

Sketch the graph of:

f(x) = xex

Sketch the graph of:

f(x) = xex

59.
GUIDELINES Example 3

A. The domain is .

B. The x- and y-intercepts are both 0.

C. Symmetry: None

A. The domain is .

B. The x- and y-intercepts are both 0.

C. Symmetry: None

60.
GUIDELINES Example 3

D. As both x and ex become large as

x

x → ∞, we have lim xe

x

However, as x → -∞, ex → 0.

D. As both x and ex become large as

x

x → ∞, we have lim xe

x

However, as x → -∞, ex → 0.

61.
GUIDELINES Example 3

So, we have an indeterminate product that

requires the use of l’Hospital’s Rule:

x x 1 x

lim xe lim x lim x lim ( e )

x x e x e x

0

Thus, the x-axis is a horizontal asymptote.

So, we have an indeterminate product that

requires the use of l’Hospital’s Rule:

x x 1 x

lim xe lim x lim x lim ( e )

x x e x e x

0

Thus, the x-axis is a horizontal asymptote.

62.
GUIDELINES Example 3

E. f’(x) = xex + ex = (x + 1) ex

As ex is always positive, we see that f’(x) > 0

when x + 1 > 0, and f’(x) < 0 when x + 1 < 0.

So, f is:

Increasing on (-1, ∞)

Decreasing on (-∞, -1)

E. f’(x) = xex + ex = (x + 1) ex

As ex is always positive, we see that f’(x) > 0

when x + 1 > 0, and f’(x) < 0 when x + 1 < 0.

So, f is:

Increasing on (-1, ∞)

Decreasing on (-∞, -1)

63.
GUIDELINES Example 3

F. Since f’(-1) = 0 and f’ changes

from negative to positive at x = -1,

f(-1) = -e-1 is a local (and absolute)

F. Since f’(-1) = 0 and f’ changes

from negative to positive at x = -1,

f(-1) = -e-1 is a local (and absolute)

64.
GUIDELINES Example 3

G. f’’(x) = (x + 1)ex + ex = (x + 2)ex

f”(x) = 0 if x > -2 and f’’(x) < 0 if x < -2.

So, f is concave upward on (-2, ∞) and concave

downward on (-∞, -2).

The inflection point is (-2, -2e-2)

G. f’’(x) = (x + 1)ex + ex = (x + 2)ex

f”(x) = 0 if x > -2 and f’’(x) < 0 if x < -2.

So, f is concave upward on (-2, ∞) and concave

downward on (-∞, -2).

The inflection point is (-2, -2e-2)

65.
GUIDELINES Example 3

H. We use this

information to sketch

the curve.

H. We use this

information to sketch

the curve.

66.
GUIDELINES Example 4

Sketch the graph of:

cos x

f ( x)

2 sin x

Sketch the graph of:

cos x

f ( x)

2 sin x

67.
GUIDELINES Example 4

A. The domain is

B. The y-intercept is f(0) = ½.

The x-intercepts occur when cos x =0,

that is, x = (2n + 1)π/2, where n is an integer.

A. The domain is

B. The y-intercept is f(0) = ½.

The x-intercepts occur when cos x =0,

that is, x = (2n + 1)π/2, where n is an integer.

68.
GUIDELINES Example 4

C. f is neither even nor odd.

However, f(x + 2π) = f(x) for all x.

Thus, f is periodic and has period 2π.

So, in what follows, we need to consider only 0 ≤ x ≤ 2π

and then extend the curve by translation in part H.

D. Asymptotes: None

C. f is neither even nor odd.

However, f(x + 2π) = f(x) for all x.

Thus, f is periodic and has period 2π.

So, in what follows, we need to consider only 0 ≤ x ≤ 2π

and then extend the curve by translation in part H.

D. Asymptotes: None

69.
GUIDELINES Example 4

E. f '( x) (2 sin x )( sin x ) cos x (cos x )

2

(2 sin x)

2sin x 1

2

(2 sin x)

Thus, f’(x) > 0 when 2 sin x + 1 < 0

sin x < -½ 7π/6 < x < 11π/6

E. f '( x) (2 sin x )( sin x ) cos x (cos x )

2

(2 sin x)

2sin x 1

2

(2 sin x)

Thus, f’(x) > 0 when 2 sin x + 1 < 0

sin x < -½ 7π/6 < x < 11π/6

70.
GUIDELINES Example 4

Thus, f is:

Increasing on (7π/6, 11π/6)

Decreasing on (0, 7π/6) and (11π/6, 2π)

Thus, f is:

Increasing on (7π/6, 11π/6)

Decreasing on (0, 7π/6) and (11π/6, 2π)

71.
GUIDELINES Example 4

F. From part E and the First Derivative

Test, we see that:

The local minimum value is f(7π/6) = -1/ 3

The local maximum value is f(11π/6) = -1/ 3

F. From part E and the First Derivative

Test, we see that:

The local minimum value is f(7π/6) = -1/ 3

The local maximum value is f(11π/6) = -1/ 3

72.
GUIDELINES Example 4

G. If we use the Quotient Rule again

and simplify, we get: 2 cos x(1 sin x)

f ''( x) 3

(2 sin x)

(2 + sin x)3 > 0 and 1 – sin x ≥ 0 for all x.

So, we know that f’’(x) > 0 when cos x < 0,

that is, π/2 < x < 3π/2.

G. If we use the Quotient Rule again

and simplify, we get: 2 cos x(1 sin x)

f ''( x) 3

(2 sin x)

(2 + sin x)3 > 0 and 1 – sin x ≥ 0 for all x.

So, we know that f’’(x) > 0 when cos x < 0,

that is, π/2 < x < 3π/2.

73.
GUIDELINES Example 4

Thus, f is concave upward on (π/2, 3π/2)

and concave downward on (0, π/2) and

(3π/2, 2π).

The inflection points are (π/2, 0) and

(3π/2, 0).

Thus, f is concave upward on (π/2, 3π/2)

and concave downward on (0, π/2) and

(3π/2, 2π).

The inflection points are (π/2, 0) and

(3π/2, 0).

74.
GUIDELINES Example 4

H. The graph of the

function restricted

to 0 ≤ x ≤ 2π is shown

H. The graph of the

function restricted

to 0 ≤ x ≤ 2π is shown

75.
GUIDELINES Example 4

Then, we extend it,

using periodicity,

to the complete graph

Then, we extend it,

using periodicity,

to the complete graph

76.
GUIDELINES Example 5

Sketch the graph of:

y = ln(4 - x2)

Sketch the graph of:

y = ln(4 - x2)

77.
GUIDELINES Example 5

A. The domain is:

{x | 4 − x2 > 0} = {x | x2 < 4}

= {x | |x| < 2}

= (−2, 2)

A. The domain is:

{x | 4 − x2 > 0} = {x | x2 < 4}

= {x | |x| < 2}

= (−2, 2)

78.
GUIDELINES Example 5

B. The y-intercept is: f(0) = ln 4

To find the x-intercept, we set:

y = ln(4 – x2) = 0

We know that ln 1 = 0.

So, we have 4 – x2 = 1 x2 = 3

Therefore, the x-intercepts are: 3

B. The y-intercept is: f(0) = ln 4

To find the x-intercept, we set:

y = ln(4 – x2) = 0

We know that ln 1 = 0.

So, we have 4 – x2 = 1 x2 = 3

Therefore, the x-intercepts are: 3

79.
GUIDELINES Example 5

C. f(-x) = f(x)

f is even.

The curve is symmetric about the y-axis.

C. f(-x) = f(x)

f is even.

The curve is symmetric about the y-axis.

80.
GUIDELINES Example 5

D. We look for vertical asymptotes

at the endpoints of the domain.

Since 4 − x2 → 0+ as x → 2- and as x → 2+,

we have:

lim ln(4 x 2 ) lim ln(4 x 2 )

x 2 x 2

Thus, the lines x = 2 and x = -2

are vertical asymptotes.

D. We look for vertical asymptotes

at the endpoints of the domain.

Since 4 − x2 → 0+ as x → 2- and as x → 2+,

we have:

lim ln(4 x 2 ) lim ln(4 x 2 )

x 2 x 2

Thus, the lines x = 2 and x = -2

are vertical asymptotes.

81.
GUIDELINES Example 5

2x

E. f '( x ) 2

4 x

f’(x) > 0 when -2 < x < 0 and f’(x) < 0 when

0 < x < 2.

So, f is:

Increasing on (-2, 0)

Decreasing on (0, 2)

2x

E. f '( x ) 2

4 x

f’(x) > 0 when -2 < x < 0 and f’(x) < 0 when

0 < x < 2.

So, f is:

Increasing on (-2, 0)

Decreasing on (0, 2)

82.
GUIDELINES Example 5

F. The only critical number is x = 0.

As f’ changes from positive to negative at 0,

f(0) = ln 4 is a local maximum by the First

Derivative Test.

F. The only critical number is x = 0.

As f’ changes from positive to negative at 0,

f(0) = ln 4 is a local maximum by the First

Derivative Test.

83.
GUIDELINES Example 5

2 2

G. f ''( x) (4 x )( 2) 2 x( 2 x) 8 2 x

2 2 2 2

(4 x ) (4 x )

Since f”(x) < 0 for all x, the curve is

concave downward on (-2, 2) and has

no inflection point.

2 2

G. f ''( x) (4 x )( 2) 2 x( 2 x) 8 2 x

2 2 2 2

(4 x ) (4 x )

Since f”(x) < 0 for all x, the curve is

concave downward on (-2, 2) and has

no inflection point.

84.
GUIDELINES Example 5

H. Using this

information, we

the curve.

H. Using this

information, we

the curve.

85.
SLANT ASYMPTOTES

Some curves have asymptotes that

are oblique—that is, neither horizontal

nor vertical.

Some curves have asymptotes that

are oblique—that is, neither horizontal

nor vertical.

86.
SLANT ASYMPTOTES

If lim[ f ( x) ( mx b)] 0

x

, then the line

y = mx + b is called a

slant asymptote.

This is because

the vertical distance

between the curve

y = f(x) and the line

y = mx + b approaches 0.

A similar situation exists

if we let x → -∞.

If lim[ f ( x) ( mx b)] 0

x

, then the line

y = mx + b is called a

slant asymptote.

This is because

the vertical distance

between the curve

y = f(x) and the line

y = mx + b approaches 0.

A similar situation exists

if we let x → -∞.

87.
SLANT ASYMPTOTES

For rational functions, slant asymptotes occur

when the degree of the numerator is one more

than the degree of the denominator.

In such a case, the equation of the slant

asymptote can be found by long division—

as in following example.

For rational functions, slant asymptotes occur

when the degree of the numerator is one more

than the degree of the denominator.

In such a case, the equation of the slant

asymptote can be found by long division—

as in following example.

88.
SLANT ASYMPTOTES Example 6

Sketch the graph of:

3

x

f ( x) 2

x 1

Sketch the graph of:

3

x

f ( x) 2

x 1

89.
SLANT ASYMPTOTES Example 6

A. The domain is: = (-∞, ∞)

B. The x- and y-intercepts are both 0.

C. As f(-x) = -f(x), f is odd and its graph is

symmetric about the origin.

A. The domain is: = (-∞, ∞)

B. The x- and y-intercepts are both 0.

C. As f(-x) = -f(x), f is odd and its graph is

symmetric about the origin.

90.
SLANT ASYMPTOTES Example 6

Since x2 + 1 is never 0, there is no vertical

Since f(x) → ∞ as x → ∞ and f(x) → -∞ as

x → - ∞, there is no horizontal asymptote.

Since x2 + 1 is never 0, there is no vertical

Since f(x) → ∞ as x → ∞ and f(x) → -∞ as

x → - ∞, there is no horizontal asymptote.

91.
SLANT ASYMPTOTES Example 6

However, long division gives:

x3 x

f ( x) 2 x 2

x 1 x 1

1

x x

f ( x) x 2 0 as x

x 1 1

1 2

x

So, the line y = x is a slant asymptote.

However, long division gives:

x3 x

f ( x) 2 x 2

x 1 x 1

1

x x

f ( x) x 2 0 as x

x 1 1

1 2

x

So, the line y = x is a slant asymptote.

92.
SLANT ASYMPTOTES Example 6

2 2 3 2 2

E. 3 x ( x 1) x 2 x x ( x 3)

f '( x) 2 2

2 2

( x 1) ( x 1)

Since f’(x) > 0 for all x (except 0), f is

increasing on (- ∞, ∞).

2 2 3 2 2

E. 3 x ( x 1) x 2 x x ( x 3)

f '( x) 2 2

2 2

( x 1) ( x 1)

Since f’(x) > 0 for all x (except 0), f is

increasing on (- ∞, ∞).

93.
SLANT ASYMPTOTES Example 6

F. Although f’(0) = 0, f’ does not

change sign at 0.

So, there is no local maximum or minimum.

F. Although f’(0) = 0, f’ does not

change sign at 0.

So, there is no local maximum or minimum.

94.
SLANT ASYMPTOTES Example 6

(4 x 3 6 x)( x 2 1) 2 ( x 4 3 x 2 ) 2( x 2 1)2 x

G. f ''( x)

( x 2 1) 4

2 x(3 x 2 )

2 3

( x 1)

Since f’’(x) = 0 when x = 0 or x = ± 3 ,

we set up the following chart.

(4 x 3 6 x)( x 2 1) 2 ( x 4 3 x 2 ) 2( x 2 1)2 x

G. f ''( x)

( x 2 1) 4

2 x(3 x 2 )

2 3

( x 1)

Since f’’(x) = 0 when x = 0 or x = ± 3 ,

we set up the following chart.

95.
SLANT ASYMPTOTES Example 6

The points of

inflection are:

3 3

(− , −¾ )

(0, 0)

(

3 3

,¾ )

The points of

inflection are:

3 3

(− , −¾ )

(0, 0)

(

3 3

,¾ )

96.
SLANT ASYMPTOTES Example 6

H. The graph of f is

H. The graph of f is