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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.
4
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)
3.
APPLICATIONS OF DIFFERENTIATION
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.
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?
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.
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.
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.
9.
SUMMARY OF CURVE SKETCHING
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.
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.
12.
SUMMARY OF CURVE SKETCHING
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.
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.
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.
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.
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.
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.
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.
20.
C. SYMMETRY—EVEN FUNCTION
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.
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.
23.
C. SYMMETRY—ODD FUNCTION
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.
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.
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.
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
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.
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
30.
D. ASYMPTOTES—SLANT
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).
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.
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.
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
35.
G. CONCAVITY AND POINTS OF INFLECTION
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
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.
38.
GUIDELINES Example 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.
40.
GUIDELINES Example 1
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
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.
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
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, ∞)
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.
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
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.
48.
GUIDELINES Example 1
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
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
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
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.
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, ∞)
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.
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.
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.
57.
GUIDELINES Example 2
H. The curve is
sketched here.
58.
GUIDELINES Example 3
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
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.
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.
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)
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)
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)
65.
GUIDELINES Example 3
H. We use this
information to sketch
the curve.
66.
GUIDELINES Example 4
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.
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
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
70.
GUIDELINES Example 4
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
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.
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).
74.
GUIDELINES Example 4
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
76.
GUIDELINES Example 5
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)
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
79.
GUIDELINES Example 5
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.
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)
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.
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.
84.
GUIDELINES Example 5
H. Using this
information, we
the curve.
85.
SLANT ASYMPTOTES
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 → -∞.
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.
88.
SLANT ASYMPTOTES Example 6
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.
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.
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.
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 (- ∞, ∞).
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.
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.
95.
SLANT ASYMPTOTES Example 6
The points of
inflection are:
3 3
(− , −¾ )
(0, 0)
(
3 3
,¾ )
96.
SLANT ASYMPTOTES Example 6
H. The graph of f is
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