Calculus Of One Real Variable By Pheng Kim Ving
Chapter 5: Applications Of The Derivative Part 1 Section 5.6: Sketching Graphs Of Functions

 

5.6
Sketching Graphs Of Functions

 

 

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1. Asymptotes

 

Horizontal Asymptotes

 

 

Fig. 1.1

 

Horizontal And Vertical Asymptotes.

 

Fig. 1.2

 

Horizontal Asymptotes.

 

 

The horizontal line y = L is called a horizontal asymptote of the graph of y = f(x) if:

 

 

 

Vertical Asymptotes

 

 

Fig. 1.3

 

Vertical Asymptotes.

 

 

The vertical line x = a is called a vertical asymptote of the graph of y = f(x) if one of the following is satisfied:

 

 

 

Oblique Asymptotes

 

The graph of f(x) = (x2 + 5x 4)/(2x 2) is sketched in Fig. 1.4. Long division of the numerator x2 + 5x 4 by the
denominator 2x 2 gives: quotient = (1/2)x + 3 and remainder = 2. So:

 

 

Fig. 1.4

 

Oblique asymptote of graph of y = f(x) = (x2 + 5x 4)/(2x 2) is
line
y = (1/2)x + 3.

 

 

 

 

Remark 1.1

 

The graph of a function can intersect a horizontal or oblique asymptote, but can never intersect a vertical asymptote (why?
hint: definition of a function).

 

One- And Two-Sided Asymptotes

 

 

iii. The graph of a function f may have two one-sided horizontal asymptotes. For example, see Fig. 1.5.

 

iv. The notions of one- and two-sided asymptotes also apply to vertical and oblique asymptotes.

 

Fig. 1.5

 

The graph of y = f(x) has two one-sided horizontal asymptotes.

 

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2. Asymptotes Of Rational Functions

 

Let f(x) = P(x)/Q(x) be a rational function, ie, a ratio or fraction of two polynomials P(x) and Q(x). Let's denote the
degree of a polynomial p by deg( p). Then:

 

 

 

The graph of f has a vertical asymptote at every point x where Q(x) = 0.

 

 

 

 

If deg(P) < deg(Q), then the line y = 0 (the x-axis) is a horizontal asymptote of the graph of f.

 

 

iii. Suppose deg(P) = deg(Q). Let a and b be the coefficients of the dominating terms of P and Q respectively; see

 

 

If deg(P) = deg(Q), then the line y = a/b, where a and b are the coefficients of the dominating terms of P and
Q respectively, is a horizontal asymptote of the graph of f. The line y = a/b cannot be the x-axis.

 

 

iv. Suppose deg(P) = deg(Q) + 1. For an example, see the discussion of the function f(x) = (x2 + 5x 4)/(2x 2)
above and its graph sketched in Fig. 1.4. Divide P by Q using long division to get:

 

 

 

 

 

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3. Sketching Graphs Of Functions

 

When sketching the graph of a function y = f(x), we have three sources of useful information: f itself, f ', and f ''. We'll
follow these suggested steps:

 

i. Information From f: Determine each of the following if any:
a. The domain of f.

b. Intercepts: x-intercepts and y-intercept.
c. Symmetry of the graph (is f even, odd, or neither?).
d. Limits at points of discontinuity and at positive and negative infinity.
e. Asymptotes: determine vertical and horizontal ones from the limits above, and oblique ones by long division if
appropriate.

ii. Information From f ':

a. Find points x where f '(x) = 0 or f '(x) doesn't exist (critical points). Calculate the value of f at each of them if
defined.
b. Determine the signs of f '; if it's not obvious to do so by simply examining the expression of f ' , draw a chart for
f '; see Section 5.3 Part 3 and Part 5.

 

iii. Information From f '':

a. Find points x where f ''(x) = 0 or f ''(x) doesn't exist. Calculate the value of f at each of them if defined.
b. Determine the signs of f ''; if it's not obvious to do so by simply examining the expression of f '' , draw a chart for
f ''; see Section 5.4 Part 4.

 

iv. Draw the chart for f, where the intervals of increase, decrease, and concavity are determined; see Section 5.4 Part 4. Indicate local maxima, local minima, and inflection points if any.

 

v. Determine additional points of the graph if helpful.

 

vi. Sketch the graph using all the information obtained above.

 

Example 3.1

 

Sketch the graph of the function:

 

 

Solution

 

Asymptotes:
long division yields:

 

 

vertical: lines x = 1 and x = 1,
horizontal: none,
oblique: line y = x.

 

First Derivative:

 

 

The chart for y'' is shown in Fig. 3.1. The chart for y is shown in Fig. 3.2. The graph of y is shown in Fig. 3.3.

 

Fig. 3.1

 

Chart For y''.

 

Fig. 3.2

 

Chart For y.

 

Fig. 3.3

 

Graph of:

 

EOS

 

Remark 3.1

 

In the charts, the double vertical line below a point x and on a row means that the function on that row isn't defined or
doesn't exist at that point.

 

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Problems & Solutions

 

1. Sketch the graph of the function y = x2(x2 1), making use of any suitable information you can obtain from the
function and its first and second derivatives.

 

Solution

 

Domain: R.

 

y = x2(x2 1) = x2(x + 1)(x 1);
y = x2(x2 1) = x4 x2.

 

Intercepts:

x-intercepts: x = 0, 1, and 1,
y-intercept: y = 0.

 

Symmetry:

y(x) = (x)4 (x)2 = x4 x2 = y(x);
function is even; graph is symmetric about y-axis.

 

Limits:

 

 

Second Derivative:

 

from y' = 4x3 2x we get:

 

 

 

 

 

 

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2. Sketch the graph of the function:

 

 

using information from the function and its first and second derivatives.

 

Solution

 

Domain: R {1, 1}.

 

Intercepts:
x-Intercepts: x = 0,
y-Intercept: y = 0.

 

Symmetry:

 

Second Derivative:

 

 

 

 

 

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3. Sketch the graph of the function:

 

 

using information from the function and its first and second derivatives.

 

Solution

 

Domain: R {1}.

 

Intercepts:

x-intercepts: x = 2,
y-intercept: y = 4.

 

Symmetry:

 

 

y' is never 0;
y' doesn't exist at x = 1 where y doesn't exist either;

 

 

 

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4. Sketch the graph of the function:

 

 

utilizing any useful information you can get from f , f ', and f ''.

 

Solution

 

Domain: R {1}.

 

 

Asymptotes:
long division yields:

 

 

vertical: line x = 1,

horizontal: none,
oblique: line y = x 1.

 

First Derivative:

 

 

 

 

 

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5. Sketch the graph of the function:

 

 

employing any useful information you can get from y , y ', and y ''.

 

Solution

 

Domain: R {1}.

 

Intercepts:

x-Intercepts: x = 0,
y-Intercept: y = 0.

 

Symmetry:

 

 

Asymptotes:

vertical: line x = 1,
horizontal: none,

oblique: none.

 

First Derivative:

 

 

y'' = 0 at x = 0,
y'' doesn't exist at x = 1 where y doesn't exist either;
if x = 0 then y = 0, point (0, 0);
sign of y'' is same as that of x/(x 1)3;
if x < 0 then x < 1 and so (x 1)3 < 0, thus y'' > 0,
if 0 < x < 1 then y'' < 0,
if 1 < x then y'' > 0.

 

 

 

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