Applications of the Derivative

Graph Sketching

Sketching involves two steps: the first involves obtaining information from the function, as outlined in earlier sections of this unit; the second involves interpreting the information in terms of the graph of the function. Once we have all the information we need, we proceed as follows:

Step 1
Plot the $x$ and $y$ intercepts, the local maxima and minima, and the inflection points.

Step 2
Draw the asymptotes using a dotted line.

Step 3
If there is a symmetry, sketch the graph on the right side of the $y$-axis first (see Step 6, below).

Step 4
Identify the intervals of increase and decrease, and the concavity.

Step 5
Join the plotted points according to the shape of the graph given by the increase and decrease and concavity.

Step 6
If there is a symmetry, reflect the graph plotted. Note that there may be more than one symmetry.

Example 5.48. We have been obtaining information about the function

$f (x) = \sqrt{x^{2} - 9}$

(see Example 5.1, and we present it in the table below.

Summary of the information about function $f (x) = \sqrt{x^{2} - 9}$ collected from previous examples in this unit
Information from $f$	Interpretation for the graph of $f$	Example
Domain $(\infty, 3] \cup [3, \infty)$	no vertical asymptote no graph between the lines $x = - 3$ and $x = 3$ no $y$-intercept	5.1
$f (3) = 0 = f (- 3)$	$x$ intercepts: $(3, 0)$ and $(- 3, 0)$	5.7
$f (- x) = f (x)$	even
$f^{'} (x) < 0$ on $(- \infty, - 3)$ $f^{'} (x) > 0$ on $(3, \infty)$	dec on $(- \infty, - 3)$ inc on $(3, \infty)$ no local extreme points	5.24
$f^{″} (x) < 0$ on domain	concave down no inflection points	5.40

The information from the table takes us to the graph in Figure 5.22, below.

Figure 5.22. Graph of the function $f (x) = \sqrt{x^{2} - 9}$

Example 5.49. For the function

$y = x^{3} - 6 x^{2} - 15 x + 4$

(see Example 5.26), we use the Intermediate Value Theorem to locate the $x$-intercepts.

Since

$y (- 3) = - 32 < 0 < 2 = y (- 2)$

and

$y (1) = - 16 < 0 < 4 = y (0),$

the function has two $x$-intercepts in the intervals $(- 3, - 2)$ and $(0, 1)$ . The $y$-intercept is $(0, 4)$ . You can check that $y (- x)$ is not equal to either $y (x)$ or $- y (x)$ . Hence, the function is not symmetric.

The rest of the information is presented in the table below.

Summary of the information about function $y = x^{3} - 6 x^{2} - 15 x + 4$ collected from previous examples in this unit
Information from $y$	Interpretation for the graph of $y$	Example
Polynomial	continuous everywhere no asymptotes
$y (- 3) < 0 < y (- 2)$ $y (1) < 0 < y (0)$	$x$ intercepts in $(- 3, - 2)$ and $(0, 1)$
$y (- x) \neq y (x)$ $y (- x) \neq - y (x)$	no symmetries
$f^{'} (x) > 0$ on $(- \infty, - 1)$ and $(5, \infty)$ $f^{'} (x) < 0$ on $(- 1, 5)$	inc on $(- \infty, - 1)$ and $(5, \infty)$ dec on $(- 1, 5)$ rel max $(- 1, 12)$ rel min $(5, - 96)$	5.26
$f^{″} (x) > 0$ on $(2, \infty)$ $f^{″} (x) < 0$ on $(- \infty, 2)$	CU on $(2, \infty)$ CD on $(- \infty, 2)$ inflection point at $(2, - 42)$	5.41 5.45

The information from the table takes us to the graph in Figure 5.23, below.

Figure 5.23. Graph of function $y = x^{3} - 6 x^{2} - 15 x + 4$ —not to scale

Example 5.50. [See Exercise 24 on page 218 of the textbook.]

The interpretation of the information given about the graph is summarized in the table below.

Summary of the interpretation of the information given by the graph of the function $y = x^{3} - 6 x^{2} - 15 x + 4$
Information from $f$		Interpretation for the graph of $f$
$f^{'} (1) = f^{'} (- 1) = 0$		horizontal tangent line at $f (1)$ and $f (- 1)$
$f^{'} (x) < 0$ on $\| x \| < 1$ $f^{'} (x) > 0$ on $1 < \| x \| < 2$ $f^{'} (x) = - 1$ on $\| x \| > 2$		dec on $(- 1, 1)$ ; inc on $(1, 2)$ and $(- 2, - 1)$ dec on $(- \infty, 2)$ and $(2, \infty)$ $f (x) = - x$ on $(- \infty, 2)$ and $(2, \infty)$ loc min at $x = - 2$ , $x = 1$ loc max at $x = - 1$ , $x = 2$ dec and linear on $(- \infty, - 2)$ and $(2, \infty)$
$f^{″} (x) < 0$ on $- 2 < x < 0$		CD on $(- 2, 0)$ CU on $(0, 2) \cup (2, \infty)$ inflection point at $(0, 1)$

The information from the table takes us to the graph in Figure 5.24, below.

Figure 5.24. Graph of the function satisfying the conditions in Exercise 24, page 218 of the textbook—not to scale

Exercises

Do Exercises 51-54 on page 96 of the textbook.
Read Examples 6 and 7 on pages 215-216.
Do Exercises 23, 25, 26, 29, 31 and 35 on page 218.
Read Examples 1, 2 and 4 on pages 234-235 and 237.
Do Exercises 11, 13, 17, 19, 21 and 23 on page 238.

Answers to Exercises