Use the guidelinesof ths section to sketch the curve.

Answered: Use the guidelinesof ths section to…

Advanced Engineering Mathematics

10th Edition

ISBN:9780470458365

Author:Erwin Kreyszig

Publisher:Erwin Kreyszig

Chapter2: Second-order Linear Odes

Section: Chapter Questions

Problem 1RQ

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Question

Use the guidelinesof ths section to sketch the curve.

Expert Solution

Solution:

The function is $y = \frac{x}{x^{2} - 4}$ .

Obtain the points where y is not defined.

Set denominator to zero: $x^{2} - 4 = 0 \Rightarrow x = \pm 2$ .

Thus, the function is not defined at x = 2 and -2.

Then, the intervals are $(- \infty, - 2), (- 2, 2) and (2, \infty)$

Now, check the function values in these intervals.

Conclusion:

On the interval $(- \infty, - 2)$ , we obtain

x	$y = \frac{x}{x^{2} - 4}$
-5	$y = \frac{- 5}{{(- 5)}^{2} - 4} = - \frac{5}{21}$
-4	$y = \frac{- 4}{{(- 4)}^{2} - 4} = - \frac{1}{3}$
-3	$y = \frac{- 3}{{(- 3)}^{2} - 4} = - \frac{3}{5}$

The function values are negative and decreasing.

On the interval $(- 2, 2)$ , we obtain

x	$y = \frac{x}{x^{2} - 4}$
-1	$y = \frac{- 1}{{(- 1)}^{2} - 4} = \frac{1}{3}$
0	$y = \frac{0}{{(0)}^{2} - 4} = 0$
1	$y = \frac{1}{{(1)}^{2} - 4} = - \frac{1}{3}$

The function values are positive and decreasing on $(- 2, 0)$ where as the function values are negative and decreasing on $(0, 2)$ .

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