Answered: Find the equation of the graph. Show…

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

Find the equation of the graph. Show steps/reasons for each aspect.

Expert Solution

Step 1

Consider the given graph.

It represents a periodic function.

The given graph resembles to that of the trigonometric function $y = \tan (t)$ .

Hence, we can conclude that the given graph represents a variation of tangent function.

Note that, the tangent function is periodic with period $π$ .

Step 2

The general form of a function which is a variation of the tangent function is given by $y = A \tan (B t)$ where A is the stretching factor and B is the value such that $p e r i o d = \frac{π}{B}$ .

From the given graph, we can observe that the function curve repeats itself in the intervals $(- \frac{π}{2}, - \frac{π}{6}), (- \frac{π}{6}, \frac{π}{6}) a n d (\frac{π}{6}, \frac{π}{2})$ .

The length of each of the above interval is the same as $(- \frac{π}{6}) - (- \frac{π}{2}) = \frac{π}{3}, (\frac{π}{6}) - (- \frac{π}{6}) = \frac{π}{3}, (\frac{π}{2}) - (\frac{π}{6}) = \frac{π}{3}$ .

Thus, the period of the given function is $\frac{π}{3}$ .

Substituting the value $p e r i o d = \frac{π}{3}$ in $p e r i o d = \frac{π}{B}$ , we get $\frac{π}{3} = \frac{π}{B}$ .

This implies that B = 3.

Substituting the value B = 3 in $y = A \tan (B t)$ , we get the function $y = A \tan (3 t)$ .

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