   Chapter 10.1, Problem 24E

Chapter
Section
Textbook Problem

Match the graphs of the parametric equations x = f(t) and y = g(t) in (a)–(d) with the parametric curves labeled I–IV. Give reasons for your choices. (a) (b) (c) (d) To determine

To match: The parametric equations x=f(t) and y=g(t) shown in graphs (a) to (d) with the parametric curves labeled I to IV.

Explanation

The parametric equations x=f(t) and y=g(t) shown in graph (a) matches with the parametric curve labeled III.

The parametric equations x=f(t) and y=g(t) shown in graph (b) matches with the parametric curve labeled I.

The parametric equations x=f(t) and y=g(t) shown in graph (c) matches with the parametric curve labeled IV.

The parametric equations x=f(t) and y=g(t) shown in graph (d) matches with the parametric curve labeled II.

Given:

The parametric equation for the variable x is as follows.

x=f(t)

The parametric equation for the variable y is as follows.

y=g(t)

Calculation:

Graph (a):

From graph (a), it is observed that the parameter x varies from (1x2) at the interval of (0t1) and parameter y varies from (1y1) at the interval of (0t1)

Plot the parametric curve for which the parameter x varies from (1x2) and y varies from (1y1) as shown below in Figure 1.

The parametric curve shown in Figure 1 resembles the parametric curve shown in graph–III. Therefore, the parametric equations x=f(t) and y=g(t) shown in graph (a) matches with the parametric curve labeled III.

Graph (b):

From graph (b), it is observed that the parameter x varies four times periodically from (2x2) at the interval of (0t1) and parameter y also varies six times periodically from (2y2) at the interval of (0t1) .

Plot the parametric curve for which the parameter x varies from (2x2) and y varies from (2y2) as shown below in Figure 2

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