Chapter 9, Problem 1RCC

(a)

To determine

To explain: The meaning of a parametric curve.

Explanation of Solution

The parametric curve is defined as the set of points $\left(x,y\right)$ of the form $x=f\left(t\right)$ and $y=g\left(t\right)$, where values of x and y varies with the increase in parameter $t$, that produces the curve.

Thus, a parametric curve has two separate functions for x and y coordinates, which are represented in terms of a variable t, called the parameter.

For example, a circle of radius r is defined by the parametric equations $x=r\cos t,y=r\sin t$, $t\in \left[a,b\right]$.

In general, the curve with parametric equations $x=f\left(t\right),y=g\left(t\right),a\le t\le b$ has initial point $\left(f\left(a\right),g\left(a\right)\right)$ and terminal point $\left(f\left(b\right),g\left(b\right)\right)$.

(b)

To determine

To explain: The method to sketch a parametric curve.

Explanation of Solution

In general, it is difficult to sketch a parametric curve that is represented by two functions.

Thus, find the values of $f\left(t\right)\text{ and }g\left(t\right)$ for different values of t. Plot those points either by hand or using a calculator.

Sometimes, the parameter t are eliminated from $f\text{ and }g$ to get the Cartesian equation relating $x\text{ and }y$.

That is, it is easier to graph the equation, than working with the original formulas of x and y in terms of t.