Chapter 9.4, Problem 1E

To determine

To fill in the blank:

The continuous functions $f$ and $g$ with the interval $I$ shows set of ordered pairs of $\left[f\left(t\right),g\left(t\right)\right]$ is $C$ . The equations given for $C$ and $t$ are and by $x=f\left(t\right)$ and $y=g\left(t\right)$ .

Answer to Problem 1E

The continuous functions $f$ and $g$ with the interval $I$ shows set of ordered pairs of $\left[f\left(t\right),g\left(t\right)\right]$ is plane curve$C$ . The equations given for $C$ and $t$ are parametric equations and parameters by $x=f\left(t\right)$ and $y=g\left(t\right)$ .

Explanation of Solution

Given information:

The incomplete sentence is:

‘The continuous functions $f$ and $g$ with the interval $I$ shows set of ordered pairs of $\left[f\left(t\right),g\left(t\right)\right]$ is $C$ . The equations given for $C$ and $t$ are and by $x=f\left(t\right)$ and $y=g\left(t\right)$ .’

A parametric curve is defined as the continuous function which is in the form of $x=f\left(t\right)$ and $y=g\left(t\right)$ .

So, these two equations are called as parametric equations for given curve which has the set of ordered pairs $\left[f\left(t\right),g\left(t\right)\right]$ .

So, the complete sentence is:

‘The continuous functions $f$ and $g$ with the interval $I$ shows set of ordered pairs of $\left[f\left(t\right),g\left(t\right)\right]$ is plane curve$C$ . The equations given for $C$ and $t$ are parametric equations and parameters by $x=f\left(t\right)$ and $y=g\left(t\right)$ .’