Chapter 10.7, Problem 2AYU

To determine

To fill: The choice in the blank for the following statement

Suppose $x=x\left(t\right)$ and $y=y\left(t\right)$ are two functions of a third variable $t$ that are defined on the same interval $I$. The graph of collection of points defined by $\left(x,y\right)=\left(x\left(t\right),y\left(t\right)\right)$ is called a (n) $\underset{\_}{}$. The variable $t$ is called a (n) $\underset{\_}{}$.

Answer to Problem 2AYU

Solution:

Suppose $x=x\left(t\right)$ and $y=y\left(t\right)$ are two functions of a third variable $t$ that are defined on the same interval $I$. The graph of collection of points defined by $\left(x,y\right)=\left(x\left(t\right),y\left(t\right)\right)$ is called a plane curve. The variable $t$ is called a parameter.

Explanation of Solution

Given information:

The statement is, “Suppose $x=x\left(t\right)$ and $y=y\left(t\right)$ are two functions of a third variable $t$ that are defined on the same interval $I$. The graph of collection of points defined by $\left(x,y\right)=\left(x\left(t\right),y\left(t\right)\right)$ is called a $\underset{\_}{}$. The variable $t$ is called a $\underset{\_}{}$.

Explanation:

Suppose $x=x\left(t\right)$ and $y=y\left(t\right)$, where $f$ and $g$ are two functions whose common domain is some interval $I$. The graph of collection of points defined by $\left(x,y\right)=\left(x\left(t\right),y\left(t\right)\right)$ is called a plane curve.

The equation $x=x\left(t\right),y=y\left(t\right)$, where $t$ is in $I$ are called parametric equations of the curve.

The variable $t$ is called a parameter.

Hence, the complete statement becomes, “Suppose $x=x\left(t\right)$ and $y=y\left(t\right)$, $y=y\left(t\right)$ are two functions of a third variable $t$ that are defined on the some interval $I$. The graph of collection of points defined by $\left(x,y\right)=\left(x\left(t\right),y\left(t\right)\right)$ is called a plane curve. The variable $t$ is called a parameter.”