Chapter 1.4, Problem 20E

(a)

To determine

To find: The graph of the parametric equations, the initial and terminal points. Also, indicate the direction in which the curve is traced.

Answer to Problem 20E

The graph of the parametric equations is shown in figure (1), initial point is $\left(0,2\right)$ and terminal point is $\left(4,0\right)$.

Explanation of Solution

Given information: The equations are $x=t^2$ and $y=\sqrt{4-t^2}$ , $0\le t\le 2$.

Calculation:

Use the following step to graph the parametric equations by graphing calculator.

Step 1: First press the “ON” button graphical calculator.

Step 2: Press the $\boxed{\text{MODE}}$ button, under FUNC and choose PAR option.

Step 3: Press $\boxed{\text{Y=}}$ , and set the parametric equations equal to ${\text{X}}_{\text{1T}}$ and ${\text{Y}}_{\text{1T}}$.

Step 4: Press $\boxed{\text{GRAPH}}$ to obtain the graph of the parametric equations and the window can be set to $\left[-1,5\right]$ by $\left[-1,3\right]$ as shown below.

  

Figure (1)

If $t=0$ , then $x=0$ and $y=2$ . So, the initial point is $\left(0,2\right)$.

If $t=2$ , then $x=4$ and $y=0$ . So, the terminal point is $\left(4,0\right)$.

Therefore, the graph of the parametric equations is shown in figure (1), initial point is $\left(0,2\right)$ and terminal point is $\left(4,0\right)$.

(b)

To determine

To find: The Cartesian equation for a curve that contains the parameterized curve and the portion of the graph of the Cartesian equation that is traced by the parameterized curve.

Answer to Problem 20E

The Cartesian equation for a curve that contains the parameterized curve is $y=\sqrt{4-x}$ and parameterized curve traces the right portion of the curve.

Explanation of Solution

Given information: The equations are $x=t^2$ and $y=\sqrt{4-t^2}$ , $0\le t\le 2$.

Calculation:

Substitute $x$ for $t^2$ in the equation $y=\sqrt{4-t^2}$.

  $y=\sqrt{4-x}$

As shown in the graph, the parameterized curve traces the right portion of the curve defined by $y=\sqrt{4-x}$.

Therefore, the Cartesian equation for a curve that contains the parameterized curve is $y=\sqrt{4-x}$ and parameterized curve traces the right portion of the curve.