Chapter 0.5, Problem 49E

(a)

To determine

To find: The parametrization cannot be used for other value of $t$ .

Answer to Problem 49E

  $\sqrt{2-t}$ would not be defined on $ℝ$

Explanation of Solution

Given information: The expression is:

  $(\sqrt{2-t},\sqrt{2+t})\text{ for }-2\le t\le 2$

Calculation:

This parametrization cannot be used for other values of $t$ because when we would substitute some lower value than-2 for $t$ then get a negative number under reciprocal $\sqrt{2+t}$ which is not defined on $ℝ$ .

Simplify bigger number than 2 then the reciprocal $\sqrt{2-t}$ would not be defined on $ℝ$ .

(b)

To determine

To find: The distance from the origin.

Answer to Problem 49E

  $2$ .

Explanation of Solution

Given information: The expression is:

  $(\sqrt{2-t},\sqrt{2+t})\text{ for }-2\le t\le 2$

Calculation:

The distance of the point $(x,y)=(0,0)$ is given by:

  $\sqrt{{(x-0)}^2+{(y-0)}^2}$

Put,

  $\begin{array}{c}x=\sqrt{2-t}\\ y=\sqrt{2+t}\end{array}$

And calculate:

  $\begin{array}{c}\sqrt{{(x-0)}^2+{(y-0)}^2}=\sqrt{{(\sqrt{2-t})}^2+{(\sqrt{2+t})}^2}\\ \begin{array}{c}=\sqrt{2-t+2+t}\\ =\sqrt{4}\\ =2\end{array}\end{array}$

Therefore, the required distance from the origin is 2.

(c)

To determine

To find: The endpoints on the graph.

Answer to Problem 49E

At $t=2:(0,2)$

At $t=-2:(2,0)$

Explanation of Solution

Given information: The expression is:

  $(\sqrt{2-t},\sqrt{2+t})\text{ for }-2\le t\le 2$

Calculation:

At $t=2$ :

  $(\sqrt{2-2},\sqrt{2+2})=(0,2)$

At $t=-2$ :

  $(\sqrt{2-(-2)},\sqrt{2-2})=(2,0)$

Therefore the required endpoints on the graph are:

At $t=2:(0,2)$

At $t=-2:(2,0)$

(d)

To determine

To find: The all other points on this curve must lie on the first quadrant.

Answer to Problem 49E

  $(\sqrt{2-(-2)},\sqrt{2-2})=(2,0)$

Explanation of Solution

Given information: The expression is:

  $(\sqrt{2-t},\sqrt{2+t})\text{ for }-2\le t\le 2$

Calculation:

All other points on the curve must lie in the first quadrant because both of the radicals must be positive. Even when substitute $t=-2$ the minimum point of domain, into the parametric curve get point:

  $(\sqrt{2-(-2)},\sqrt{2-2})=(2,0)$

(e)

To determine

To find: From (a) through (d) give a complete geometric description of the graph.

Answer to Problem 49E

The graph is shown:

  

Explanation of Solution

Given information: The expression is:

  $(\sqrt{2-t},\sqrt{2+t})\text{ for }-2\le t\le 2$

Graph:

Every point on this parametrized curve, as seen in portion b), is 2 distances from the origin. This ought to connect us to the circle. All of the points on this curve, however, must be located in the first quadrant as of part d). As a result, anticipate that our graph will only be a circle with a radius of two that is centered at the origin.