Chapter 8.4, Problem 69E

(a)

To determine

To show: that the points on all four of these curves satisfy the same rectangular coordinate equation.

Answer to Problem 69E

The points on all four of these curves satisfy the same rectangular coordinates equations.

Explanation of Solution

Given:

  $\begin{array}{l}C:x=t,y=t^2\\ D:x=\sqrt{t},y=t,t\ge 0\\ E:x=sint,y={\sin }^{2}t\\ F:x=3^t,y=3^{2t}\end{array}$

Calculation:

Show that the points on all four of these curves satisfy the same rectangular

coordinates equations.

For the curve C , $x=t$ and $y=t^2$ .

On substituting $t=x$ in $y=t^2$ ,

  $y=x^2$ .

For the curve D , $x=\sqrt{t}$ and $y=t,t\ge 0$

On substituting $\sin t=x$ in $y={\sin }^{2}t$ ,

  $\begin{array}{l}\begin{array}{l}x=\sqrt{y}\\ x^2=y\end{array}\hfill \\ y=x^2\hfill \end{array}$

For the curve E , $x=sint$ and $y={\sin }^{2}t$ .

On substituting $\sin t=x$ in $y={\sin }^{2}t$ ,

  $y=x^2$

For the curve F , $x=3^t$ and $y=3^{2t}$ .

On substituting $3^t=x$ in $y=3^{2t}$ ,

  $y=x^2$

On comparing all the rectangular coordinates equation of the given curves, see that all are same.

Thus, the points on all four of these curves satisfy the same rectangular coordinates equations.

Conclusion:

Therefore, the points on all four of these curves satisfy the same rectangular coordinates equations.

(b)

To determine

To draw: the graph of each curve and explain how the curves differ from one another.

Answer to Problem 69E

On comparing all the graphs, see that the curves C and E represent the same curve as parabola for all x . And, the curves D and F represent the same curve as parabola for all $x\ge 0$ .

Explanation of Solution

Given:

  $\begin{array}{l}C:x=t,y=t^2\\ D:x=\sqrt{t},y=t,t\ge 0\\ E:x=sint,y={\sin }^{2}t\\ F:x=3^t,y=3^{2t}\end{array}$

Calculation:

Draw the graph of the curves C, D, E and F and explain how the curves differ from one another.

For the curve C, $x=t$ and $y=t^2$ . To draw the graph of this parametric equation, enter these parametric equations in the graphing calculator that displays the graph as shown below.

  

For the curve D, $x=\sqrt{t}$ and $y=t,t\ge 0$ . To draw the graph of this parametric equation, enter these parametric equations in the graphing calculator that displays the graph as shown below.

  

For the curve E, $x=sint$ and $y={\sin }^{2}t$ . To draw the graph of this parametric equation, enter these parametric equations in the graphing calculator, which displays the graph as shown below.

  

For the curve F, $x=3^t$ and $y=3^{2t}$ . To draw the graph of this parametric equation, enter these parametric equations in the graphing calculator, which displays the graph as shown below.

  

On comparing all the graphs, see that the curves C and E represent the same curve as parabola for all x . And, the curves D and F represent the same curve as parabola for all $x\ge 0$ .

Conclusion:

Therefore, on comparing all the graphs, see that the curves C and E represent the same curve as parabola for all x . And, the curves D and F represent the same curve as parabola for all $x\ge 0$ .