Chapter 10.7, Problem 43AYU

In Problems 43 and 44, the parametric equations of four curves are given. Graph each of them, indicating the orientation.

$C_1:x=t,y=t^2;-4\le t\le 4$

$C_2:x=\cos t,y=1-{\sin }^{2}t;0\le t\le \pi$

$C_3:x=e^t,y=e^{2t};0\le t\le \ln 4$

$C_4:x=\sqrt{t},y=t;0\le t\le 16$

To determine

To graph: The four curves defined by the given parametric equations.

Answer to Problem 43AYU

a.

Explanation of Solution

Given:

$C_1:x=t,y=t²;-4\le t\le 4$

Graph:

The graph of $C_1:x=t,y=t²;-4\le t\le 4$ is plotted.

To determine

To graph: The four curves defined by the given parametric equations.

Answer to Problem 43AYU

b.

Explanation of Solution

Given:

$C_2:x=\text{ cos}t,y=1-\text{ sin}²t;0\le t\le \pi$

Graph:

The graph of $C_2:x=\text{ cos}t,y=1-\text{ sin}²t;0\le t\le \pi$ is plotted.

To determine

To graph: The four curves defined by the given parametric equations.

Answer to Problem 43AYU

c.

Explanation of Solution

Given:

$C_3:x=e^t,y=e^{2t},0\le t\le \text{ ln}4$

Graph:

The graph of $C_3:x=e^t,y=e^{2t},0\le t\le \text{ ln}4$ is plotted.

To determine

To graph: The four curves defined by the given parametric equations.

Answer to Problem 43AYU

d.

Explanation of Solution

Given:

$C_4:x=\sqrt{t},y=t,0\le t\le 16$

Graph:

The graph of $C_4:x=\sqrt{t},y=t,0\le t\le 16$ is plotted.

The graphs of $C_1:x=t,y=t²;-4\le t\le 4$.

$C_2:x=\text{ cos}t,y=1-\text{ sin}²t;0\le t\le \pi$

$C_3:x=e^t,y=e^{2t},0\le t\le \text{ ln}4$

$C_4:x=\sqrt{t},y=t,0\le t\le 16$ are plotted and the orientations are also shown.