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Expert-verified Found in: Page 774 ### Calculus

Book edition 1st
Author(s) Peter Kohn, Laura Taalman
Pages 1155 pages
ISBN 9781429241861 # Sketch the curves defined by the given sets of parametric equations. Indicate the direction of motion on each curve. $x={t}^{2}y={t}^{3},t\in \left[-2,2\right]$

The Curve plot See the step by step solution

## Step 1: Given information

The parametric curves, $x={t}^{2}y={t}^{3},t\in \left[-2,2\right]$

## Step 2: Calculation

The goal is to draw the parametric curve.

Assume$-2,0,1,2$ when drawing the graph for the parametric equations.

Find the values of $x,y$ by substituting different $t$ values in the parametric equations.

The point $\left(x,y\right)$ When $t=-2$ is,

$\left(x,y\right)=\left({t}^{2},{t}^{3}\right)$

$\left(x,y\right)=\left(\left(-2{\right)}^{2},\left(-2{\right)}^{3}\right)$ [since by substituting $t=-2]$

$\left(x,y\right)=\left(4,-8\right)$ simplify

The point $\left(x,y\right)$ When $t=-1$ is,

$\left(x,y\right)=\left({t}^{2},{t}^{3}\right)\phantom{\rule{0ex}{0ex}}\left(x,y\right)=\left(\left(-1{\right)}^{2},\left(-1{\right)}^{3}\right)\left[\text{since by substituting}t=-1\right]\phantom{\rule{0ex}{0ex}}\left(x,y\right)=\left(1,-1\right)\text{simplify}$

The point $\left(x,y\right)$ When $t=0$ is,

$\left(x,y\right)=\left({t}^{2},{t}^{3}\right)$

$\left(x,y\right)=\left(\left(0{\right)}^{2},\left(0{\right)}^{3}\right)\left[$ since by substituting $t=0\right]$

$\left(x,y\right)=\left(0,0\right)$ simplify

The point $\left(x,y\right)$ When $t=1$ is,

$\left(x,y\right)=\left({t}^{2},{t}^{3}\right)\phantom{\rule{0ex}{0ex}}\left(x,y\right)=\left({1}^{2},{1}^{3}\right)\left[\text{since by substituting}t=1\right]\phantom{\rule{0ex}{0ex}}\left(x,y\right)=\left(1,1\right)\text{simplify}$

The point $\left(x,y\right)$ When $t=2$ is,

$\left(x,y\right)=\left({t}^{2},{t}^{3}\right)\phantom{\rule{0ex}{0ex}}\left(x,y\right)=\left({2}^{2},{2}^{3}\right)\left[\text{since by substituting}t=2\right]\phantom{\rule{0ex}{0ex}}\left(x,y\right)=\left(4,8\right)\text{simplify}$

## Step 3: Calculation

The tabular representation of the points is as follows, The graphical representation is shown below, Therefore, the solution is the required graph. ### Want to see more solutions like these? 