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Q 1.

Expert-verifiedFound in: Page 774

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. $x={t}^{2}y={t}^{3},t\in [-2,2]$

The Curve plot

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

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 $(x,y)$ When $t=-2$ is,

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

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

$(x,y)=(4,-8)$ simplify

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

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

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

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

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

$(x,y)=(0,0)$ simplify

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

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

$(x,y)=\left({t}^{2},{t}^{3}\right)\phantom{\rule{0ex}{0ex}}(x,y)=\left({2}^{2},{2}^{3}\right)[\text{since by substituting}t=2]\phantom{\rule{0ex}{0ex}}(x,y)=(4,8)\text{simplify}$The tabular representation of the points is as follows,

The graphical representation is shown below,

Therefore, the solution is the required graph.

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