Q 1. - Calculus

Question

Sketch the curves defined by the given sets of parametric equations. Indicate the direction of motion on each curve.  $x=t^{2}y=t^3,t\in [-2,2]$

Step-by-Step Solution

Answer

The Curve plot  

Step 1: Given information

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

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 $(x, y)$ 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, 

$$\begin{gathered} (x,y)=\left(t^2,t^3\right) \\ (x,y)=\left((-1{)}^2,(-1{)}^3\right)[\text{ since by substituting }t=-1] \\ (x,y)=(1,-1)\text{ simplify } \end{gathered}$$

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,

$$\begin{gathered} (x,y)=\left(t^2,t^3\right) \\ (x,y)=\left(1^2,1^3\right) [\text{ since by substituting }t=1] \\ (x,y)=(1,1)\text{ simplify } \end{gathered}$$

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

$$\begin{gathered} (x,y)=\left(t^2,t^3\right) \\ (x,y)=\left(2^2,2^3\right) [\text{ since by substituting }t=2] \\ (x,y)=(4,8)\text{ simplify } \end{gathered}$$

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.