Q. 1 - Calculus

Question

Sketching parametric equations: Sketch the curves defined by the given sets of parametric equations. Indicate the direction of motion on each curve. 

1. $x=t^2$, $y=t^3$, $t\in \left[-2,2\right]$.

Step-by-Step Solution

Answer

The required graph is as follows, 

The direction of the curve is upward.

Step 1. Given Information.

The equations are $x=t^2$ and $y=t^3$ such that $t\in \left[-2,2\right]$.

Step 2. Explanation.

Using equations, make a table that contains the coordinates of several points on the curve.

t	-2	-1	0	1	2
$x=t^2$	4	1	0	1	4
$y=t^3$	-8	-1	0	1	8

 

Step 3. Find the direction.

Find derivatives of the parametric equations to check the direction of motion along the parametric curve with increasing values of t.

$\frac{dx}{dt}=2t$

$\frac{dy}{dt}=3t^2$

For all values of t , $\frac{dx}{dt}>0$ so curve is increasing.

Step 4. The graph.

The graph is as follows.