Q2 TF. - Calculus

Question

Parametric curves: Imagine the curve traced in the xy-plane by

the coordinates (x, y) = (3z + 1, $z^2$ − 4) as z varies, where the

parameter z is a function of time t.

If the parameter z moves at 3 units per second and

z = 0 when t = 0, plot the points (x, y) in the plane

that correspond to t = 0, 1, 2, 3, and 4.

Step-by-Step Solution

Answer

Step 1. Given Information

$(x,y)=(3z+1,z^2-4)$

Step 2. Calculation

Since, $\frac{dz}{dt}=3$

Integrating both sides

$z=3t+c,c=$integration constant

At $t=0,z=0$$t=0,z=0$

Therefore, $c=0$

Then $z=3t$

Now for, $t=0, z=0$

and $(x,y)=(1,-4)$

Now for, $t=1,z=3$

and $(x,y)=(10,5)$

Now for, $t=1,z=3$

and $(x,y)=(19,32)$

Now for, $t=3,z=9$

and $(x,y)=(28,77)$

Now for, $t=4,z=12$

and $(x,y)=(37,140)$

Step 3. Graph