Question

Find an equation of the tangent(s) to the curve at a given point. Then graph the curve and the tangent(s).

x = 6sin(t), y = t^2 + t; (0,0)

How do I do this without the value of t? I don't see a way to eliminate the parameter.

Answer 1

Oh also if you can, how do you find d2y/d2x for parametrics?

Answer 2

The nice thing about parametric equations is that you don't have to eliminate the parameter.

dy/dx = dy/dt / dx/dt
dy/dx = (2t+1)/6cos(t)

This is zero at t = -1/2 and has a vertical asymptote whenever t = an odd multiple of pi/2.

The graph is a kind of double wave sine surve. It wanders back and forth in the x direction, but the escillations get longer in the y direction, since y uses t^2, instead of just t.

Mosey on over to wolfram dot com to see the big picture. Type in a command like

graph x=6*sin(t), y=t^2 + t, t=-5..5

try

graph x=6*sin(t), y=t^2 + t, t=-2..1.5 to see the horizontal tangent at t=-1/2