Expert: Ahmed Salami - 12/27/2009

Question
I can't seem to match up my answer to the correct one. *cries* So I was wondering if you could do it step by step, so I could see where I went wrong.

An object moving along a curve in the xy-plane has position (x(t), y(t)) at time t with

dx/dt = cos(t^3)  and   dy/dt= 3sin(t^2)

for 0 less than or equal to t, t less than or equal to 3 (basically in between or equal to 0 or 3). At time t=2, the object is at position (4,5)

Write an equation for the line tangent to the curve at (4,5)



I believe you have to set up an equation dy/dx= 3sin(t^2)/cos(t^3)  but I don't fully understand why.


Thanks very much for any help you provide! :D

Answer
Hi Devi,
The functions x(t) and y(t) are known as parametric equations with t as a parameter. The function of the curve along which the object is moving is actually of the form y(x) i.e y as a function of x.
We also know that
dx/dt = cos(t³)  and   dy/dt= 3sin(t²)
for 0 ≤ t ≤ 3
We need to write an equation of the tangent line at a point (x,y), that is why we need dy/dx. And of course,
dy/dx = (dy/dt) / (dx/dt)
     = 3sin(t²)/cos(t³)
At t = 2
dy/dx = 3sin(2²)/cos(2³)
     = 3sin(4)/cos(8)
     = 3(-0.7568)/(-0.1455)
     = 15.6
And the equation of the tangent at (4,5) would be
(y - 5)/(x - 4) = 15.6
y - 5 = 15.6(x - 4)
     = 15.6x - 62.4
y = 15.6x - 57.4
Notice here that sines and cosines have been evaluated in radians rather than degrees!

Regards