Is the particle travelling in the direction of greatest increase in temperature

[f.debby]

Homework Statement

A particle travels across a heated plate according to the path f(t) = (t^2, sint) at the time t seconds. The temperature T(x,y) at position (x,y) on the plate is given by T(x,y)= 200E^[-(x^2 + y^2)] degrees Celsius. At time t=pi seconds, is the particle traveling in the direction of greatest increase in temperature on the plate from position (pi^2, 0)? Explain why or why not.

Homework Equations

The Attempt at a Solution

I have that at time t=pi the particle's rate of change in temperature is -800(pi^3)e^[-(pi^2+1)] - 400e^[-(pi^2+1)] or -0.479498546
Also, I found the the gradient of T is the direction of the maximal rate of increase
so i found that the gradient of T(pi^2,0) = <-400pi*e^(-pi^2), 0>
But now i don't know what to do.

 

[Dick]

You've found that the gradient vector of T is horizontal at (pi^2,0). That only way the particle can be traveling in the direction of greatest temperature increase if if it's tangent vector at t=pi is parallel to the gradient vector. Is it?

 

[f.debby]

hmmm okay thankyou:), but I am unsure of how to calculate the tangent vector?

thanks for your help

 

[Dick]

hmmm okay thankyou:), but I am unsure of how to calculate the tangent vector?

thanks for your help

The tangent vector is the vector f'(t).

 

[f.debby]

Oh okay:)! So then it wouldn't be parallel to the gradient vector because f'(pi) = <2pi, -1> which is not a multiple of the gradient i calculated. So, the particle isn't traveling in the direction of greatest temperature increase.

Thanks so much!