Question

Calc III/Multivariable calc problem, involves gradients?:
The temperature at the point (x,y) on a metal plate is modeled by
T(x, y) = 400e^(-(x^2 + y)/2),
x is greater than or equal to 0
y is greater than or equal to 0.

a) Find the directions of no change in heat on the plate from the point (3,5)
b) Find the direction of greatest increase in heat from the point (3,5).

Answer 1

\[ \Large
T(x,y)= 400e^{-(x^2+y)/2}\\ \Large

\bigtriangledown T(x,y) = <\frac{\partial T}{\partial x}, \frac{\partial T}{\partial y}> \\ \Large

\bigtriangledown T(3,5) = <?,?> \text{ is the direction of greatest increase at (3,5)}
\]

Answer 2

So I find the tangent first which is r'(t)/|r'(t) right? and then I find the partial derivative of that with respect to x and y?

Answer 3

Wasn't the gradiant just (gradiant)f(x,y) = <fx, fy>?

Answer 4

Yes. (gradiant)f(x,y) = <fx, fy>. Here the function is T(x,y) and hence I am using T(x,y) instead of f(x,y).

Answer 5

How wouldI find the directions of no change in heat?

Answer 6

<fx , fy> dot <a,b> = 0, where <a,b> is a unit vector.

so solve and <a,b> = <-1,6> or <1,-6>