Gradient of T: Fly Bug to Warm Quickly

[bengaltiger14]

Homework Statement

The temperature in a room is given by T(x,y,z)=2x^2+3y^2 - 4z. A flying bug located at P(1,1,2) in the room desires to fly in such a direction that it will warm as soon as possible. In what direction must the bug fly?

Taking the gradient of T: 4x(x hat)+6y(y hat)-4(z hat). Do I plug the P(1,1,2) into that equation to determine the direction? Doing this, the answer would be. 1/SQRT(68) * [4(x hat) + 6(y hat)-4(z hat)]

But what if P=(2,2,2). Would if be: 1/SQRT(224) * [8(x hat)+12(y hat) + 4(z hat)]

 

[LowlyPion]

Homework Statement

The temperature in a room is given by T(x,y,z)=2x^2+3y^2 - 4z. A flying bug located at P(1,1,2) in the room desires to fly in such a direction that it will warm as soon as possible. In what direction must the bug fly?

Taking the gradient of T: 4x(x hat)+6y(y hat)-4(z hat). Do I plug the P(1,1,2) into that equation to determine the direction? Doing this, the answer would be. 1/SQRT(68) * [4(x hat) + 6(y hat)-4(z hat)]

But what if P=(2,2,2). Would if be: 1/SQRT(224) * [8(x hat)+12(y hat) + 4(z hat)]

I'd say basically correct. Looks like a typo in the last term.

∇T of 2x²+ 3y² - 4z = (4x,6y,-4)