Calculus for enginerring

Question

Suppose that T=4x²-4xy+4y² be the temperature of the point (x,y) on the circle x=cost,y=sint ,o<t<2Π. Find the maximum and minimum values of T on the circle.

Answer 1

Replace x by cost and y by sint:

T = 4cos²t - 4costsint + 4sin²t = 4(cos²t + sin²t) - 2(2costsint) = 4 - 2sin(2t)

Since -1 ≤ sin(2t) ≤ 1, 2 ≥ -2sin(2t) ≥ -2

So, 2 ≤ 4 - 2sin(2t) ≤ 6

On the given circle, the maximum value of T is 6 and the minimum is 2.