Question
The Celsius temperature at a point (x,y) on a large metal plate is given by T(x,y) = 80+(x+1)^2(y-2)^2+(x-2)^2. Find direction of the plate at the point (2,1). Please show work. I need to study this i dont need a link to a website
Explanation / Answer
a) T(x,y) = 135xy / (1+x²+y²) ----> don't forget parentheses ?T/?x = (135y(1+x²+y²) - 135xy(2x)) / (1+x²+y²)² . . . . = 135y (1+x²+y² - 2x²) / (1+x²+y²)² . . . . = 135y (1-x²+y²) / (1+x²+y²)² ?T/?x = (135x(1+x²+y²) - 135xy(2y)) / (1+x²+y²)² . . . . = 135x (1+x²+y² - 2y²) / (1+x²+y²)² . . . . = 135x (1+x²-y²) / (1+x²+y²)² ?T = < ?T/?x, ?T/?y > ?T = < 135y(1-x²+y²)/(1+x²+y²)², 135x(1+x²-y²)/(1+x²+y²)² > ?T(1,1) = < 135(1-1+1)/(1+1+1)², 135(1+1-1)/(1+1+1)² > ?T(1,1) = < 135/9, 135/9 > = < 15, 15 > Now we calculate rate at which temperature is changing at point (1,1) in direction of vector v = 2i - j = < 2, -1 > First we need to find unit vector in direction of vector v: u = < 2, -1 > / v5 = < 2/v5, -1/v5 > ?uT(1,1) = < 15, 15 > • < 2/v5, -1/v5 > = 6v5 - 3v5 = 3v5 Temperature is INCREASING at a rate of 3v5 degrees/m b) Temperature is increasing most rapidly in direction of gradient ?T(1,1) = < 15, 15 > and is decreasing most rapidly in opposite direction = < -15, -15 > Unit vector in direction of most rapid decrease = < -1/v2, -1/v2 > In that direction, temperature is decreasing at a rate of ?uT(1,1) = < 15, 15 > = -15/v2 - 15/v2 = -15v2 So temperature is DECREASING at a rate of 15v2 degrees/m