Question

The temperature on an unevenly heated metal plate positioned in the first quadrant of the xy-plane is given by c(x,y) = 25xy 25 (x -1)2 (y -1)2 1. Assume that temperature is measured in degrees celsius and that x and y are each measured in inches. Determine ∂c ∂x and ∂c ∂y.

Answer 1

To determine the partial derivatives ∂c/∂x and ∂c/∂y of the given temperature function c(x, y), we differentiate the polynomial terms with respect to x and y, treating the other variable as a constant.

Step-by-step explanation:

To determine the partial derivatives ∂c/∂x and ∂c/∂y, we need to take the partial derivative of the given temperature function c(x, y) with respect to x and y.

To find ∂c/∂x, we differentiate the polynomial term 25(x-1)^2(y-1)^2 with respect to x, treating y as a constant. The derivative of (x-1)^2 with respect to x is 2(x-1), and the derivative of y^2 with respect to x is 0 since y is a constant. Therefore, ∂c/∂x = 25y(2(x-1)) = 50y(x-1).

To find ∂c/∂y, we differentiate the polynomial term 25(x-1)^2(y-1)^2 with respect to y, treating x as a constant. The derivative of (y-1)^2 with respect to y is 2(y-1), and the derivative of x^2 with respect to y is 0 since x is a constant. Therefore, ∂c/∂y = 25x(2(y-1)) = 50x(y-1).