Question

Find the domain of Q(x,y)=ln(x²+y²xy​)

Sketch several level curves for each of the following functions of two variables. (a) f(x,y)=y−x²−1
(b) f(x,y)=e⁻ˣ²⁻ʸ²

Answer 1

To find the domain of Q(x, y) = ln(x²+y²-xy), we need to consider the values of x and y that will make the natural logarithm function defined. The domain of Q(x, y) is the entire plane except for the line y = 2x.

Step-by-step explanation:

To find the domain of Q(x, y) = ln(x²+y²-xy), we need to consider the values of x and y that will make the natural logarithm function defined. The natural logarithm is defined only for positive numbers, so we need to ensure that the expression inside the logarithm is positive. The expression inside the logarithm, x²+y²-xy, must be greater than 0. By rearranging the terms, we can find the domain of Q(x, y) as follows:

x²+y²-xy > 0

x²-xy+y² > 0

(x-y/2)²+(3y/2)² > 0

This inequality holds true for all values of x and y, except when both (x-y/2) and (3y/2) equal to zero. Therefore, the domain of Q(x, y) is the entire plane except for the line y = 2x.