Question

Find the slope in the x-direction at the point P(0,2, f(0,2)) on the graph of f when

f(x, y)=2(y²-x²) ln (x+y) .

Answer 1

To find the slope in the x-direction of the function f(x, y) at the point P(0,2), we compute the partial derivative of f with respect to x and evaluate it at P, yielding a slope of 4.

Step-by-step explanation:

The question asks us to find the slope in the x-direction at point P(0,2, f(0,2)) on the graph of the function f(x, y) = 2(y² - x²) ln (x + y).

To do this, we need to compute the partial derivative of the function with respect to x and evaluate it at point P.

Steps to Find the Slope in the x-direction

1. Compute the partial derivative of the function with respect to x:
   fx(x, y) = ∂f/∂x.
2. Evaluate the partial derivative at point P(0,2).

Let's perform the computations:

1. The partial derivative of f with respect to x is:
   fx(x, y) = ∂/∂x [2(y² - x²) ln (x + y)] = 2(ln (x + y) * -2x) + 2(y² - x²) * 1/(x + y).
2. Evaluating this at P(0,2) yields zero since ln(0+2) is defined but the term -2x becomes 0 and (y² - x²) becomes 4, thus we get:
   fx(0,2) = 2(ln (2) * 0) + 2(4) * 1/(0 + 2) = 0 + 4 = 4.

The slope in the x-direction at P(0,2) is 4.