Question

Write the Linear Approximation to f(x, y) = x(1+y)-1 at (a,b) = (12,1) in the form f(a+h,b+ k) = f(a,b) + f1(a,b)h + fy(a,b)k (Give an exact answer. Use decimal notation and fractions where needed.) f(12+h, 1+k)

Answer 1

To write the linear approximation to f(x, y) = x(1+y)-1 at (a,b) = (12,1), we find the first-order partial derivatives of f with respect to x and y and substitute the values into the linear approximation formula.

Step-by-step explanation:

To write the linear approximation to f(x, y) = x(1+y)-1 at (a,b) = (12,1), we need to find the first-order partial derivatives of f with respect to x and y. Then, we substitute the values of a, b, h, and k into the linear approximation formula.

The partial derivative with respect to x is f1(a,b) = 1 + y, and the partial derivative with respect to y is fy(a,b) = x.

Substituting the values, the linear approximation is f(12+h, 1+k) ≈ f(12,1) + (1 + 1)(h) + (12)(k).