Question

Write the Linear Approximation to f(x, y) = x(1+y)-¹ at (a, b) = (18, 2) in the form

f(a+h,b+k) f(a, b) + fz (a, b)h + f(a, b)k
(Give an exact answer. Use decimal notation and fractions where needed.)
f(18+h, 2+k)----------------------- ~
Use it to estimate 17.97/3.05.

Answer 1

The linear approximation of the function f(x, y) at the point (18, 2) leads to the approximation formula f(18+h, 2+k) ≈ 9 + 1/3h - 9k, which can be used to estimate values close to the point of linearization, such as 17.97/3.05.

Step-by-step explanation:

The linear approximation to f(x, y) = x(1+y)-1 at (a, b) = (18, 2) can be found by computing the partial derivatives of f with respect to x and y at the point (18, 2). The formula for linear approximation is f(a+h, b+k) ≈ f(a, b) + fx(a, b)h + fy(a, b)k.

The partial derivative of f with respect to x is fx(x, y) = (1+y)-1 and with respect to y is fy(x, y) = -x(1+y)-2. Evaluating them at (18, 2) gives fx(18, 2) = 1/3 and fy(18, 2) = -9. The linear approximation is f(18+h, 2+k) ≈ 9 + 1/3h - 9k.

To estimate 17.97/3.05, we let h = -0.03 (17.97 = 18 - 0.03) and k = 1.05 (3.05 = 2 + 1.05), and substitute into the linear approximation f(18+h, 2+k) to get the estimate.