Question

Suppose that f(x,y) is a smooth function and that its partial derivatives have the values, fx(2,8)=−4 and fy(2,8)=−5. Given that f(2,8)=6, use this information to estimate the value of f(3,9). Note this is analogous to finding the tangent line approximation to a function of one variable. In fancy terms, it is the first Taylor approximation.

a. Estimate of (integer value) f(2,9):
b. Estimate of (integer value) f(3,8):
c. Estimate of (integer value) f(3,9):

Answer 1

Using the first-order Taylor approximation, the estimated values for the smooth function f(x, y) are f(2,9)=1, f(3,8)=2, and f(3,9)=-3 based on the given partial derivatives and function value at (2,8).

Step-by-step explanation:

The student's question involves estimating the value of a smooth function f(x, y) at a new point using the given partial derivatives and function value at a known point. This is analogous to using a first-order Taylor approximation or linear approximation for functions of multiple variables.

To estimate f(2,9), we start with the known value f(2,8)=6 and adjust it by the change in y, which is 1, multiplied by the partial derivative with respect to y, fy(2,8)=-5. Hence, the estimate will be 6 + 1*(-5) = 1.

To estimate f(3,8), we again start with f(2,8)=6, but now adjust by the change in x, which is 1, multiplied by the partial derivative with respect to x, fx(2,8)=-4. The estimate will be 6 + 1*(-4) = 2.

Finally, to estimate f(3,9), we need to adjust in both the x and y directions from (2,8). We take the estimate for f(3,8), which is 2, and adjust it for the change in y, hence 2 + 1*(-5) = -3. Therefore, the estimated value of f(3,9) is -3.