Question

Suppose the f(x,y) is a differentiable function and that its partial derivatives have the values fx​(1,3)=4 and fy​(1,3)=−5. Given that f(1,3)=0, use this information to estimate the following values:

a. f(2,3)≈
b. f(1,4)≈
c. f(2,4)≈​

Answer 1

To estimate the values f(2,3), f(1,4), and f(2,4) of the differentiable function f(x,y), the linear approximation is used based on the given derivatives fx(1,3)=4, fy(1,3)=-5, and value f(1,3)=0, resulting in estimates of 4, -5, and -1, respectively.

Step-by-step explanation:

The question involves estimating the values of a differentiable function f(x,y) at different points using given partial derivatives and function value. Given fx(1,3)=4 and fy(1,3)=-5, along with f(1,3)=0, the linear approximation to estimate f(2,3) would be f(1,3) + fx(1,3)∗(2-1), estimating f(1,4) would be f(1,3) + fy(1,3)∗(4-3), and to estimate f(2,4), we can add both changes: f(1,3) + fx(1,3)∗(2-1) + fy(1,3)∗(4-3).

This utilizes the concept of a linear approximation or tangent plane approximation for multivariable functions, which is an application of the first-order Taylor series expansion.

• a. f(2,3) ≈ f(1,3) + fx(1,3)∗(2-1) = 0 + 4∗(1) = 4
• b. f(1,4) ≈ f(1,3) + fy(1,3)∗(4-3) = 0 + (-5)∗(1) = -5
• c. f(2,4) ≈ f(1,3) + fx(1,3)∗(2-1) + fy(1,3)∗(4-3) = 0 + 4∗(1) + (-5)∗(1) = -1