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.
Estimate of (integer value) f(2,9):

Estimate of (integer value) f(3,8):

Estimate of (integer value) f(3,9):

Answer 1

The Taylor expansion of a smooth function is given by a formula in point of the plane (x₀,y₀)

$f(x,y) = \sum^{\infty}_{k=0} \sum_{i=0}^{i=k} \frac{f^{(i,k-i)}(x_0,y_0)}{k!}(dx)^{k}(dy)^{k-i}$

Use it:

$f(3,9) = f(2,8) + f_x(2,8)(3-2) + f_y(2,8)(9-8)= -3$

$f(3,8) = f(2,8) + f_x(2,8)(3-2)= 2$

$f(2,9) = f(2,8) + f_y(2,8)(9-8) =1$