Question

Please help me solve the following problem. Suppose that a function f(x,y) is differentiable at the point (3,4) with fx(3,4)=2 and fy(3,4)=-1. If f(3,4)=5, estimate the value of f(3.01,3.98).

Answer 1

@dan815

Answer 2

@hartnn

Answer 3

i would try using the total differential
\[df=f_x dx + f_y dy\]

Answer 4

because it's estimation, \[\Delta f(x,y)\approx f_x(x,y) \Delta x+f_y(x,y)\Delta y\] where
\[\Delta f(x,y) = f(x+\Delta x,y+\Delta y)-f(x,y)\]

Answer 5

how do i use the total differential if i'm not given a function?

Answer 6

in the equation, the function is of little importance. it is the value of the function (and also the partial derivatives) that matter.

let \[x=3,y=4\]
now, put \[f(x+\Delta x,y+\Delta y)=f(3.01, 3.98)\]

Answer 7

here's the approximation equation
\[f(x+\Delta x, y+\Delta y)\approx f_x(x,y) \Delta x+ f_y(x,y) \Delta y+f(x,y)\]
let \[x=3,y=4, \Delta x=0.01, \Delta y=-0.02\]
sub these values

Answer 8

\[\large f(3+0.01,4-0.02)\approx f_x(3,4)(0.01)+f_y(3,4)(-0.02) +f(3,4)\]
the expression on the right hand side can be evaluated, since the value of the function and the partial derivatives are given