Suppose that f(x,y) is a smooth function and that its partial derivatives have the values fx(-6,9)=1 and fy(-6,9)=4. Given that f(-6,9)=-7, use this information to estimate the following values:
Estimate of (integer value) f(-6,10)
Estimate of (integer value) f(-5,9)
Estimate of (integer value) f(-5,10)

This is what I have tried so far, but I am missing something.

-7 = 1(x+6) + 4(y-9)
-7 = x+6+ 4y-36
f(x,y) = x +4y -23

Then I just plugged in the x and y values to get the integers. What am I doing wrong?

Question

Suppose that f(x,y) is a smooth function and that its partial derivatives have the values fx(-6,9)=1 and fy(-6,9)=4. Given that f(-6,9)=-7, use this information to estimate the following values:
Estimate of (integer value) f(-6,10)
Estimate of (integer value) f(-5,9)
Estimate of (integer value) f(-5,10)

This is what I have tried so far, but I am missing something.

-7 = 1(x+6) + 4(y-9)
-7 = x+6+ 4y-36
f(x,y) = x +4y -23

Then I just plugged in the x and y values to get the integers. What am I doing wrong?

irishboy123 · Answer

they want  you to use the derivatives to estimate/interpolate

So
$f(-6, 10) = f(-6, 9) + 1* f_y(-6, 9)$

turbodog · Answer

That makes sense, but I don't think I completely understand what I should do next. Do you just plug in the values of the f and fy and solve?

irishboy123 · Answer

yes, plug and play

in 2_d you are doing this

|dw:1473611194315:dw|

$f(x + h) \approx f(x) + \mathbf{slope} * h $

ie $f(x + h) \approx f(x) + h f'(x)  $

so you're guessing the red point is pretty close to the purple point using linear approximation

in your question you're now doing that with an extra dimension.