Suppose that f(x,y) is a smooth function and that its partial derivatives have the values, fx(4,−7)=−3 and fy(4,−7)=1. Given that f(4,−7)=−8, use this information to estimate the value of f(5,−6). 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(4,−6)
b) Estimate of (integer value) f(5,−7)
c) Estimate of (integer value) f(5,−6)

Question

Suppose that f(x,y) is a smooth function and that its partial derivatives have the values, fx(4,−7)=−3 and fy(4,−7)=1. Given that f(4,−7)=−8, use this information to estimate the value of f(5,−6). 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(4,−6)
b) Estimate of (integer value) f(5,−7)
c) Estimate of (integer value) f(5,−6)

anonymous · Answer

I believe you have to use something like: $$
L(x,y)-f(x_0,y_0)=f_x(x_0,y_0)(x-x_0)+f_y(x_0,y_0)(y-y_0)
$$Where $$
f(x,y)\approx L(x,y)
$$

anonymous · Answer

And we'd want to use $(x_0,y_0) = (4,-7)$

anonymous · Answer

Get it?

anonymous · Answer

$$
L(x,y)-f(4,−7)=f_x(4,−7)(x-(4))+f_y(4,−7)(y-(-7))
$$

anonymous · Answer

$$
L(x,y)-(-8)=(-3)(x-4)+(1)(y+7)
$$

anonymous · Answer

@sammimarcs That should really be enough to get you thinking.

kainui · Answer

Yeah, I'd basically just try to make it a line and realize that when you take partial derivatives, you're keeping the other variable constant so it's like two lines going on there that you need to keep track of.

Actually, I think it might be possible to just do all one variable and then do all the other variable separately rather than all in one equation if you wanted to. Good luck!

anonymous · Answer

Thanks so much!