I'm a little confused by the following question:
You know that a function f(x, y, z) satisfies f(0, 0, 0) = 33 and $D_{<1,1,1>/ √ 3}$ f(0, 0, 0) = 4/√3 and $D_{<1,1,0>/ √2}$ f(0, 0, 0) = 7/√2 and $D_{<1,2,2>/3}$f(0, 0, 0) = 12. Estimate f(0.01, −0.001, 0.1) using linearization.
So my thought process is that we can make up some phony equation that satisfies the first f(0,0,0), say x2 + y2 + z2 + 33 = f(x,y,z). However, after this I am awfully stuck: how do we use directional derivatives to help with linearization?
Thanks for the help.
Consider a 1D case of approximation: $$ g(x_0)\approx g(0)+g'(0)x_0 $$ All you need to approximate $g(x_0)$ is knowing $g(0)$, $g'(0)$ and $x_0$. You don't need to know the "expression" of $g$.
The 3D case is similar: $$ f(a)\approx f(0)+f_x(0)a_1+f_y(0)a_2+f_z(0)a_3 $$ where $a=(a_1,a_2,a_3)$. Now you see that you need to find $f_x(0), f_y(0), f_z(0)$. How?
Note that you have the assumptions for three directional derivatives. On the other hand, $$ D_vf(0)=\nabla f(0)\cdot v $$ which give you a hint that you would need to solve a linear equation so that you can find $$ \nabla f(0)=(f_x(0),f_y(0),f_z(0)) $$