Question

Find the linearizationL(x,y,z) of the function f(x,y,z) =sin(xy)/z at the point (pi/2,1,1).

Answer 1

Given, $f:\mathbb{R}^3\rightarrow\mathbb{R}$ to be a real valued map $(x,y,z)\mapsto\frac{\sin(xy)}{z}$ , now we have to find Linearization of $f(x,y,z)$ at $(a,b,c)=(\pi/2,1,1)$

Since, from Taylor expansion we get

$$L(x,y,z)=f(a,b,c)+\sum_{x,yz}(x-a)f_x(a,b,c)$$

Now,

$$f_x(x,yz)=\frac{y}{z}\cos(xy)\implies f_x(\pi/2,1,1)=0\\ f_y(x,y,z)=\frac{x}{z}\cos(xy)\implies f_y(\pi/2,1,1)=0\\ f_z(x,y,z)=-\frac{1}{z^2}\sin(xy)\implies f_z(\pi/2,1,1)=-1$$

Thus,

$$L(x,y,z)=1+(x-\pi/2)\cdot 0+(y-1)\cdot 0+(z-1)\cdot(-1)\\ \implies L(x,y,z)=-z+2$$