91Ó°ÊÓ

Explain why the function is differentiable at the given point. Then find the linearization of the function at that point. \(f(x, y, z)=e^{-x y} \cos z, \quad(0,2,0)\)

Find the first partial derivatives of the function. $$f(x, t)=\sqrt{x} \ln t$$

Find the local maximum and minimum values and saddle point(s) of the function. If you have three-dimensional graphing software, graph the function with a domain and viewpoint that reveal all the important aspects of the function. \(f(x, y)=y \cos x\)

Let \(W(s, t)=F(u(s, t), v(s, t)),\) where \(F, u,\) and \(v\) are differentiable, and $$\begin{array}{ll}{u(1,0)=2} & {v(1,0)=3} \\ {u_{s}(1,0)=-2} & {v_{s}(1,0)=5} \\ {u_{t}(1,0)=6} & {v_{t}(1,0)=4} \\ {F_{u}(2,3)=-1} & {F_{v}(2,3)=10}\end{array}$$ Find \(W_{s}(1,0)\) and \(W_{t}(1,0)\)

Verify the linear approximation at \((0,0)\) \(\frac{2 x+3}{4 y+1} \approx 3+2 x-12 y\)

Find the local maximum and minimum values and saddle point(s) of the function. If you have three-dimensional graphing software, graph the function with a domain and viewpoint that reveal all the important aspects of the function. \(f(x, y)=\left(x^{2}+y^{2}\right) e^{y^{x}-x^{2}}\)

Find the first partial derivatives of the function. $$z=(2 x+3 y)^{10}$$

Find the directional derivative of the function at the given point in the direction of the vector \(v\) . \(g(p, q)=p^{4}-p^{2} q^{3}, \quad(2,1), \quad \mathbf{v}=[1,3]\)

Find and sketch the domain of the function. \(f(x, y)=\sqrt{2 x-y}\)

Find the first partial derivatives of the function. $$z=\tan x y$$