91Ó°ÊÓ

Find \(D_{\mathrm{u}} f\) at \(P\) $$ f(x, y, z)=4 x^{5} y^{2} z^{3} ; \quad P(2,-1,1) ; \mathbf{u}=\frac{1}{3} \mathbf{i}+\frac{2}{3} \mathbf{j}-\frac{2}{3} \mathbf{k} $$

Find an equation for the tangent plane and parametric equations for the normal line to the surface at the point \(P\). $$ x^{2}-x y z=56 ; P(-4,5,2) $$

Evaluate the indicated partial derivatives. $$ z=\left(x^{2}+5 x-2 y\right)^{8} ; \frac{\partial z}{\partial x}, \frac{\partial z}{\partial y} $$

Use an appropriate form of the chain rule to find \(d z / d t\). $$ z=e^{1-x y} ; x=t^{1 / 3}, y=t^{3} $$

Use limit laws and continuity properties to evaluate the limit. $$ \lim _{(x, y) \rightarrow(0,0)} \ln \left(1+x^{2} y^{3}\right) $$

Use Lagrange multipliers to find the maximum and minimum values of \(f\) subject to the given constraint. Also, find the points at which these extreme values occur. $$ f(x, y)=x y ; 4 x^{2}+8 y^{2}=16 $$

These exercises are concerned with functions of two variables. \begin{aligned} &\text { Find } F(g(x), h(y)) \text { if } F(x, y)=x e^{x y}, g(x)=x^{3} \text { , and }\\\ &h(y)=3 y+1 \end{aligned}

Find an equation for the tangent plane and parametric equations for the normal line to the surface at the point \(P\). $$ z=x^{2}+y^{2} ; \quad P(2,-3,13) $$

Use limit laws and continuity properties to evaluate the limit. $$ \lim _{(x, y) \rightarrow(4,-2)} x \sqrt[3]{y^{3}+2 x} $$

Find \(D_{\mathrm{u}} f\) at \(P\) $$ f(x, y, z)=y e^{x z}+z^{2} ; P(0,2,3) ; \mathbf{u}=\frac{2}{7} \mathbf{i}-\frac{3}{7} \mathbf{j}+\frac{6}{7} \mathbf{k} $$