91Ó°ÊÓ

Sketch the graph of the function. $$ f(x, y)=x^{2} $$

Use a graph or level curves or both to estimate the local maximum and minimum values and saddle point(s) of the function. Then use calculus to find these values precisely. $$ f(x, y)=(x-y) e^{-x^{2}-y^{2}} $$

Find the maximum rate of change of \(f\) at the given point and the direction in which it occurs. $$ f(x, y, z)=x /(y+z), \quad(8,1,3) $$

Find the differential of the function. \(z=e^{-2 x} \cos 2 \pi t\)

Find \(h(x, y)=g(f(x, y))\) and the set of points at which \(h\) is continuous. $$ g(t)=t^{2}+\sqrt{t}, \quad f(x, y)=2 x+3 y-6 $$

Find the first partial derivatives of the function. $$ g(u, v)=\left(u^{2} v-v^{3}\right)^{5} $$

Use the Chain Rule to find the indicated partial derivatives. $$ \begin{array}{l}{N=\frac{p+q}{p+r}, \quad p=u+v w, \quad q=v+u w, \quad r=w+u v} \\ {\frac{\partial N}{\partial u}, \frac{\partial N}{\partial v}, \frac{\partial N}{\partial w} \quad \text { when } u=2, v=3, w=4}\end{array} $$

Consider the problem of minimizing the function \(f(x, y)=x\) on the curve \(y^{2}+x^{4}-x^{3}=0\) (a piriform). (a) Try using Lagrange multipliers to solve the problem. (b) Show that the minimum value is \(f(0,0)=0\) but the Lagrange condition \(\nabla f(0,0)=\lambda \nabla g(0,0)\) is not satisfied for any value of \(\lambda .\) (c) Explain why Lagrange multipliers fail to find the minimum value in this case.

Use a graph or level curves or both to estimate the local maximum and minimum values and saddle point(s) of the function. Then use calculus to find these values precisely. $$ \begin{array}{l}{f(x, y)=\sin x+\sin y+\sin (x+y)} \\ {0 \leqslant x \leqslant 2 \pi, 0 \leqslant y \leqslant 2 \pi}\end{array} $$

Sketch the graph of the function. $$ f(x, y)=10-4 x-5 y $$