91Ó°ÊÓ

Describe in words the level curves of the paraboloid \(z=x^{2}+y^{2}.\)

Use Theorem 12.7 to find the following derivatives. When feasible, express your answer in terms of the independent variable. $$d z / d t, \text { where } z=x^{2}+y^{3}, x=t^{2}, \text { and } y=t$$

Write the approximate change formula for a function \(z=f(x, y)\) at the point \((a, b)\) in terms of differentials.

Use Lagrange multipliers to find the maximum and minimum values of \(f\) (when they exist) subject to the given constraint. $$f(x, y)=x+y \text { subject to } x^{2}-x y+y^{2}=1$$

Use Lagrange multipliers to find the maximum and minimum values of \(f\) (when they exist) subject to the given constraint. $$f(x, y)=x^{2}+y^{2} \text { subject to } 2 x^{2}+3 x y+2 y^{2}=7$$

What is the procedure for locating absolute maximum and minimum values on a closed bounded domain?

Let \(R\) be the unit disk \(\left\\{(x, y): x^{2}+y^{2} \leq 1\right\\}\) with (0,0) removed. Is (0,0) a boundary point of \(R ?\) Is \(R\) open or closed?

What is the name of the surface defined by the equation \(y=\frac{x^{2}}{4}+\frac{z^{2}}{8} ?\)

Use the limit definition of partial derivatives to evaluate \(f_{x}(x, y)\) and \(f_{y}(x, y)\) for each of the following functions. $$f(x, y)=x+y^{2}+4$$

Use Theorem 12.7 to find the following derivatives. When feasible, express your answer in terms of the independent variable. $$d z / d t, \text { where } z=x y^{2}, x=t^{2}, \text { and } y=t$$