91Ó°ÊÓ

Find and sketch the level curves \(f(x, y)=c\) on the same set of coordinate axes for the given values of \(c .\) We refer to these level curves as a contour map. $$f(x, y)=\sqrt{25-x^{2}-y^{2}}, \quad c=0,1,2,3,4$$

Find the limits by rewriting the fractions first. $$\lim _{(x, y) \rightarrow(2,-4) \atop x \neq-4, x \neq x^{2}} \frac{y+4}{x^{2} y-x y+4 x^{2}-4 x}$$

Draw a branch diagram and write a Chain Rule formula for each derivative. $$\begin{aligned} &\frac{\partial w}{\partial x} \text { and } \frac{\partial w}{\partial y} \text { for } w=f(r, s, t), \quad r=g(x, y), \quad s=h(x, y),\\\&t=k(x, y) \end{aligned}$$

Find all the local maxima, local minima, and saddle points of the functions. $$f(x, y)=x^{3}+y^{3}+3 x^{2}-3 y^{2}-8$$

Find \(\partial f / \partial x\) and \(\partial f / \partial y\). \(f(x, y)=\sin ^{2}(x-3 y)\)

Find parametric equations for the line tangent to the curve of intersection of the surfaces at the given point. $$\begin{array}{l}\text { Surfaces: } x^{3}+3 x^{2} y^{2}+y^{3}+4 x y-z^{2}=0 \\\\\quad x^{2}+y^{2}+z^{2}=11 \\\\\text { Point: } \quad(1,1,3)\end{array}$$

Find all the local maxima, local minima, and saddle points of the functions. $$f(x, y)=x^{3}+3 x y^{2}-15 x+y^{3}-15 y$$

Find the point on the plane \(x+2 y+3 z=13\) closest to the point \((1,1,1).\)

Find the limits by rewriting the fractions first. $$\lim _{(x, y) \rightarrow(0,0)\atop x \neq y} \frac{x-y+2 \sqrt{x}-2 \sqrt{y}}{\sqrt{x}-\sqrt{y}}$$

Draw a branch diagram and write a Chain Rule formula for each derivative. $$\frac{\partial w}{\partial u} \text { and } \frac{\partial w}{\partial v} \text { for } w=g(x, y), \quad x=h(u, v), \quad y=k(u, v)$$