91Ó°ÊÓ

Find the specific function values. \(f(x, y)=x^{2}+x y^{3}\) a. \(f(0,0)\) b. \(f(-1,1)\) c. \(f(2,3)\) d. \(f(-3,-2)\)

In Exercises find \(\partial f / \partial x\) and \(\partial f / \partial y\). $$f(x, y)=2 x^{2}-3 y-4$$

Find the gradient of the function at the given point. Then sketch the gradient together with the level curve that passes through the point. $$f(x, y)=y-x, \quad(2,1)$$

Find the limits. $$\lim _{(x, y) \rightarrow(0,0)} \frac{3 x^{2}-y^{2}+5}{x^{2}+y^{2}+2}$$

(a) express \(d w / d t\) as a function of \(t,\) both by using the Chain Rule and by expressing \(w\) in terms of \(t\) and differentiating directly with respect to \(t .\) Then (b) evaluate \(d w / d t\) at the given value of \(t\) \(w=x^{2}+y^{2}, \quad x=\cos t, \quad y=\sin t ; \quad t=\pi\)

Find equations for the (a) tangent plane and (b) normal line at the point \(P_{0}\) on the given surface. $$x^{2}+y^{2}+z^{2}=3, \quad P_{0}(1,1,1)$$

Find all the local maxima, local minima, and saddle points of the functions. $$f(x, y)=x^{2}+x y+y^{2}+3 x-3 y+4$$

Find the points on the ellipse \(x^{2}+2 y^{2}=1\) where \(f(x, y)=x y\) has its extreme values.

Find equations for the (a) tangent plane and (b) normal line at the point \(P_{0}\) on the given surface. $$x^{2}+y^{2}-z^{2}=18, \quad P_{0}(3,5,-4)$$

Find all the local maxima, local minima, and saddle points of the functions. $$f(x, y)=2 x y-5 x^{2}-2 y^{2}+4 x+4 y-4$$