91Ó°ÊÓ

Find the extreme values of \(f(x, y)=x y\) subject to the constraint \(g(x, y)=x^{2}+y^{2}-10=0.\)

(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 \(t\). $$w=x^{2}+y^{2}, \quad x=\cos t+\sin t, \quad y=\cos t-\sin t ; \quad t=0$$

Use Taylor's formula for \(f(x, y)\) at the origin to find quadratic and cubic approximations of \(f\) near the origin. $$f(x, y)=e^{x} \cos y$$

Find the specific function values. $$f(x, y)=\sin (x y)$$ a. \(f\left(2, \frac{\pi}{6}\right)\) b. \(f\left(-3, \frac{\pi}{12}\right)\) c. \(f\left(\pi, \frac{1}{4}\right)\) d. \(f\left(-\frac{\pi}{2},-7\right)\)

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

Begin by drawing a diagram that shows the relations among the variables. If \(w=x^{2}+y-z+\sin t\) and \(x+y=t,\) find a. \(\left(\frac{\partial w}{\partial y}\right)_{x, z}\) b. \(\left(\frac{\partial w}{\partial y}\right)_{z, t} \quad\) c. \(\left(\frac{\partial w}{\partial z}\right)_{x, y}\) d. \(\left(\frac{\partial w}{\partial z}\right)_{y, t}\) e. \(\left(\frac{\partial w}{\partial t}\right)_{x, z} \quad\) f. \(\left(\frac{\partial w}{\partial t}\right)_{y, z}\)

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

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

Find the limits. $$\lim _{(x, y) \rightarrow(0,4)} \frac{x}{\sqrt{y}}$$

(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 \(t\). $$w=\frac{x}{z}+\frac{y}{z}, \quad x=\cos ^{2} t, \quad y=\sin ^{2} t, \quad z=1 / t ; \quad t=3$$