91Ó°ÊÓ

Let \(f(x, y)=x^{2} y+1 .\) Find (a) \(f(2,1)\) (b) \(f(1,2)\) (c) \(f(0,0)\) (d) \(f(1,-3)\) (e) \(f(3 a, a)\) (f) \(f(a b, a-b)\)

Locate all absolute maxima and minima, if any, by inspection. Then check your answers using calculus. $$ \begin{array}{l}{\text { (a) } f(x, y)=(x-2)^{2}+(y+1)^{2}} \\ {\text { (b) } f(x, y)=1-x^{2}-y^{2}} \\ {\text { (c) } f(x, y)=x+2 y-5}\end{array} $$

Suppose that a function \(f(x, y)\) is differentiable at the point \((3,4)\) with \(f_{x}(3,4)=2\) and \(f_{y}(3,4)=-1 .\) If \(f(3,4)=5,\) estimate the value of \(f(3.01,3.98) .\)

Consider the ellipsoid \(x^{2}+y^{2}+4 z^{2}=12 .\) (a) Find an equation of the tangent plane to the ellipsoid at the point \((2,2,1) .\) (b) Find parametric equations of the line that is normal to the ellipsoid at the point \((2,2,1) .\) (c) Find the acute angle that the tangent plane at the point \((2,2,1)\) makes with the \(x y\) -plane.

Find \(D_{\mathfrak{u}} f\) at \(P\). $$ f(x, y)=(1+x y)^{3 / 2} ; P(3,1) ; \mathbf{u}=\frac{1}{\sqrt{2}} \mathbf{i}+\frac{1}{\sqrt{2}} \mathbf{j} $$

Use an appropriate form of the chain rule to find \(d z / d t\) $$ z=3 x^{2} y^{3} ; x=t^{4}, y=t^{2} $$

Let \(f(x, y)=3 x^{3} y^{2} .\) Find $$ \begin{array}{lll}{\text { (a) } f_{x}(x, y)} & {\text { (b) } f_{y}(x, y)} & {\text { (c) } f_{x}(1, y)} \\ {\text { (d) } f_{x}(x, 1)} & {\text { (e) } f_{y}(1, y)} & {\text { (f) } f_{y}(x, 1)} \\ {\text { (g) } f_{x}(1,2)} & {\text { (h) } f_{y}(1,2)}\end{array} $$

Use limit laws and continuity properties to evaluate the limit. $$ \lim _{(x, y) \rightarrow(1,3)}\left(4 x y^{2}-x\right) $$

Consider the surface \(x z-y z^{3}+y z^{2}=2\) (a) Find an equation of the tangent plane to the surface at the point \((2,-1,1)\) (b) Find parametric equations of the line that is normal to the surface at the point \((2,-1,1) .\) (c) Find the acute angle that the tangent plane at the point \((2,-1,1)\) makes with the \(x y\) -plane.

These exercises are concerned with functions of two variables. Let \(f(x, y)=x+\sqrt[3]{x y} .\) Find $$ \begin{array}{llll}{\text { (a) } f\left(t, t^{2}\right)} & {\text { (b) } f\left(x, x^{2}\right)} & {\text { (c) } f\left(2 y^{2}, 4 y\right)} & {}\end{array} $$