Chapter 13

Find all critical points of the following functions. $$f(x, y)=x^{4}+y^{4}-16 x y$$

Evaluate the following limits. $$\lim _{(x, y) \rightarrow(-3,3)}\left(4 x^{2}-y^{2}\right)$$

Find an equation of the plane tangent to the following surfaces at the given points. $$x y \sin z=1 ;(1,2, \pi / 6) \text { and }(-2,-1,5 \pi / 6)$$

Computing gradients Compute the gradient of the following functions and evaluate it at the given point \(P\). $$f(x, y)=x e^{2 x y} ; P(1,0)$$

Find an equation of the plane that passes through the point \(P_{0}\) with a normal vector \(\mathbf{n}\). $$P_{0}(2,3,0) ; \mathbf{n}=\langle-1,2,-3\rangle$$

Find the domain of the following functions. $$f(x, y)=\sqrt{25-x^{2}-y^{2}}.$$

Use Theorem 7 to find the following derivatives. When feasible, express your answer in terms of the independent variable. $$d w / d t, \text { where } w=x y \sin z, x=t^{2}, y=4 t^{3}, \text { and } z=t+1$$

Find the first partial derivatives of the following functions. $$g(x, y)=\cos 2 x y$$

Lagrange multipliers in two variables Use Lagrange multipliers to find the maximum and minimum values of \(f\) (when they exist) subject to the given constraint. $$f(x, y)=x y+x+y \text { subject to } x^{2} y^{2}=4$$

Find all critical points of the following functions. $$f(x, y)=x^{3} / 3-y^{3} / 3+3 x y$$

Evaluate the following limits. $$\lim _{(x, y) \rightarrow(2,-1)}\left(x y^{8}-3 x^{2} y^{3}\right)$$

Find an equation of the plane tangent to the following surfaces at the given points. $$y z e^{x z}-8=0 ;(0,2,4) \text { and }(0,-8,-1)$$

Computing gradients Compute the gradient of the following functions and evaluate it at the given point \(P\). $$f(x, y)=\sin (3 x+2 y) ; P(\pi, 3 \pi / 2)$$

Find an equation of the plane that passes through the point \(P_{0}\) with a normal vector \(\mathbf{n}\). $$P_{0}(1,2,-3) ; \mathbf{n}=\langle-1,4,-3\rangle$$

Find the domain of the following functions. $$f(x, y)=\frac{1}{\sqrt{x^{2}+y^{2}-25}}.$$

Use Theorem 7 to find the following derivatives. When feasible, express your answer in terms of the independent variable. $$\begin{aligned} &d Q / d t, \text { where } Q=\sqrt{x^{2}+y^{2}+z^{2}}, x=\sin t, y=\cos t, \text { and } &z=\cos t \end{aligned}$$

Find the first partial derivatives of the following functions. $$h(x, y)=\left(y^{2}+1\right) e^{x}$$

Lagrange multipliers in three variables Use Lagrange multipliers to find the maximum and minimum values of \(f\) (when they exist) subject to the given constraint. $$f(x, y, z)=x+3 y-z \text { subject to } x^{2}+y^{2}+z^{2}=4$$

Find all critical points of the following functions. $$f(x, y)=x^{4}-2 x^{2}+y^{2}-4 y+5$$

Evaluate the following limits. $$\lim _{(x, y) \rightarrow(0, \pi)} \frac{\cos x y+\sin x y}{2 y}$$

Find an equation of the plane tangent to the following surfaces at the given points. $$z^{2}-x^{2} / 16-y^{2} / 9-1=0 ;(4,3,-\sqrt{3}) \text { and }(-8,9, \sqrt{14})$$

Computing gradients Compute the gradient of the following functions and evaluate it at the given point \(P\). $$F(x, y)=e^{-x^{2}-2 y^{2}} ; P(-1,2)$$

Find the equation of the plane that is parallel to the vectors \langle 1,0,1\rangle and \(\langle 0,2,1\rangle,\) passing through the point (1,2,3)

Find the domain of the following functions. $$f(x, y)=\sin \frac{x}{y}.$$

Use Theorem 7 to find the following derivatives. When feasible, express your answer in terms of the independent variable. $$\begin{aligned}&d U / d t, \text { where } U=\ln (x+y+z), x=t, y=t^{2}, \text { and }&z=t^{3}\end{aligned}$$

Find the first partial derivatives of the following functions. $$f(x, y)=e^{x^{2} y}$$

Lagrange multipliers in three variables Use Lagrange multipliers to find the maximum and minimum values of \(f\) (when they exist) subject to the given constraint. $$f(x, y, z)=x y z \text { subject to } x^{2}+2 y^{2}+4 z^{2}=9$$

Find all critical points of the following functions. $$f(x, y)=x^{2}+x y-2 x-y+1$$

Find an equation of the plane tangent to the following surfaces at the given points. $$2 x+y^{2}-z^{2}=0 ;(0,1,1) \text { and }(4,1,-3)$$

Computing gradients Compute the gradient of the following functions and evaluate it at the given point \(P\). $$h(x, y)=\ln \left(1+x^{2}+2 y^{2}\right) ; P(2,-3)$$

Find the equation of the plane that is parallel to the vectors \langle 1,-3,1\rangle and \(\langle 4,2,0\rangle,\) passing through the point (3,0,-2)

Find the domain of the following functions. $$f(x, y)=\frac{12}{y^{2}-x^{2}}.$$

Use Theorem 7 to find the following derivatives. When feasible, express your answer in terms of the independent variable. $$d V / d t, \text { where } V=\frac{x-y}{y+z}, x=t, y=2 t, \text { and } z=3 t$$

Find the first partial derivatives of the following functions. $$f(s, t)=\frac{s-t}{s+t}$$

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

Find all critical points of the following functions. $$f(x, y)=x^{2}+6 x+y^{2}+8$$

Find an equation of the plane tangent to the following surfaces at the given points. $$z=4-2 x^{2}-y^{2} ;(2,2,-8) \text { and }(-1,-1,1)$$

Compute the directional derivative of the following functions at the given point P in the direction of the given vector. Be sure to use a unit vector for the direction vector. $$f(x, y)=x^{2}-y^{2} ; P(-1,-3) ;\left\langle\frac{3}{5},-\frac{4}{5}\right\rangle$$

Find an equation of the following planes. The plane passing through the points \((1,0,3),(0,4,2),\) and (1,1,1)

Find the domain of the following functions. $$g(x, y)=\ln \left(x^{2}-y\right).$$

The volume of a right circular cylinder with radius \(r\) and height \(h\) is \(V=\pi r^{2} h\). a. Assume that \(r\) and \(h\) are functions of \(t .\) Find \(V^{\prime}(t)\). b. Suppose that \(r=e^{t}\) and \(h=e^{-2 t},\) for \(t \geq 0 .\) Use part (a) to find \(V^{\prime}(t)\). c. Does the volume of the cylinder in part (b) increase or decrease as \(t\) increases?

Find the first partial derivatives of the following functions. $$f(w, z)=\frac{w}{w^{2}+z^{2}}$$

Find all critical points of the following functions. $$f(x, y)=e^{x^{2} y^{2}-2 x y^{2}+y^{2}}$$

Evaluate the following limits. $$\lim _{(x, y) \rightarrow(1,-1)} \frac{10 x y-2 y^{2}}{x^{2}+y^{2}}$$

Find an equation of the plane tangent to the following surfaces at the given points. $$z=2+2 x^{2}+\frac{y^{2}}{2} ;\left(-\frac{1}{2}, 1,3\right) \text { and }(3,-2,22)$$

Compute the directional derivative of the following functions at the given point P in the direction of the given vector. Be sure to use a unit vector for the direction vector. $$f(x, y)=3 x^{2}+y^{3} ; P(3,2) ;\left\langle\frac{5}{13}, \frac{12}{13}\right\rangle$$

Find an equation of the following planes. The plane passing through the points \((-1,1,1),(0,0,2),\) and (3,-1,-2)

Find the domain of the following functions. $$f(x, y)=\sin ^{-1}\left(y-x^{2}\right).$$

The volume of a pyramid with a square base \(x\) units on a side and a height of \(h\) is \(V=\frac{1}{3} x^{2} h\). a. Assume that \(x\) and \(h\) are functions of \(t\). Find \(V^{\prime}(t)\). b. Suppose that \(x=t /(t+1)\) and \(h=1 /(t+1),\) for \(t \geq 0\) Use part (a) to find \(V^{\prime}(t)\). c. Does the volume of the pyramid in part (b) increase or decrease as \(t\) increases?

Find the first partial derivatives of the following functions. $$g(x, z)=x \ln \left(z^{2}+x^{2}\right)$$