Chapter 12

Find the directional derivative of \(f\) at the point \(\mathbf{p}\) in the direction of \(\mathbf{a}\). \(f(x, y)=x^{2} y ; \mathbf{p}=(1,2) ; \mathbf{a}=3 \mathbf{i}-4 \mathbf{j}\)

Find the minimum of \(f(x, y)=x^{2}+y^{2}\) subject to the constraint \(g(x, y)=x y-3=0\)

Find all critical points. Indicate whether each such point gives a local maximum or a local minimum, or whether it is a saddle point. Hint: Use Theorem \(\mathrm{C} .\) \(f(x, y)=x^{2}+4 y^{2}-4 x\)

In Problems 1-6, find \(d w / d t\) by using the Chain Rule. Express your final answer in terms of \(t\). $$ w=x^{2} y^{3}, x=t^{3}, y=t^{2} $$

Find the equation of the tangent plane to the given surface at the indicated point. \(x^{2}+y^{2}+z^{2}=16 ;(2,3, \sqrt{3})\)

Find the gradient \(\nabla f\). $$ f(x, y)=x^{2} y+3 x y $$

Find the indicated limit or state that it does not exist. \(\lim _{(x, y) \rightarrow(1,3)}\left(3 x^{2} y-x y^{3}\right)\)

Find all first partial derivatives of each function. \(f(x, y)=(2 x-y)^{4}\)

Let \(f(x, y)=x^{2} y+\sqrt{y}\). Find each value. (a) \(f(2,1)\) (b) \(f(3,0)\) (c) \(f(1,4)\) (d) \(f\left(a, a^{4}\right)\) (e) \(f\left(1 / x, x^{4}\right)\) (f) \(f(2,-4)\) What is the natural domain for this function?

Find the directional derivative of \(f\) at the point \(\mathbf{p}\) in the direction of \(\mathbf{a}\). \(f(x, y)=y^{2} \ln x ; \mathbf{p}=(1,4) ; \mathbf{a}=\mathbf{i}-\mathbf{j}\)

Find the maximum of \(f(x, y)=x y\) subject to the constraint \(g(x, y)=4 x^{2}+9 y^{2}-36=0\)

Find \(d w / d t\) by using the Chain Rule. Express your final answer in terms of \(t\). $$ w=x^{2} y-y^{2} x ; x=\cos t, y=\sin t $$

Find the equation of the tangent plane to the given surface at the indicated point. \(8 x^{2}+y^{2}+8 z^{2}=16 ;(1,2, \sqrt{2} / 2)\)

Find the gradient \(\nabla f\). $$ f(x, y)=x^{3} y-y^{3} $$

Find the indicated limit or state that it does not exist. \(\lim _{(x, y) \rightarrow(-2,1)}\left(x y^{3}-x y+3 y^{2}\right)\)

Find all first partial derivatives of each function. \(f(x, y)=\left(4 x-y^{2}\right)^{3 / 2}\)

Find the directional derivative of \(f\) at the point \(\mathbf{p}\) in the direction of \(\mathbf{a}\). \(f(x, y)=2 x^{2}+x y-y^{2} ; \mathbf{p}=(3,-2) ; \mathbf{a}=\mathbf{i}-\mathbf{j}\)

Find the maximum of \(f(x, y)=4 x^{2}-4 x y+y^{2}\) subject to the constraint \(x^{2}+y^{2}=1\).

Find \(d w / d t\) by using the Chain Rule. Express your final answer in terms of \(t\). $$ w=e^{x} \sin y+e^{y} \sin x ; x=3 t, y=2 t $$

Find the equation of the tangent plane to the given surface at the indicated point. \(x^{2}-y^{2}+z^{2}+1=0 ;(1,3, \sqrt{7})\)

Find the gradient \(\nabla f\). $$ f(x, y)=x e^{x y} $$

Find the indicated limit or state that it does not exist. \(\lim _{(x, y) \rightarrow(2, \pi)}\left[x \cos ^{2}(x y)-\sin (x y / 3)\right]\)

Find all first partial derivatives of each function. \(f(x, y)=\frac{x^{2}-y^{2}}{x y}\)

Find the directional derivative of \(f\) at the point \(\mathbf{p}\) in the direction of \(\mathbf{a}\). \(f(x, y)=x^{2}-3 x y+2 y^{2} ; \mathbf{p}=(-1,2) ; \mathbf{a}=2 \mathbf{i}-\mathbf{j}\)

Find the minimum of \(f(x, y)=x^{2}+4 x y+y^{2}\) subject to the constraint \(x-y-6=0\).

Find \(d w / d t\) by using the Chain Rule. Express your final answer in terms of \(t\). $$ w=\ln (x / y) ; x=\tan t, y=\sec ^{2} t $$

Find the equation of the tangent plane to the given surface at the indicated point. \(x^{2}+y^{2}-z^{2}=4 ;(2,1,1)\)

Find the gradient \(\nabla f\). $$ f(x, y)=x^{2} y \cos y $$

Find the indicated limit or state that it does not exist. \(\lim _{(x, y) \rightarrow(1,2)} \frac{x^{3}-3 x^{2} y+3 x y^{2}-y^{3}}{y-2 x^{2}}\)

Find all first partial derivatives of each function. \(f(x, y)=e^{x} \cos y\)

Let \(g(x, y, z)=\sqrt{x \cos y}+z^{2} .\) Find each value. (a) \(g(4,0,2)\) (b) \(g(-9, \pi, 3)\) (c) \(g(2, \pi / 3,-1)\) (d) \(g(3,6,1.2)\)

Find the directional derivative of \(f\) at the point \(\mathbf{p}\) in the direction of \(\mathbf{a}\). \(f(x, y)=e^{x} \sin y ; \mathbf{p}=(0, \pi / 4) ; \mathbf{a}=\mathbf{i}+\sqrt{3} \mathbf{j}\)

Find the minimum of \(f(x, y, z)=x^{2}+y^{2}+z^{2}\) subject to the constraint \(x+3 y-2 z=12\).

Find all critical points. Indicate whether each such point gives a local maximum or a local minimum, or whether it is a saddle point. Hint: Use Theorem \(\mathrm{C} .\) \(f(x, y)=x y\)

Find \(d w / d t\) by using the Chain Rule. Express your final answer in terms of \(t\). $$ w=\sin \left(x y z^{2}\right) ; x=t^{3}, y=t^{2}, z=t $$

Find the equation of the tangent plane to the given surface at the indicated point. \(z=\frac{x^{2}}{4}+\frac{y^{2}}{4} ;(2,2,2)\)

Find the gradient \(\nabla f\). $$ f(x, y)=x^{2} y /(x+y) $$

Find the indicated limit or state that it does not exist. \(\lim _{(x, y) \rightarrow(-1,2)} \frac{x y-y^{3}}{(x+y+1)^{2}}\)

Find all first partial derivatives of each function. \(f(x, y)=e^{y} \sin x\)

Find \(F(f(t), g(t))\) if \(F(x, y)=x^{2} y\) and \(f(t)=t \cos t\), \(g(t)=\sec ^{2} t\)

Find the directional derivative of \(f\) at the point \(\mathbf{p}\) in the direction of \(\mathbf{a}\). \(f(x, y)=e^{-x y} ; \mathbf{p}=(1,-1) ; \mathbf{a}=-\mathbf{i}+\sqrt{3} \mathbf{j}\)

Find the minimum of \(f(x, y, z)=4 x-2 y+3 z\) subject to the constraint \(2 x^{2}+y^{2}-3 z=0\)

Find \(d w / d t\) by using the Chain Rule. Express your final answer in terms of \(t\). $$ w=x y+y z+x z ; x=t^{2}, y=1-t^{2}, z=1-t $$

Find the equation of the tangent plane to the given surface at the indicated point. \(z=x e^{-2 y} ;(1,0,1)\)

Find the gradient \(\nabla f\). $$ f(x, y)=\sin ^{3}\left(x^{2} y\right) $$

Find the indicated limit or state that it does not exist. \(\lim _{(x, y) \rightarrow(0,0)} \frac{x y+\cos x}{x y-\cos x}\)

Find all first partial derivatives of each function. \(f(x, y)=\left(3 x^{2}+y^{2}\right)^{-1 / 3}\)

Find \(F(f(t), g(t))\) if \(F(x, y)=e^{x}+y^{2}\) and \(f(t)=\ln t^{2}\) \(g(t)=e^{t / 2} .\)

Find the directional derivative of \(f\) at the point \(\mathbf{p}\) in the direction of \(\mathbf{a}\). \(f(x, y, z)=x^{3} y-y^{2} z^{2} ; \mathbf{p}=(-2,1,3) ; \mathbf{a}=\mathbf{i}-2 \mathbf{j}+2 \mathbf{k}\)

What are the dimensions of the rectangular box, open at the top, that has maximum volume when the surface area is \(48 ?\)