Chapter 12

In Problems \(1-8\), find the equation of the tangent plane to the given surface at the indicated point. $$ x^{2}+y^{2}+z^{2}=16 ;(2,3, \sqrt{3}) $$

In Problems 1-8, find the directional derivative of \(f\) at the point \(\mathbf{p}\) in the direction of \(\mathbf{a}\). \(f(x, y)=x^{2} y ; \mathbf{p}=(1,2) ; \mathbf{a}=3 \mathbf{i}-4 \mathbf{j}\)

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

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?

In Problems \(1-16\), find the indicated limit or state that it does not exist. 1\. \(\lim _{(x, y) \rightarrow(1,3)}\left(3 x^{2} y-x y^{3}\right)\)

In Problems 1-16, find all first partial derivatives of each function. \(f(x, y)=(2 x-y)^{4}\)

$$ \text { In Problems 1-10, find the gradient } \nabla f \text {. } $$ $$ f(x, y)=x^{2} y+3 x y $$

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

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

In Problems 1-8, find the directional derivative of \(f\) at the point \(\mathbf{p}\) in the direction of \(\mathbf{a}\). \(f(x, y)=y^{2} \ln x ; \mathbf{p}=(1,4) ; \mathbf{a}=\mathbf{i}-\mathbf{j}\)

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

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

\(\lim _{(x, y) \rightarrow(-2,1)}\left(x y^{3}-x y+3 y^{2}\right)\)

In Problems 1-16, find all first partial derivatives of each function. \(f(x, y)=\left(4 x-y^{2}\right)^{3 / 2}\)

$$ \text { In Problems 1-10, find the gradient } \nabla f \text {. } $$ $$ f(x, y)=x^{3} y-y^{3} $$

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

In Problems \(1-8\), find the equation of the tangent plane to the given surface at the indicated point. $$ x^{2}-y^{2}+z^{2}+1=0 ;(1,3, \sqrt{7}) $$

In Problems 1-8, find the directional derivative of \(f\) at the point \(\mathbf{p}\) in the direction of \(\mathbf{a}\). \(f(x, y)=2 x^{2}+x y-y^{2} ; \mathbf{p}=(3,-2) ; \mathbf{a}=\mathbf{i}-\mathbf{j}\)

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

Let \(g(x, y, z)=x^{2} \sin y z\). Find each value. (a) \(g(1, \pi, 2)\) (b) \(g(2,1, \pi / 6)\) (c) \(g(4,2, \pi / 4)\) (d) \(g(\pi, \pi, \pi)\)

\(\lim _{(x, y) \rightarrow(2, \pi)}\left[x \cos ^{2}(x y)-\sin (x y / 3)\right]\)

In Problems 1-16, find all first partial derivatives of each function. \(f(x, y)=\frac{x^{2}-y^{2}}{x y}\)

$$ \text { In Problems 1-10, find the gradient } \nabla f \text {. } $$ $$ f(x, y)=x e^{x y} $$

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

In Problems \(1-8\), find the equation of the tangent plane to the given surface at the indicated point. $$ x^{2}+y^{2}-z^{2}=4 ;(2,1,1) $$

In Problems 1-8, find the directional derivative of \(f\) at the point \(\mathbf{p}\) in the direction of \(\mathbf{a}\). \(f(x, y)=x^{2}-3 x y+2 y^{2} ; \mathbf{p}=(-1,2) ; \mathbf{a}=2 \mathbf{i}-\mathbf{j}\)

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

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) \(g(3,6,1.2)\)

$$ f(x, y)=x y^{2}-6 x^{2}-3 y^{2} $$

\(\lim _{(x, y) \rightarrow(1,2)} \frac{x^{3}-3 x^{2} y+3 x y^{2}-y^{3}}{y-2 x^{2}}\)

In Problems 1-16, find all first partial derivatives of each function. \(f(x, y)=e^{x} \cos y\)

$$ \text { In Problems 1-10, find the gradient } \nabla f \text {. } $$ $$ f(x, y)=x^{2} y \cos y $$

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

In Problems \(1-8\), find the equation of the tangent plane to the given surface at the indicated point. $$ z=\frac{x^{2}}{4}+\frac{y^{2}}{4} ;(2,2,2) $$

In Problems 1-8, find the directional derivative of \(f\) at the point \(\mathbf{p}\) in the direction of \(\mathbf{a}\). \(f(x, y)=e^{x} \sin y ; \mathbf{p}=(0, \pi / 4) ; \mathbf{a}=\mathbf{i}+\sqrt{3} \mathbf{j}\)

In Problems 1-6, find dw/dt 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 \(F(f(t), g(t))\) if \(F(x, y)=x^{2} y\) and \(f(t)=t \cos t\), \(g(t)=\sec ^{2} t .\)

In Problems 1-16, find all first partial derivatives of each function. \(f(x, y)=e^{y} \sin x\)

$$ \text { In Problems 1-10, find the gradient } \nabla f \text {. } $$ $$ f(x, y)=x^{2} y /(x+y) $$

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

In Problems \(1-8\), find the equation of the tangent plane to the given surface at the indicated point. $$ z=x e^{-2 y} ;(1,0,1) $$

In Problems 1-8, find the directional derivative of \(f\) at the point \(\mathbf{p}\) in the direction of \(\mathbf{a}\). \(f(x, y)=e^{-x y} ; \mathbf{p}=(1,-1) ; \mathbf{a}=-\mathbf{i}+\sqrt{3} \mathbf{j}\)

In Problems 1-6, find dw/dt 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 \(F(f(t), g(t))\) if \(F(x, y)=e^{x}+y^{2}\) and \(f(t)=\ln t^{2}\), \(g(t)=e^{t / 2} .\)

In Problems 1-16, find all first partial derivatives of each function. \(f(x, y)=\left(3 x^{2}+y^{2}\right)^{-1 / 3}\)

$$ \text { In Problems 1-10, find the gradient } \nabla f \text {. } $$ $$ f(x, y)=\sin ^{3}\left(x^{2} y\right) $$

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

In Problems \(1-8\), find the equation of the tangent plane to the given surface at the indicated point. $$ z=2 e^{3 y} \cos 2 x ;(\pi / 3,0,-1) $$

In Problems 1-8, find the directional derivative of \(f\) at the point \(\mathbf{p}\) in the direction of \(\mathbf{a}\). \(f(x, y, z)=x^{3} y-y^{2} z^{2} ; \mathbf{p}=(-2,1,3) ; \mathbf{a}=\mathbf{i}-2 \mathbf{j}+2 \mathbf{k}\)

In Problems 7-12, find \(\partial w / \partial t\) by using the Chain Rule. Express your final answer in terms of \(s\) and \(t\). $$ w=x^{2} y ; x=s t, y=s-t $$