Chapter 12

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

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

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