Problem 2
Find \(d w / d t\) by using the Chain Rule. Express your final answer in terms of \(t\). $$ w=x^{2} y-y^{2} x ; x=\cos t, y=\sin t $$
Problem 2
Find the equation of the tangent plane to the given surface at the indicated point. \(8 x^{2}+y^{2}+8 z^{2}=16 ;(1,2, \sqrt{2} / 2)\)
Problem 2
Find the maximum of \(f(x, y)=x y\) subject to the constraint \(g(x, y)=4 x^{2}+9 y^{2}-36=0\)
Problem 2
Find the directional derivative of \(f\) at the point \(\mathbf{p}\) in the direction of \(\mathbf{a}\). \(f(x, y)=y^{2} \ln x ; \mathbf{p}=(1,4) ; \mathbf{a}=\mathbf{i}-\mathbf{j}\)
Problem 2
Find all first partial derivatives of each function. \(f(x, y)=\left(4 x-y^{2}\right)^{3 / 2}\)
Problem 2
Find the gradient \(\nabla f\). $$ f(x, y)=x^{3} y-y^{3} $$
Problem 3
Find all first partial derivatives of each function. \(f(x, y)=\frac{x^{2}-y^{2}}{x y}\)
Problem 3
Find \(d w / d t\) by using the Chain Rule. Express your final answer in terms of \(t\). $$ w=e^{x} \sin y+e^{y} \sin x ; x=3 t, y=2 t $$
Problem 3
Find the indicated limit or state that it does not exist. \(\lim _{(x, y) \rightarrow(2, \pi)}\left[x \cos ^{2}(x y)-\sin (x y / 3)\right]\)
Problem 3
Find the directional derivative of \(f\) at the point \(\mathbf{p}\) in the direction of \(\mathbf{a}\). \(f(x, y)=2 x^{2}+x y-y^{2} ; \mathbf{p}=(3,-2) ; \mathbf{a}=\mathbf{i}-\mathbf{j}\)
