Problem 1
Find the directional derivative of \(f\) at the point \(\mathbf{p}\) in the direction of \(\mathbf{a}\). \(f(x, y)=x^{2} y ; \mathbf{p}=(1,2) ; \mathbf{a}=3 \mathbf{i}-4 \mathbf{j}\)
Problem 1
Find the minimum of \(f(x, y)=x^{2}+y^{2}\) subject to the constraint \(g(x, y)=x y-3=0\)
Problem 1
Find the gradient \(\nabla f\). $$ f(x, y)=x^{2} y+3 x y $$
Problem 1
Find the equation of the tangent plane to the given surface at the indicated point. \(x^{2}+y^{2}+z^{2}=16 ;(2,3, \sqrt{3})\)
Problem 1
Find the indicated limit or state that it does not exist. \(\lim _{(x, y) \rightarrow(1,3)}\left(3 x^{2} y-x y^{3}\right)\)
Problem 1
In Problems 1-6, find \(d w / d t\) by using the Chain Rule. Express your final answer in terms of \(t\). $$ w=x^{2} y^{3}, x=t^{3}, y=t^{2} $$
Problem 1
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 \(\mathrm{C} .\) \(f(x, y)=x^{2}+4 y^{2}-4 x\)
Problem 1
Find all first partial derivatives of each function. \(f(x, y)=(2 x-y)^{4}\)
Problem 1
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?
Problem 2
Find all first partial derivatives of each function. \(f(x, y)=\left(4 x-y^{2}\right)^{3 / 2}\)
