Directional Derivatives and the Gradient:

1. Find the directional derivative \(D_u f(x,y) \) of \( f(x,y)=x^2+3xy-y^3\) in the direction of \( {\bf u}=\langle \cos \theta, \sin \theta\rangle \), given \(\theta = \cos^{-1}\frac{3}{5} \). What is \( D_u(1,-1)\) ?   solution

2. Find the gradient \(\nabla f(x,y) \) of each of the following functions.   (a) \(f(x,y)=x^3+xy^3 \)    (b) \(f(x,y)=e^{xy}+1 \)    solution

3. Find the directional derivative  of \( f(x,y)=e^x \cos y \) at the point \(P(0, \pi/2)\) in the direction of \(Q(0, 1+\pi/2)\).   solution

4. Find the directional derivative of  \( f(x,y, z)=x^2y + y^2 z \)   at the  point \(P = (2, -1, 3) \) in the direction of \({\bf PQ} \) where \(Q=(3, 1, 1) \).   solution

5. Find the direction for which the directional derivative of \(f(x,y)=4x-xy+3y^2 \) at \((-1,2) \) is a maximum. What is the maximum value?  solution

6. Find  equations for (a) the tangent plane, and (b) the normal line to the surface \( x^2+10xyz+y^2+8z^2=0 \) at the point \((-1,-1,-1) \).   solution