In-class work for 22 June 2012
1.) Find the directional derivative of $f(x,y)=x^2y^3-y^4$ at the point $(2,1)$ in the direction indicated by the angle $\theta=\frac{\pi}{4}$.

2.) Find the gradient of...
    a.) $f(x,y) = \sin(2x+3y)$
    b.) $f(x,y) = \frac{y^2}{x}$
    c.) $f(x,y) = \sqrt{x+yz}$

3.) Find the directional derivative of the function at the given point in the direction of vector $\vec{v}$:
    a.) $f(x,y) = 1+2x \sqrt{y}$, $(3,4), \vec{v} = < 4,-3 >$
    b.) $f(x,y)=\arctan{xy}$, $(1,2)$, $\vec{v} = < 5,10 >$
    c.) $f(x,y,z) = ( x + 2y + 3z)^{\frac{3}{2}}, (1,1,2), \vec{v} = <2, -1 >$

4.) Find the maximum rate of change of $f$ at the given point and specify the direction in which it occurs.
    a.) $f(x,y) = \frac{y^2}{x}$, $(2,4)$
    b.) $f(x,y) = \sin(x,y)$, $(1,0)$
    c.) $f(x,y,z) = \sqrt{x^2+y^2+z^2}$, $(3,6,-2)$

5.) The temperature at a point $(x,y,z)$ is given by $T(x,y,z)=200e^{-x^2-3y^2-9z^2}$ where $T$ is measured in degrees Celcius and $x,y,z$ in meters.
    a.) Find the rate of change of $T$ at $(1,2,2)$ in the direction toward the point $(2,1,3)$.
    b.) Show that at any point in the ball the direction of greatest increase in temperature is given by a vector that points toward the origin. (Does this make physical sense?)

6.) Find the equations of the tangent plane AND the normal line to the given surface at the specified point.

    a.) $2(x-2)^2 + (y-1)^2 + (z-3)^2 = 10$, $(3,3,5)$
    b.) $x - z = 4 \arctan(yz)$, $(1 + \pi, 1, 1)$