Observation 6 - Strike Week Info + Cont. Partial Derivatives

Strike Week

Gradient/Directional Derivative/Tangent Plane problems for them on dropbox
But he is not going to do any lecture at all next week
Despite this, he wants them to follow the reading schedule posted

This Week

They need to;

They're reading up to all the gradient and directional derivative stuff we covered last workshop
His office hours have changed (should be changed on dropbox)
They also gotta quiz tomorrow

He's giving them hints too on the HW

14.3.83: Compute Partials Implicitly
14.3.96: Compute Partials Implicitly
14.3.99: Given the ellipsoid, parametrize the tangent line to the cross section at $y = 2$ .

New Stuff

They first looked at a surface $S : z = f (x, y)$ , and given ${\vec{v}}_{x} = ⟨ 1, 0, f_{x} ⟩$ , and similar for $y$ , they can get the tangent plane by doing:

{\vec{v}}_{x} \times {\vec{v}}_{y}

Really they first looked at the directional derivative. Looking at an intuitive definition for rise/run using ${\vec{v}}_{x}$ and similar for ${\vec{v}}_{y}$ .

I think a cool exercise is to show that the gradient always points upwards. They can do this using the definition that the directional derivative is:

D_{\vec{⟨ a, b ⟩}} f (x, y) = ⟨ a, b, a f_{x} + b f_{y} ⟩

But they should try to plug in $a = f_{x}$ and $b = f_{y}$ .