# Math 141 - Week 5 Notes Monday, February 8, 1999

Topics for this week -

1. Tangent planes and differentials
2. More about partial derivatives -- the chain rule
3. Directional derivatives, the gradient of a function, normal lines

Examples -

1. Find the differential of z = 3x2y - y3. Then find the tangent plane to the graph of z at the point (3,2,46). Also find the equations of the normal line at that point.
2. If x = t2 and y = 3t3, and z = x2 - y3, then calculate dz/dt.
3. Find the gradient of the function z in the first example above. At (3,2), what is the directional derivative of z in the direction of the vector <1,1> ? In what direction does z increase the fastest at that point? How fast?

Maple - To plot curves in three dimensions, use "spacecurve" (pretty much like parametric plotting in 2D). For graphs of functions of two variables, use "plot3d", for implicitly defined functions use "implicitplot3d" -- Note that "spacecurve" and "implicitplot3d" are in the "plots" library.

Fifth Homework Assignment - due on FRIDAY, February 19.