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.

  1. Reading: In Chapter 12, read sections 12.4, 12.5, 12.6.
  2. More reading: From the Lab Manual, the sections on 3d plotting and implicit 3d plotting (pp 117-124). Also look at the solved problems on pp 169-171.
  3. Chapter 12 problems: Make sure you can do all of the Core problems for sections 12.4 12.5 and 12.6. But write up only the following to be handed in: