Math 141 - Week 6 Notes Monday, February 15, 1999

Topics for this week -

  1. Directional derivatives, the gradient of a function, normal lines
  2. Maximum and minimum values
  3. Lagrange Multiplier

Topics for next week -

  1. Double integrals over rectangles
  2. Iterated Integrals
  3. Double integrals over general regions
  4. Double integrals in polar coordinates

Examples -

  1. Find the gradient of the function z=3x2y - y3. 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?
  2. Find the maximum rate of change of f(x,y)=ln(x2+y2) at (1,2) and the direction in which it occurs.
  3. Find the tangent line and the normal line to the level curve x2+4y2=8 at the point (2,1).
  4. Find the tangent plane and the normal line to the level surface x2-2y2-3z2+xyz=4 at the point (3,-2,-1).
  5. Find the local maximum and minimum values and saddle points of the function
  6. Find the absolute maximum and minimum values of f(x,y)=yx1/2-y2-x+6y on the set D={(x,y)| x=0..9,y=0..5}.
  7. (Fall 97 #17) Find the maximum of f(x,y)=2x2-y2 on the set D={(x,y) | 2x2+y2=0..1}
  8. Find the shortest distance from (0,0) to the curve xy=1

Sixth Homework Assignment - due on FRIDAY, February 26.

  1. Reading: In Chapter 12, read sections 12.6,12.7,12.8.
  2. Go over all materials on Chapter 12.
  3. Chapter 12 problems: Make sure you can do all of the Core problems for sections 12.6, 12.7 and 12.8. But write up only the following to be handed in ( from the old final exams):

The idea behind Lagrange Multiplier