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
   • f(x,y)=yx^1/2-y²-x+6y
   • f(x,y)=2x³+xy²+5x²+y²
6. Find the absolute maximum and minimum values of f(x,y)=yx^1/2-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):
• Fall 94: #11,12,13;
• Spring 95: #16,17;
• Fall 95: #7,11,17;
• Spring 96: #10,11,12;
• Fall 96:#14,15,16;
• Spring 97; #14;
• Fall 97: #12,13,17,18.