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)=yx1/2-y2-x+6y
• f(x,y)=2x3+xy2+5x2+y2
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.