Math 141 - Week 6 Notes Monday, February 15, 1999
Topics for this week -
- Directional derivatives, the gradient of a function, normal
lines
- Maximum and minimum values
- Lagrange Multiplier
Topics for next week -
- Double integrals over rectangles
- Iterated Integrals
- Double integrals over general regions
- Double integrals in polar coordinates
Examples -
- Find the gradient of the function z=3x^{2}y - y^{3}. 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?
- Find the maximum rate of change of
f(x,y)=ln(x^{2}+y^{2}) at (1,2) and the direction in
which it occurs.
- Find the tangent line and the normal line to the level curve
x^{2}+4y^{2}=8 at the point (2,1).
- Find the tangent plane and the normal line to the level surface
x^{2}-2y^{2}-3z^{2}+xyz=4 at the point
(3,-2,-1).
- Find the local maximum and minimum values and saddle points of the
function
- f(x,y)=yx^{1/2}-y^{2}-x+6y
- f(x,y)=2x^{3}+xy^{2}+5x^{2}+y^{2}
- Find the absolute maximum and minimum values of
f(x,y)=yx^{1/2}-y^{2}-x+6y on the set
D={(x,y)| x=0..9,y=0..5}.
- (Fall 97 #17) Find the maximum of
f(x,y)=2x^{2}-y^{2} on the set
D={(x,y) | 2x^{2}+y^{2}=0..1}
- Find the shortest distance from (0,0) to the curve xy=1
Sixth Homework Assignment - due on FRIDAY, February
26.
- Reading: In Chapter 12, read sections
12.6,12.7,12.8.
- Go over all materials on Chapter 12.
- 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.
The idea behind Lagrange Multiplier