[Ask students to remind me to return HW at the end of
class. Sound them out about moving the midterm exam to
Nov. 2. Hand out candy.]
Section 11.6: Directional derivatives and the gradient
vector (concluded)
If f 0, then f points in the direction of most
[Collect section summaries.]
Section 11.8: Lagrange multipliers
Main ideas?
.?.
.?.
Constrained optimization problems
The geometric justification for the method of
Lagrange multipliers
Extension of the method for two-constraint problems
Suppose (x0,y0
Section 12.5: Triple integrals
Main ideas?
.?.
.?.
Triple Riemann sums.
The basic definition of a triple integral.
The triple Fubini theorem.
The triple integral on a general domain.
The various types of volume domain, and how to set
up the volume in
Math 241, Problem Set #6: Solutions to additional problems
A. Suppose f (x, y ) and g (x, y ) are dierentiable at (a, b). Must the function h(x, y ) = f (x, y ) + g (x, y ) be dierentiable at (a, b)? Your solution
should make explicit use of the denition
Math 241, Problem Set #5: Solutions to additional problems
A. Recall that in the metric system of measurement, position has units of
meters and time has units of seconds. Determine the metric unit unit
of curvature, using the formula (9) on page 572. Do t
Math 241, Problem Set #4: Solutions to additional problems
A. Show that for the helix h(t) = a cos t, a sin t, bt , the vector derivative
makes a constant angle with the z -axis. What is this angle if a = 1 and
b = 3?
The vector derivative is h (t) = a si
Math 241, Problem Set #3: Solutions to additional problems
A. Given points P = Q in R3 , describe geometrically the set of all points
R such that P Q P R = 0.
Certainly R = P is such a point. If R = P , then the vector P R is
non-zero, and we know that fo
Math 241, Problem Set #2: Solutions to additional problems
A. (a) Show algebraically that if a, b, c are non-zero vectors in the x, y plane such that a b = b c = 0, then a and c are parallel. (For
simplicity, you may assume that all components of the vect
Math 241, Problem Set #1: Solutions to additional problems
A. Two ies y past one another between time 0 and time 1. At time
t (with 0 t 1), one y has position (0, 0, t) and the other has
position (1, 1 t, 0). At what instant are they closest, and how far