Math 212
Solutions to Homework #14
Spring 2008
Solutions to Homework #14 from section 3.4. Please email me if you detect any errors in
these solutions.
2. Find extrema of f (x, y ) = x y subject to g (x, y ) = x2 y 2 = 2.
Solution. Checking to see when g
Math 212
Select Solutions to Homework #13 due 4-21-08
Spring 2008
3.3 #10. Solution. f /x = sin y and f /y = 1 + x cos y . The rst is zero when y = k
where k is any integer. Now cos(k ) equals 1 if k is even and 1 if k is odd. So we get an
inninte number
Select Solutions to Homework #12 due 4-11-08
Math 212
Spring 2008
8.2 #7. Let F = r (i + j + k) where r = xi + y j + z k is the position vector. Calculate
F .dS , where S is the portion of the surface of a sphere dened by x2 + y 2 + z 2 = 1
S
and x + y +
Select Solutions to Homework #11 due 3-28-08
Math 212
Spring 2008
7.4 #6. Find the area of the portion of the unit sphere that is cut out by the cone z
x2 + y 2 .
Solution. In problem 1 we are given a parametrization of the unit sphere: : [0, 2 ] [0, ]
Select Solutions to Homework #10 due 3-21-08
Math 212
Spring 2008
7.2 #12. Let F = (z 3 + 2xy )i + x2 j + 3xz 2 k. Show that the integral of F around the
circumference of the unit square with verticies (1, 1) is zero. Let c : [a, b] IR3 be a path
paramete
Math 212
Select Solutions to Homework #9 due 3-14-08
Spring 2008
6.3
#8. Calculate the mass of the object bounded by the cylinder x2 + y 2 = 2x and the cone
x2 + y 2 = z 2 if the density is given by (x, y, z ) = x2 + y 2 .
Call the region W . In x, y, z c
Math 212
Select Solutions to Homework #6 due 2/15/08
Spring 2008
4.4
#6. The ow lines are given by c(t) = (c1 e3t , c2 et ). The vector eld and the ow lines:
Figure 1: Vector eld given by F = 3xi y j
10
5
10
5
5
10
5
10
Figure 2: Flow lines in the vector
Select Solutions to Homework #5 due February 8
Math 212
Spring 2008
2.6
#24(b). f (c(t) = cos2 t + 4 sin2 t = 1 + 3 sin2 t. The max and min will occur at the critical
points. In other words, when f (t) = 0. f (t) = 6 sin t cos. The zeros of this function
Solutions to Homework #4 due Feb. 1
Math 212
Spring 2008
2.5 Properties of the Derivative.
#5Verify the rst special case of the chain rule for the composition f c in each of the cases:
a. f (x, y ) = xy , c(t) = (et , cos t). In order to verify this, we j
Homework #3 due January 25
Math 212
Spring 2008
2.2 Limits and Continuity
Caution: It is NOT enough to prove that a limit is a specic number just by showing that
the limit is that number if you approach it from dierent directions. Many people made this
mi