Hwk No. 
Date Assigned 
Assignment 
Date Due 
1 
8/25 

8/29 
2 
8/27 
 11.2 # 7, 13, 70
 11.3 # 1, 12, 14, 16, 25

9/3 
3 
9/3 

9/8 
4 
9/8 
 Let u(t)=p+at and v(t)=q+bt, where p, q, a and b are vectors in three dimensional
space. Let theta(u,v) be the angle between u and v. Show that in the limit
as t > (+ infinity) or ( infinity), cos(theta(u,v)) > cos(theta(a,b)).
 11.5 # 14 (Explain how you match the equations with the figures.)
 11.5 # 3 (show the curve lies on the given cone)
 11.5 # 11, 31, and 42

9/12 
5 
9/12 
 11.6 # 6 (arc length)
 11.6 # 27 (curvature, 2D)
 11.6 # 34 (curvature, 3D)
 11.6 # 64 One step in deriving the relations of Newton's and Kepler's Laws.

9/15 
6 
9/15 
 11.7 # 9 (3D graph sketching)
 11.7 # 42, 43, 48 (characterizing surfaces via planar intersections)
 11.7 # 51 (Intersection of two surfaces)

9/19 
7 
9/17 
 11.8 # 19 (Convert Cartesian coordinates to cylindrical and polar)
 11.8 # 35 (Graph of an equation in cylindrical coords)
 11.8 # 55 (Describing a surface in cylindrical & spherical coords)
 11.8 # 59 (Flying from Fairbanks to St. Petersburg)

9/22 
8 
9/25 
Having recovered from the exam on 9/24 ... Please note the change in due date!
 12.2 #5, 12.2 #15 (Finding a function's domain.)
 12.2 #30 (Drawing a graph.)
 12.2 #32 (Drawing level curves.)
 12.2 #44 (Drawing a level surface.)
 12.2 #5358 (Matching. Be sure to explain how you arrive at each match.)
 12.3 # 29 (finding limits)
 12.3 # 31 (finding a region of continuity)
 12.3 # 42 (finding a limit  similar to example from class)
 12.3 # 52 (a more challenging limit)

WED. 10/1 
9 
9/29 
 12.4 # 4550 (matching practice)
 12.4 # 56 (heat equation in 2D)
 12.4 # 58 (Laplace's equation in 2D)
 12.4 # 65 (finding the point at which a surface is horizontal)

10/3 
10 
10/1 
 12.5 # 21 (finding the highest or lowest point on a surface  justify your answer carefully)
 12.5 # 31 (minimizing the distance to a plane, within a region)
 12.5 # 46 (constrained maximum volume problem  continued from handout)
 12.5 # 59 (another volume maximization problem, slightly more involved)
 * 12.5 # 47, 48, 49, 54

10/6 
11 
10/6 
 12.6 # 28 (no calculator allowed for problems from section 12.6)
 12.6 # 31 (estimating the location of a point on a curve)
 12.6 # 34 (estimating the maximum error of a compound measurement)
 * 12.6 # 29, 35, 40, 42

10/10 
12 
10/8 
 12.7 # 31 (Use the chain rule to find the plane tangent to a surface)
 12.7 # 38 (Dependence of resistance on three parallel resistances)
 12.7 # 53 (Hint: use equation 10; see the implicit function theorem)
 12.7 # 54 (Apply the implicit function theorem!)

10/13 
13 
10/13 
 12.8 # 21 (directional derivatives)
 12.8 # 30 (using the gradient to find the line tangent to a curve)
 12.8 # 27 (another directional derivative problem, for practice!)

10/17 
14 
10/15 
 12.9 # 17 (optimizing a function of 3 variables, given 2 constraints)
 12.9 # 37 (inscribed circle of maximal area, using Lagrange multipliers)

10/22 


    F A L L    B R E A K    10/2021
 
15 
10/22 
 12.10 # 1 (classifying critical points of f(x,y))
 12.10 # 4 (classifying critical points of f(x,y))
 12.10 # 23 (a case where the discriminant is zero)

10/24 
16 
10/27 
 13.1 # 12 (practice w/ double integrals)
 13.1 # 19 (practice w/ double integrals)
 13.1 # 32 (check that the order of integration doesn't matter)
 Show that if z(x,y)=f(x)g(y), then the double integral of z(x,y)dxdy
over the region a<=x<=b and c<=y<=d is equal to the product of the integral of f(x)
over [a,b], times the integral of g(y) over [c,d].

10/29 
17 
10/29 
 13.2 # 3 (double integral where boundary involves one of the variables)
 13.2 # 25
 13.3 # 13
 13.3 # 27
 13.4 # 11
 13.4 # 15
 13.4 # 35 (try using Pappus' Theorem instead of the hint!)
 * (from class): Find the area under the surface z=1(x^2+y^2) bounded by the region z >= 0.

11/3 
18 
11/3 
 13.5 # 15
 13.5 # 43
 Suppose z=f(x,y) is a function of (x^2+y^2) only. Assume that f(x,y)>=0
as long as (x^2+y^2) >= a^2. Calculate the volume of the solid formed by the
surface with height z=f(x,y) over the region (x^2+y^2) <= a^2, in two different
ways:
 Integrate a system of cylindrical shells of thickness "dr", height
z and circumference 2*pi*r.
 Use Pappus' theorem.
Then show that these two approaches give the same answer.
 13.6 #1 (triple integral over a rectangular region)
 13.6 #7 (triple integral over a more complicated region)
 13.6 #38 (moment of inertia of a sphere)
 13.6 #40 (another moment of inertia)

11/7 
19 
11/5 
 13.7 # 1 (centroid of a shape with cylindrical symmetry)
 13.7 # 2 (moment of inertia of same)
 13.7 # 30 (volume of a solid defined in spherical coordinates)
 13.7 # 31 (moment of inertia of a solid with some rotational symmetry)

11/10 
20 
11/10 
Surface area integrals:
 13.8 #1
 13.8 #3
 13.8 #9
 13.8 #13

11/14 
21 
11/12 
 13.9 # 1 Practice finding the Jacobian
 13.9 # 3 More practice finding the Jacobian
 13.9 # 7 Jacobian & area
 13.9 # 9 Finding an area in the plane
 13.9 # 15 Finding a volume in 3 space
 13.9 # 19 Using ChangeofVariables with a nonconstant function
 13.9 # 20 Another look at the ellipsoid problem
 13.9 # 21 Another look at the ellipsoid, continued.

11/17 
22 
11/23 
 14.1 # 7 (sketching a vector field)
 14.1 # 1114 (matching gradients with vector fields)
 14.1 # 15 (finding curl and divergence)
 14.1 # 32 (prove that div(curl(F)) = 0)
 14.2 # 3 (2 kinds of line integrals: ds, and dx or dy)
 14.2 # 11 (line integral of T dot F ds)
 14.2 # 17 line integral of f(x,y,z) ds along curve C
 14.2 # 1920 (centroid & moments of inertia of a piece of wire)
 14.2 # 37 work moving on a sphere in a central force field

11/26 
optional 
 
 14.3 # 1,3,5,17,25,30,31,36
 14.4 # 1,17,21,34

 