Final
190s -- 2
180s -- 10
170s -- 11
160s -- 7
150s -- 2
Course Grade
A -- 17
A- -- 3
B+ -- 6
B -- 6
Have a good vacation, get some sleep, and feel free to keep in touch with me for advice on your mathematical questions.
Bruce Reznick
This is the home page for Math 242, Calculus II, Section D1H, which meets MWF 11:00-11:50 in 243 ALTGELD, and its companion hour, MATH 249, Section R1H, which meets T 11:00-11:50 in 241 ALTGELD.
My intention is to provide, at the very least, an archive for all of the TeX-d handouts in the course and a guide to the semester, class-by-class.
There is a free ``Tutoring Room" for all sections of Math 242, which meets MTuWTh 7-9 in 245 Everitt Lab. Don't ask for help on Honors Questions!
W 8/25 -- Distribution of Class Organization, Syllabusand How to solve it guide. We went through Section 11.1. First homework, due on Friday 8/27: 11.1 -- 9,14,17,18,23,24,29, 30, 36, 37 (even ones graded; odd ones ungraded, all due) Plus HQ1 (a) 11.1 -- 54. (b) -- Show that it is impossible for there to be 5 non-zero vectors vj in the plane with the property that for any two different i and j, vi.vj is <= 0. Note: both (a) and (b) can be solved with a well-chosen picture and a few sentences.
F 8/27 -- Section 11.2. Homework due M 8/30: (p.788) 2,3,19,22,26,27,
43,44, HQ2: 69,70. We covered most of the section. Material not in
the book included a discussion of 3 of the 5 Platonic Solids: the
cube, tetrahedron and octahedron, which can both be defined in terms
of the cube. There will be more on Platonic Solids on Tuesday.
M 8/30 -- More on section 11.2, including a handout on the length
of the cross-product and the Cauchy-Schwarz inequality in n variables.
We began a discussion of section 11.3 on the cross-product, which will
continue on Wednesday. Homework due W 9/1: 11.2 (p. 789) -- 36, 67;
11.3 (p.796) -- 1,2,7,8,12; HQ 11.2 -- 64. As always, the even ones
are graded.
T 8/31 -- Platonic solids. Here are a few interesting links:
Platonic
Solids. This site, mathworld.wolfram.com, gives an excellent and
free on-line mathematical encyclopedia.
Geometry in Art and Architecture
W 9/1 -- Finished 11.3, started on 11.4, with a discussion of lines.
Homework due F 9/3: 11.3 -- 3,4,14,15,16, HQ4 (discussed at top).
F 9/3 -- More on lines and planes in R^3. They are groovy and
beautiful. Homework due W 9/8: 11.4 -- 3,4,7,8,11,12,13,14,17,18,
21,22,31,32. No Honors Question, but a Bonus HQ due M 9/13 (extra
credit and no hints or help beyond the statement; do this if you
are interested.): The unit cube contains two tetrahedra, T1, with
vertices (0,0,0), (1,1,0), (1,0,1), (0,1,1) and T2, with vertices
(1,0,0), (0,1,0), (0,0,1) (1,1,1). Determine the intersection
T1 ^ T2. Tell me what it looks like, what its vertices are (hint, it's
a polyhedron of some kind) and its volume. Pictures are nice. You
may want to find the equations of the faces of T1 and T2 first.
M 9/6 -- Labor Day (no class). Last break for 10 weeks! But go to
Lunation
to see an animated version of a month in the life of the moon.
T 9/7 -- First, there was about 10 minutes of review and questions
about lines and planes. After
that, I spoke about solving the equation u x v = w, for v, given
non-zero vectors u, w with u.w = 0. There will be a handout available
Wednesday for those who don't make it.
W 9/8 -- Done with lines and planes, unless you have further
questions. We began 11.5. Homework due F 9/10; 11.4 -- 35, 36, 39, 40,
43, 44, 49, 51 HQ: 11.4 -- 50, 52.
Three questions about this homework: Q1. A student writes: "After you have found the direction vector of the line that is the intersection of two planes, in order to get a point on the actual line can one just set one of the variables, let's say x, to zero and then solve the resulting system of equations? A1. Sure, unless the line has no points with x=0. Any point on the line will do. Q2. A student writes: "I'm having trouble finding the line formed by the intersection of two planes, could you send me some info on how to go about those problems?" A2. Think of it as solving two equations. The equation of one plane gives one variable in terms of the other two. Plug this into the other plane to give a second variable in terms of a third. It's true that the direction vector of the line of intersection will be perpendicular to the normals of the two planes, but you still need to find a point on the line of intersection. (See Q1/A1.) Q3. A student writes: "When I am trying to find out whether a line intersects or is parallel with a plane, do I use the normal vector in some way?" A3. The best way is to think about the plane as an equation. If you have the parametric form of a line, it tells you (x,y,z) as functions of t. You can plug it into the equation of the plane and then you can solve (or not) for t. I notice now that the book didn't give an example of this kind of thing. I'll work out an example in class beyond the ones in the text.
F 9/10 -- Example noted above worked out in class at great length:
x + 3y + z = 4 and 2x - y + z = 6 and their intersection. Then on
to section 11.5, ending up with the general case of projectiles on
Earth. HW 11.4 -- 48, 11.5 -- 7, 8, 13, 14, 18, 19, 27, 28.
M 9/13 -- We finished up on 11.5 and began 11.6. Fun stuff, but not
trivial, and worthy of extra attention. Homework in a handout:
1. (Old exam problem) Show that the lines L1: x = 3 + 2y, y = 4, z = 2
- t and L2: x - 1 = y/2 = z-1 do not intersect and find parallel
planes P1 and P2 so that L1 lies in P1 and L2 lies in P2.; 2. p.847
#11 (not to hand in); 11.5 -- 31, 32, 58, 63, 64, 11.6 -- 1, 4.
As usual, hand in the even ones.
T 9/14 -- Some review of lines and planes with a couple of old
homeworks, and then I couldn't help myself: a preview of some of
the fun stuff in 11.6.
W 9/15 -- HW9, Due Friday 9/17 (or sooner if you prefer!): 11.6 --
7,8,22,23,33,34,42,43, HQ: (This is essentially a 2-dimensional
problem.) Gravity on a planet is switched on at t=0, and gives
an acceleration of a(t) = -6t j. A projectile is launched from
the origin at time t=0 with an initial velocity of v(0) =
v0 * cos(theta) * i + v0 * sin(theta) * j. (a) Determine the
horizontal distance traveled by the projectile before it lands.
(b) Determine the angle theta, 0 < theta < pi/2, which maximizes this
distance. For some reason (instructor error), I forgot to write up
11.6 -- 1,4 on the last solutions. For your information, the answer to
11.6 -- 4 was 3/2 + log 2. As general points: (i) I will expect you
to be able to integrate t + 1/t on an exam, without a calculator and
(ii) The preferred answer is nearly always "analytic", such as 3/2 +
log 2, rather than "numerical" such as 2.193.. HW10, due Monday, was
also distributed in class.
F 9/17 -- 11.6 all day; pretty much done. HW10: 11.6 -- 13, 14, 29,
30, 47, 48, 55, HQ 11.6 -- 54.
M 9/20 -- Mostly discussion; about the Putnam mathematical
competition UIUC math
competitions; about recent homeworks and other things. The first test
will either be F 10/1 or M 10/4, to be decided later. I gave my
patented hand-waving tour through the conic sections.
T 9/21 -- A stab at Kepler's laws, if there's time after your questions.
W 9/22 -- OK, this class was different and I know it freaked some
people out. What I was doing was justifying that the various surfaces
discussed in the section are the only surfaces of interest that arise.
What YOU have to be able to do is to work the homework problems, and
on an exam, match pictures of surfaces to their equations. HW11:
11.7 -- 5,6,19,20,25,26,27,28,35,36. (We also reviewed osculating
circles one last time.)
Questions: A student writes: I had a question about this week's homework. In most of these problems we are supposed to be drawing what these equations look like. I am wondering how detailed you want these graphs to be? My artistic abilities are not the greatest, so I just wanted to know what you were expecting. My reply: I would not expect your graphs to be drawn any better than I could draw on the blackboard. The point is not to be accurate, but to make you think a little bit about what the surfaces should look like. Another student writes: On the problems do you want us just to say the basic shape of the equation and give a drawing of that shape, or do you want us to go into more detail like semi-axis and other more descriptive measures? My reply: Just give a general shape.
F 9/24 -- Planned: cylindrical and spherical coordinates.
Last hw (#12) before Test 1: 11.7 -- 41, 42, 51, 52; 11.8 -- 3, 4, 9,
10, 17, 18 HQ A particle travels on the surface of a sphere using
phi = theta = t, so r(t) = (sin t * cos t, sin^2 t, cos t). Determine
the vectors T and N and the curvature kappa at t = pi/4.
M 9/27 -- Start on Chapter 12, which won't be on the test. No
homework. Review questions will be distributed, for discussion on
W 9/29. You can always ask questions.
T 9/28 -- Review, but NO discussion of the review questions. The main
topics were lines and planes and projections and traces. The first
two will be on the test, the second two won't be, at least on the
first test.
W 9/29 -- Review questions. Any questions about the exam are posted
above.
A student writes: I have difficulty in solving problems that
deal with three planes. Problems number 49 and 51 in section
11.4 deal with three planes and their intersection. I looked
at the solution to these problems but I am still confused as
to how to approach problems like these.
My reply: The way to approach lines and planes is to look at it this way:
What you need to determine a line is either 2 points or 1 point
and a direction for the line to go.
What you need to determine a plane is either 3 points or 1 point
and a normal direction.
In doing these problems, your job is to transform the information
given into information about the line or plane. You often need the
fact that if a vector a is perpendicular to both b and c, and b and
c are not parallel, then a is a multiple of b x c.
In 49, you are given a plane through a point, and you are told
the intersection of the plane with the xy-plane (ie z = 0), so
you get more points in the plane and this tells you what the plane is.
In 51, you are given a plane through 2 points. This gives a direction in
the plane which must be perpendicular to the normal vector n. The plane is
also parallel to a line, which means that the direction of the line is
also perpendicular to n. This is now enough to give you n.
M 10/4 -- Moving on to Ch. 12. Level curves, continuity and limits.
HW 13, due 10/6: 12.2 -- 11, 12, 25, 26, 35, 36, 43, 44; 12.3 -- 5, 6,
17, 18, 37, 38.
T 10/5 -- Any test questions you may have and more funny examples
of limits and by funny I don't mean funny-ha-ha, I mean funny strange.
W 10/6 -- Partial derivatives and tangent planes. Went by quickly,
so be sure to ask questions. HW 14, due 10/8: 12.3 -- 52, 53;
12.4 -- 5, 6, 7, 8, 11, 12, 31, 32, 33, 34.
F 10/8 -- Partial derivatives, an interlude with 12.4 -- 58 and the
Laplace equation, which will be very important in your later
mathematical studies. A beginning of max/min. HW15, due 10/11: 12.4 -- 21, 22,
23, 24, 55, 56; 12.5 -- 3, 4, 7, 8, 13, 14. HQ 12.4 -- 66.(corrected,
sigh.)
M 10/11 -- More on max/min problems and various examples worked out.
HW 16, due 10/13: 12.5 -- 23, 24, 30, 31, 39, 40, 45, 46, 52.
T 10/12 -- A couple of questions on limits, how to describe
points on a plane and 12.4 -- #74. There are notes, which I'll bring
on W 10/13.
A student writes: how do you minimize the distance between a point and
a plane equation? i
understand how we did the two line equations in class but im not sure how to
apply it to the plane.
My reply: This came up during the 11:00 class. The basic idea is that
if (x,y,z) is
a point on the plane ax + by + cz = d and c is not 0, then z =
(d-ax-by)/c. There are also geometric ways of doing problems like this,
and we'll talk about them tomorrow.
W 10/13 -- Some muddled remarks on differentials, plus cogent comments on max/min and the arithmetic-geometric inequality and an introduction to the star of the rest of the semester: the gradient HW 17, due 10/15: 12.5 -- 43, 44, 53, 54, 62; 12.6 -- 3, 4, 31, 32; HQ 12.5-- 70. (Hint: be careful!)
F 10/15 -- Mostly went over recent homework. The application of
these sections to line-fitting (a future handout.) No specific HW
due on Monday, but you are welcome to turn in problems for me to
look at and correct.
M 10/18 -- The chain rule and its various applications, including
to Laplace's equation. Test 2 is tentatively W 11/3, but we'll talk
about it in class on W. A handout on curve-fitting. HW 18, due 10/20:
12.5 -- 26; 12.6 -- 11, 12, 20; 12.7 -- 4, 5, 10, 19, 20, 30, 31. No
HQ, but hand in the even ones.
T 10/19 -- The arithmetic-geoemtric inequality and some
examples of finding extreme values. There will be a handout eventually.
W 10/20 -- Test 2 is now set for F 11/5.
The chain rule in matrix form, the gradient in several
of its manifestations. HW19:12.7 -- 37, 38, 49, 50;
12.8 -- 5, 6, 13, 14, 21, 22, 31, 32, HQ 12.8 -- 53 (yes, it's odd.)
F 10/22 -- More on the gradient, and an introduction to our friend,
Lagrange Multipliers. More on Monday. HW20: 12.8 -- 47, 48, 50, 55;
12.9 -- 1, 2, 7, 8, 9, 10.
M 10/25 -- More Lagrange multipliers, and now with more than 1
constraint. HW21: 12.9 -- 13, 14, 15, 16, 40, 41, 44, 49, 50.
UGLY ANSWER ALERT: Problem #40 is nasty, and you'll get full credit if
you set it up right. (There is a 4th degree polynomial with no "nice"
solutions.) Problem #50 also has some square roots involved, but they
are not quite as bad.
T 10/26 -- I'll do the surmounted window via Lagrange
multipliers and answer any other questions you might have.
W 10/27 -- Catching up: a handout on the surmounted window,
more Lagrange multipliers and an
unsuccessful attempt to explain the second derivative tests of various
kinds
F 10/29 -- A handout on second derivative tests, and a few examples
of it in action. HW22: 12.10 -- 12.10 --3, 4, 9, 10, 17, 18, 19, 20, 26.
Review questions for the second exam distributed. I'll leave a few
copies of the handouts in an envelope on my door (327 Altgeld). the
building is open at times during the weekend.
M 11/1 -- A few questions answered about Test 2, then introduction
to the final topic of the semester: multiple integrals.
HW 23: 13.1 -- 3, 4, 11, 12, 16, 17, 21, 22, 31, 34. This is due
on M 11/8.
T 11/2 -- Open review of Ch. 12. I'll bring all old handouts.
W 11/3 -- Discussion of the review questions.
F 11/5 -- Test Two
M 11/8 -- Back in the saddle again. Through 13.2 -- HW 24: 13.2 --
5, 6, 11, 12, 15, 16, 21, 22, 25, 26, 31, 32. Tuesday's meeting will
be devoted to a review of the exam, and any questions that may have
already arisen about multiple integrals.
T 11/9 -- Some exam 2 questions gone over in some detail. A little bit
about how to reverse the order of integration.
W 11/10 -- Area and volume as iterated double integrals and a first
brush with integration in polar coordinates, and the mysterious fudge
factor. HW 25: 13.2 -- 29, 34; 13.3 -- 3, 4, 7, 8, 15, 16, {31, 32
-- just set up the integrals, don't evaluate!}, HQ 13.3 -- 37.
F 11/12 -- Polar coordinates in all their glory, plus the definite
integral of e^(-x^2) from 0 to infinity, done more carefully than
in the book. (There will eventually be a handout.)
HW 26: 13.4 -- 9, 12, 13, 14, 15, 16, 27, 28, 29, 32.
M 11/15 -- Applications of double integrals; mainly centroids.
HW 27: 13.4 -- 38, 39; 13.5 -- 7, 8, 13, 17, 18, 24, 48, 51. (Sorry
this wasn't up earlier.)
There's a typo in 13.5 - #18. The region *should be* bounded by x =
1 and x = e (not x = 0 and x = e), as well as y =0 and y = ln
x. You'll probably want to integrate by parts a few times, or look up
an integral. (I won't throw something like this on the test.)
T 11/16 -- An introduction to the gamma function; handout available after break.
W 11/17 -- Solution to 13.5 -- 48, which didn't show up on the
solution sheet for some reason (prof needs vacation too.)
Surface area for z = f(x,y) and introduction to triple integrals (just
like double integrals only in 3D). HW 28:
13.6 -- 1, 4, 11, 12, 17, 18 (You only have to do these with one order
of the variables, and when a sketch is called for, I've often picked
a problem where there's already one in the book. You should still
sketch for practice.); 13.8 -- 3, 4, 7, 8, 10, 11.
F 11/19 -- For the few people who showed up, I did a few old homeworks
and gave a brief, fast sketch of the rest of the semester.
Thanksgiving
M 11/29 -- General discussion of triple integrals and an introduction
to cylindrical coordinates, with spherical coordinates to follow on
Wednesday. HW 29: 13.6 -- 17, 18, 36; 13.7 -- 5, 6, 7, 8, 11, 12.
T 11/30 -- Review, plus an amusing proof using double integrals of
a combinatorial fact about decomposing rectangles into smaller
rectangles.
From the email sent to everybody: Several people asked me about 13.6 -- #17,18, which are assigned on homework 29, even though I also assigned them on homework 28. Well, if you did it, don't cross it out; if you didn't do it, you don't have to do it again. I will check your answer and correct the problem, but it won't count in the grade, and it's too late for me to assign a replacement.
W 12/1 -- Spherical coordinates and surface area. HW 30 was on
a separate sheet. It's 13.7 -- 19, 20, 21, 22, 23, 24 & HQ. Note
that 19=23 and 20=24, but with cylindrical and spherical coordinates
separately. For 20=24, a particular cone is not specified: choose
the one in 19, with the vertex at the origin, the sides determined
by z^2 = r^2 = x^2 + y^2 (you'll have to figure out phi for yourself.)
and with base lying on the plane z = 1. The HQ is, in its corrected
version, as follows; using the spherical coordinate parameterizaiton
of the sphere of radius 2: ie (x = 2 sin u cos v, y = 2 sin u sin v, z
= 2 cos u), determine the area of that portion of the sphere x^2 + y^2
+ z^2 = 4 which lies above the plane z = 1.
F 12/3 -- Introduction to changes of two and three variables. Review
questions passed out. No more graded homework this semester.
M 12/6 -- Review questions, distributed on Friday, will be discussed on
Monday. There will be solutions this time. The stest will cover
chapter 13.1 through 13.8, with the only surface area questions
involving z = f(x,y).
T 12/7 -- More review.
W 12/8 -- Third Hour Exam.
F 12/10 -- Last day. Test 3 will be returned, course review given,
and ICES form distributed.
S 12/11 -- Reading Day (Extended Office Hours): 1:30 -- 4:00 pm.
S 12/18 -- Final Exam, 8:00-11:00 AM. This is a terrible time, but
there is nothing that we can do about it.