Below is a tentative list of homework problems for Math 105. The first set
are the assigned problems and are to be handed in for a grade; the second set
are optional. In general, HW is due each class, and we will typically cover on
the order of one section a day. All problems are worth 10 points; I will drop
whatever assignment helps your HW average the most. For the suggested problems:
these are not to be handed in, though of course I and the
TAs are happy to chat with you about them. The purpose of these problems is to
provide both guided, additional practice for those who want it, as well as to
post a few more challenging problems. I will add due dates as the semester
progresses.
HOMEWORK: Homework problems listed below;
suggested problems collected together at the end.
- Due Wednesday, Feb 9:
- Read: Section 11.1, 11.2. There is no class on Monday. Use that time to
read the material and make sure your Calc I/II is fresh. Feel free to check
out the review videos: part 1
part 2.
-
Video of the week:
Fibonacci numbers (click
here for more on the Fibonacci numbers).
-
Slides on the course mechanics.
- HW problems:
(1) What is wrong with the following argument
(from Mathematical
Fallacies, Flaws, and Flimflam - by
Edward Barbeau): There is no point
on the parabola 16y = x2 closest
to (0,5). This is because the distance-squared from (0,5) to a point (x,y) on
the parabola is x2 +
(y-5)2. As 16y = x2 the distance-squared is f(y) = 16y +
(y-5)2. As df / dy = 2y+6, there is only one critical point, at y =
-3; however, there is no x such that (x,-3) is on the parabola. Thus there is
no shortest distance! (2) Compute the derivative of cos(sin(3x2 +
2x ln x)). Note that if you can do this derivative correctly,
your knowledge of derivatives should
be
fine for the course. (3) Let f(x)
= x2 + 8x + 16 and g(x)
= x2+2x-8. Compute the limits as x goes to 0, 3 and ∞
of f(x)+g(x), f(x)g(x) and f(x)/g(x).
-
Due Friday, Feb 11:
-
Read: 11.2,
11.3.
-
HW
problems: Section 11.1: Page 823: #9, #18, #38, #42. Section
11.2: Page 833: #1, #39, and also find the cosine of the angle between a
= <2, 5, -4> and b = <1, -2, -3>.
-
Due Monday, Feb 14:
-
Read: 11.3,
11.4
-
Video of
the week:
light
cycle scene from Tron.
-
HW problems: Section 11.2: Question 1: The corollary on page 830 states two
vectors are perpendicular if and only if their dot product is zero. Find a
non-zero vector, say u, that is perpendicular to <1,1,1>. (Extra
credit: find another vector perpendicular to <1,1,1> and the vector u
that you just found.) Question 2: Consider a triangle with sides of length 4,
5 and 6. Which two sides surround the largest angle, and what is the cosine of
that angle? Section 11.3: Question 3: Find the determinant of the 2x2 matrix
; in other
words, we filled in the entries with the numbers 1, 2, 3 and 4 in that order,
row by row. Similarly, find the determinant of the 3x3 matrix
; in other words, we
fill in the numbers by 1, 2, 3, 4, 5, 6, 7, 8, 9.(Extra credit: find a nice
formula for the determinant of the n x n matrix where the entries are 1, 2,
..., n2 filled as above, and prove your claim.) Question 4: Find
the area of the parallelogram with vertices (0,0), (2,4), (1,6), (3,10).
-
Due Wednesday, Feb 16:
- Read 11.4, 11.8 (skip sections 11.5, 11.6, 11.7)
- HW problems:
Section 11.3: Page 842: #1, #5, #11, #12.
-
Due Monday, Feb 21 (NO CLASS FRIDAY
-- Mathematica/TeX workshop):
- Read 12.1, 12.2, 12.3
- Section 11.4: Page 849: #1, #2, #3, #22. Section 11.8: Page 893: #1, #26.
Extra Credit: Section 11.8: #55.
-
Due Wednesday, Feb 23:
-
Due Friday, Feb 25:
- Read 12.4.
- HW: Section 12.3: Page 917: #24, #38, #54 (hint: limit of sin(t)/t
is 1 as t tends to 0). Section 12.4: Page 928: #1, #4, #5.
-
Due Monday, Feb 28:
- Read 12.5, 12.6.
- HW: Section 12.4: Page 928: #22, #25, #33, #36, #63 (is this surprising?).
-
Due Wednesday, Mar 2:
-
Due Friday, Mar 4:
- Read 12.6 and 12.7, and if you wish my notes on the
multivariate chain rule.
- HW:
From my
notes on the Method of Least Squares: Exercise 3.3, Exercise 3.9.
Section 12.6: Page 949: #18, #23. Note for the problems from Section 12.6, it
is equivalent to evaluating the tangent plane at Q, and thus this may be done
without reading Section 12.6!
-
Due Monday, Mar 7:
- Read 12.7, 12.8.
- HW: Section
12.7: Page 960: #2, #5, #8, #34, #41.
- Section 12.6:
Extra Credit:
Let f(x) = exp(-1/x2) if |x| > 0 and 0 if x = 0.
Prove that f(n)(0) = 0 (i.e., that all the derivatives at the
origin are zero). This implies the Taylor series approximation to f(x) is
the function which is identically zero. As f(x) = 0 only for x=0, this means
the Taylor series (which converges for all x) only agrees with the function at
x=0, a very unimpressive feat (as it is forced to agree there).
-
Due Wednesday, Mar 9:
- Read notes and book to prepare for midterm on Friday.
- Video of the week: Mandelbrot zoom:
video 1,
video 2.
Here's a
cubic fractal zoom.
- HW: (1) Use Newton's Method to find a rational number that estimates the
square-root of 5 correctly to at least 4 decimal places;
click here for a Mathematica
program to do Newton's method for sqrt(3). (2) Let w(r,s,t) = f(u(r,s,t),
v(r,s,t)) with f(u,v) = u^2 + v^2, u(r,s,t) = t cos(rs) and v(r,s,t) = t
sin(rs). Find the partial derivatives of w with respect to r, s and t both by
direct substitution (which is very nice here!) and by the chain rule. (3)
Write (1/2, sqrt(3)/2) in polar coordinates. (4) Find the tangent plane to z =
f(x,y) with f(x,y) = x^2 y + sqrt(x+y) at (1,3), and approximate the function
at (.9,1.2).
-
Due Friday, Mar 11:
-
Due Wednesday, March 16, 2011:
- Read Section 12.9.
- Video of the week:
Trick
shots in pool (directions and constraints!).
- HW: Section
12.8: Page 971: #3, #10, #11, #19, #21.
- Due Friday, March 18, 2011:
- Due Monday, April 4, 2011:
- Read: 13.1, 13.2.
- No written homework. Use the time to catch up on the course. You may do
any late HW for 90% credit. Read the next few sections of Chapter 13 so that,
when we return, you'll be ahead of the game.
- Due Wednesday, April 6, 2011:
- Read: 13.1, 13.2.
- HW: Section 13.1: Page 1004: #15, #24, #25, #37.
- Extra Credit: Let G(x) = Integral_{t = 0 to x^3} g(t) dt. Find a nice
formula for G'(x) in terms of the functions in this problem.
- Due Friday, April 8, 2011:
- Read: 13.2, 13.3.
- HW:
Section 13.2: Page 1011: #4, #11. Also let f(x) = x^3 - 4 x^2 + cos(2x^3) +
sin(x+1701). Find a finite B such that |f'(x)| <= B for all x in [2,3]. Do not
try to find the smallest B that works -- just find a B that works.
- Due Monday, April 11, 2011: Next Midterm will be Wednesday, April 20th
- Due Wednesday, April 13, 2011: Next
Midterm will be Wednesday, April 20th
- Read 13.4, 13.7.
- Video of the week:
- HW: Section 13.3: Page 1018: #13, #42 (hint: notice that the region that
matters lies above a circle). Section 13.4: Page 1026: #13. Also do:
Integral_{y = 0 to 1} Integral_{x = -y to y} x^9 y^8 dx dy.
- The following problem is due on Friday, April 15th: Compute Integral_{y =
0 to 1} Integral_{x = -y to y} sin(x y) exp(x^2 y^2) dx dy.
- Due Friday, April 15, 2011: Next
Midterm will be Wednesday, April 20th:
practice exam
soln to
practice exam
- Read 13.7, 13.9. Click here for my handout on the
Change of Variable formula.
- Video of the week:
Pacman
(actually Ms. Pacman).
- HW: Page 1026: #4 (note the region of integration is theta goes from -pi/4
to pi/4, and r goes from 0 to 2 Cos[2 theta]). Do the following problems
as well: (1) Compute Integral_{y = 0 to 1} Integral_{x = -y to y} sin(x y) exp(x^2 y^2) dx dy.
(2) Let f(x,y,z) = cos(xy + z^2). Find (Df)(x,y,z). (3) Find the maximum value
of f(x,y) = xy given that g(x,y) = x^2 + 4 y^2 = 1.
- Due Monday, April 18, 2011: Next
Midterm will be Wednesday, April 20th:
practice exam
soln to
practice exam
- Due Wednesday, April 20, 2011
- Due Monday, April 25, 2011
- Slides for Birthday Problem
- Read
multivariable calculus (Cain and Herod)
and my lecture notes.
- Homework: Due Monday, April 25: Do not use the same sequence for two
different problems. (1) Give an example of a
sequence {a_n}_{n=1 to oo} that diverges. (2) Give an example of a sequence of
distinct terms a_n such that the sequence {a_n}_{n=1 to oo} converges. (3)
Give an example of a sequence of distinct terms a_n such that |a_n| < 2011 and
the sequence {a_n}_{n=1 to oo} does not converge. Extra Credit 1: give
a product of infinitely many distinct, positive terms such that the product
converges to a number c with 0 < c < oo. Extra Credit 2 (hard): Let {a_n}_{n
= 1 to oo} be a sequence of positive numbers such that Sum_{n=1 to oo} 1/a_n
converges. Let B_n = (1/n) Sum_{k= 1 to n} a_k. Prove Sum_{n=1 to oo} 1/B_n
converges. Hint: how the sum of the B_n’s is largest if the a_n’s are an
increasing sequence, and then deal with that case.
- Due Wednesday, April 27, 2011 (Remember:
Math/Stats Ice Cream Registration Info Session: Tuesday in Paresky Basement)
- Due Friday, April 29, 2011
- Due Monday, May 2, 2011
- Due Wednesday, May 4, 2011
- Due Friday, May 6, 2011
-
For
Taylor series, see my handout here (essentially just pages 2 and 3).
- Homework: Cain and Herod: Page 10.10, #20. Also do: (1) Compute the first
five terms of the Taylor series expansion of ln(1-x) (the natural logarithm of
x) about x = 0, and conjecture the answer for the full Taylor series. (2)
Compute the first five terms of the Taylor series expansion of ln(1+x) (the
natural logarithm of x) about x = 0, and conjecture the answer for the full
Taylor series. Hint: you can get (2) immediately from (1).
- Due Monday, May 9, 2011
- If you want, take the last quiz,
quiz 5. Please do the MSRC evaluation if you have gone there.
- I will discuss multivariable Taylor series in class on Monday. I do
not want to go into the detail and depth that the book does; instead I
will show you a very elementary way that handles almost all multivariate
Taylor series you'll find.
- Due Wednesday, May 11, 2011
- Last HW Assignment: (1) Find the second order Taylor series expansion of
cos(xy) about (0,0). (2) Find the second order Taylor series expansion of
cos(sqrt(x+y)) about (0,0). Find the second order Taylor series expansion of
cos(x^3 y^4) about (0,0).
- Start reviewing the material from the semester and bring questions to
class on Wednesday. You are strongly encouraged to look at the practice exams,
the solutions to HW problems (this year and last year) and quizzes, and the
review sheets.
- Due Friday, May 13, 2011
- The last class is optional. It will be on material that will not be on the
final exam, namely the big theorems from Multivariable Calculus: Green's
Theorem, Gauss' Theorem and Stokes' Theorem. These are massive generalizations
of the Fundamental Theorem of Calculus. They're worth knowing, especially if
you are considering a math or physics major. To do the subjects justice
requires two weeks; we'll have 50 minutes and thus be moving rapidly. I will
not be insulted if anyone is not here, but if you're interested on where
multivariable goes, please come along for the ride. There are two reasons I
want to do this. One, obviously, is that these are important theorems. The
other is that this is a great way to review many of the concepts of the
course.
- If you plan on coming to the lecture, a good summary is my
lecture notes on Green's
theorem.
- The video of the week is a TED lecture
(these are great talks, and worth hearing). We'll do
Malcolm Gladwell on Spaghetti Sauce (but only a snippet starting around 5
minutes).
Suggested
Problems and Extra Credit Problems for Math 105: The suggested problems are not
to be turned in, but are for your own personal edification or for additional
practice, though of course I and the TAs are happy to chat about these (or any)
problems. If you submit an extra credit problem, please clearly mark that it is
an extra credit problem.
- Introduction: THREE Extra Credit Problems:
(1) Let N be a
large integer. How should we divide N into positive integers ai such
that the product of the ai is
as large as possible. Redo the problem when N and the ai need
not be integers. (2) Without using any computer,
calculator or computing by brute force, determine which is larger: eπ or
πe. (In other words, find out which is larger without actually
determining the values of eπ or
πe). If you're
interested in formulas for π,
see also my paper A
probabilistic proof of Wallis' formula for π, which
appeared in the American
Mathematical Monthly (there are
a lot of good articles in this magazine, many of which are accessible to
freshmen). (3) Prove that the product of the slopes of
two perpendicular lines in the plane that are not parallel to the coordinate
axes is -1. What is the generalization of this to lines in three-dimensional
space? What is the analogue of the product of the slopes of the line equaling
-1?
- Section 11.1: Page 823: Is #38 true for all points (i.e., if you take any
three points in the plane)?
- Section 11.2: Page 833: #59, #61.
- Section 11.3: Page 842: #7, #17a.
- Section 11.4: Page 849: #25, #54, #58, #60.
- Section 11.8: Page 893: #33, #53. Extra Credit: #55.
- Section 12.2: Page 908: #41, #43, #45.
- Section 12.3: Page 917: #41, #51, #55.
- Section 12.4: Page 981: #55, #57, #58, #68.
- Section 12.5: Page 940: #10, #17, #46.
- Section 12.6:
Extra Credit:
Let f(x) = exp(-1/x2) if |x| > 0 and 0 if x = 0.
Prove that f(n)(0) = 0 (i.e., that all the derivatives at the
origin are zero). Show this implies the Taylor series approximation to f(x) is
the function which is identically zero. As f(x) = 0 only for x=0, this means
the Taylor series (which converges for all x) only agrees with the function at
x=0, a very unimpressive feat (as it is forced to agree there).
- Section 12.7: Page 960: #38, #53.
- Section 12.8: Page 971: #29, #40, #41, #60.
- Section 12.9: Page 981: #36, #37, #47, #49, #62 (important).
- Section 13.1: Page 1004: #33.
- Section 13.2: Page 1011: #41, #44, #49.
- Section 13.3: Page 1018: #29.
- Section 13.4: Page 1026: #7, #34.
- Section 13.7: Page 1056: #47, #48. (Extra credit for solving both of
these.)
- Section 13.9: Page 1070: #10, #28, #29.
-
From
multivariable calculus (Cain and Herod): Exercise 1 (page 10.3).
Extra
credit:
Find a series where the ratio test provides no information on whether or not
it converges but the root test says whether or not it converges or diverges.
-
Problems
leading up to Green's Theorem TBD.
