Monday Jan. 5
Solve, for practice, by tomorrow's lecture: Section 12.2: all problems 1 through 24; among these the following need to be turned in for grading:
Solve, write up the solutions, and turn in for grading on Monday, Jan. 12: Section 12.2: 2, 4, 8, 14, 20, 24
Tuesday Jan. 6
Evaluate the limits, and, for the problems maked with *, explain your result using the order of growth at infinity:
for x->infinity we have ln x << x^n << x^(n+p) << e^x
Solve, for practice, by tomorrow's lecture: Section 12.3: 1*, 3, 5, 11, 13*, 15*, 19*, 27,33,35, 39
Solve, write up the solutions, and turn in for grading on Monday, Jan. 12: Section 12.3: 2, 4*,6*,8*,14*,34,36, 44
W Jan. 7
Solve, for practice, by tomorrow's lecture: Section 12.4: 1,3,5,7,9,10,11
Solve, write up the solutions, and turn in for grading on Monday, Jan. 12: Section 12.4: 2,4,6,8,12
R Jan. 8
Solve, for practice: Section 12.4: 13,15
Solve, write up the solutions, and turn in on Monday, Jan. 12: Section 12.4: 14,16, 24a,c. Use 23 to decide if 12 is convergent.
Bonus problem (extra 3 pts) 24d
F Jan. 9
Solve for practice: Section 13.2: 1,3,5. Read: What are the real numbers? pages 3-11. (You do not need to solve the problems there.)
Solve, write up the solutions, and turn in on Monday, Jan. 12: Sec. 13.2: 4, 6, 10
Problems due Tuesday Jan 20:
M Jan. 12
See how the sequence (1+1/n)^n converges to e. (Please scroll down paciently, there are a few empty pages for an unclear reason...) The program uses MAPLE.
Solve, for practice: Sec. 13.3: 2h,j
Solve, write up the solutions, and turn in on Tuesday, Jan. 20: 13.2: 8, 12. 13.3: 2f,g,i,k,l. Solve also the following: use the limit of the sum of a geometric progression to write as a fraction the numbers whose decimal representations are: A=0.3333333... ; B=0.666666... ; C=3.1212121212... ; D=1.7222222.... . Show that 0.9999999.... is in fact 1.
T Jan. 13
Solve, for practice: Sec. 13.2: 9bc, 13.3: 5a-g, 10ac
Solve, write up the solutions, and turn in on Tuesday, Jan. 20: 13.2: 8, 9a; 13.3: 4; 10d
W Jan. 14 Topics discussed today: Telescopic series (example 4 page 441). We discussed that the harmonic series with exponent p: sum of (1/n^p) converges for p>1, diverges for p<=1 (just like the improper integral). Sec. 13.5: the comparison test (with proof) and The limit comparison test (without proof). The latter is very useful. But: how does one find a nice sequence to compare to? Use the order of growth! Example: does the series sum of (n+1)^2/(n^3+5)^2 converge or diverge? We have a_n=(n+1)^2/(n^3+5)^2 behaves like n^2/n^6=1/n^4 so I use limit comparison test with b_n=1/n^4. We have lim (a_n/b_n)=1 (not zero, not infinite) so, since the series sum(b_n) converges, then the series sum(a_n) converges too!
Solve, for practice: Sec. 13.4: 1, 3a-c, 7ab. 13:5: 1a-f (use any method), 3, 7, 11, 13, 17, 19
Solve, write up the solutions, and turn in on Tuesday, Jan. 20: 13.4: 2b-i, 4. 13.5: 2, 4, 6, 12, 14, 18
R Jan. 15
Solve, for practice: Sec. 13.6: all odd-numbered 1-9
Solve, write up the solutions, and turn in on Tuesday, Jan. 20: 13.6: 2,4,6,8
F Jan. 16 Topics discussed today: Example 2 in 13.6 and Sec 13.7.
Solve, for practice: Sec. 13.6: 13. Sec. 13.7: 1-23 odd numbered (except for 15 which is a bonus problem) and 1h page 454 (in sec 13.5)
Solve, write up the solutions, and turn in on Tuesday, Jan. 20: 13.6:12. Sec. 13.7: 2-22 even numbered
Bonus problem (extra 2 points) 13.7: 15
The write-ups for the problems listed below are due Monday, Jan. 26. Solving the problems is due after each lecture.
T Jan. 20 Topics: Sec 13.8
Solve, for practice: Sec. 13.8: 1-25 among these some are due to be written up, as seen below, 27 (please do it). Be careful when working with series: read the text of 32 and 33 as cautionary tales...
Solve, write up the solutions, and turn in on Monday, Jan. 26: 13.8: 2, 4, 8, 10, 18, 22
W Jan. 21 Topics: Sec 14.4 Taylor series
Solve, for practice: 14.4: 1,5,7,9,11
Solve, write up the solutions, and turn in on Monday, Jan. 26: 14.4: 6,8,10,12, 13a
R Jan. 22 Topics: 14.8 (complex numbers and Euler's formula) and start 14.2.
Solve, for practice: 14.2: 1 to 20, among which
Solve, write up the solutions, and turn in on Monday, Jan. 26: 14.2: 2,6,10,14
F Jan. 23 Topics: finish 14.2, and start perhaps, finish) 14.3.
Solve, for practice: 14.2: 1 to 29, among which some need to be written up, 14.3: 3
Solve, write up the solutions, and turn in on Monday, Jan. 26: 14.2: 22, 26. 14.3: 2, 4, 5
The write-ups for the problems listed below are due Monday, Feb.2. Solving the problems is due after each lecture.
M Jan. 26 Topics discussed today: More on power series: 14.3, 14.4, 14.7 (we do not discusss division of power series).
Solve, for practice: Sec. 14.7: 3, 12, 33, 35b
Solve, write up the solutions, and turn in on Monday, Feb.2: Sec.14.3: 6, 8. Sec.14.4: 13b, 14, 15a. Sec.14.7: 4, 6, 28, 35a
T Jan. 27 Topics discussed: 14.6. See some examples worked in class calculated using MAPLE.
Solve, for practice: Sec. 14.6: 1, 5
Solve, write up the solutions, and turn in on Monday, Jan. Feb.2: Sec.14.6: 2a, 4
R Jan. 29 Topics discussed: more examples with power series.
F Jan. 30 Review.
Solve, for practice: see the additional problems for Chapters 12, 13, 14.
Solve, write up the solutions, and turn in on Monday, Feb.2:
Additional problems for Chapters 12 (p.424): 1,4, 100, 101,120.
Also: find which of the following sequences converge (find the limit if they do): a_n=(1+(-1)^n)/ln n; b_n=(2+(-1)^n)*(n+1)/n; c_n=(-1)^n/(2^n); d_n=3-(-1)^n. . (Note: c^n denotes the number c raised to the power n).
Additional problems for Chapters 13 (p.470): 4a, 35 a,b,c,f,k,l, 51.
Additional problems for Chapters 14 (p.523): 1a (use the root test), 4a,b (use any method), 8c.
Also: Approximate the function f(x)=cos(4x) by its Taylor polynomial at x=0 finding all the terms of degree 4 and lower.
Bonus problems (extra 2 points each): 64 at p.476, 5 at page 524
The following write-ups are due Monday, Feb.9.
M Feb. 2 The first midterm test.
T Feb. 3 Sec. 15.3 (we do not discuss the directrix)
Solve, write up the solutions, and turn in on M Feb.9: 15.3: 3
W Feb. 4 Sec. 15.4 (we do not discuss eccentriciy or directrix) How to draw a hyperbola. Looking into a hyperbolic mirror.
Solve, for practice: Sec. 15.4: 1,3,4,5,6,7,17,19
Solve, write up the solutions, and turn in on M Feb.9: 15.4: 2,8,18,20,25
R Feb. 5 Sec. 15.2 (we do not discuss eccentriciy or directrix) , 15.6
Solve, for practice: Sec. 15.2: 1, plot the parabolas in 3. Sec. 15.6: 1-11, 15 among which some will have to be written up.
Solve, write up the solutions, and turn in on M Feb.9: 15.2: 1a,f; plot the parabolas 3b,e; 9. Sec. 15.6: 2,4,6, 13, 15 (for problems 2,4,6 only).
F Feb. 6 Sec. 16.1
Solve, for practice: Sec. 16.1: 1, 5, 9b
Solve, write up the solutions, and turn in on M Feb.9: 16.1: 1j,l, 4, 6a
The following write-ups are due Monday, Feb.16.
M Feb. 9 Sec. 16.3 (no conics)
Solve, for practice: Sec. 16.3:
Solve, write up the solutions, and turn in on M Feb.16: 16.3: 4,6,18 (this is how conics look like in polar coordinates!)
T Feb. 10 Sec. 16.4
Solve, for practice: Sec. 16.4: 2, 3a, 17
Solve, write up the solutions, and turn in on M Feb.16: 16.4: 4, 10 (sketch the spiral, too), 13, 15
W Feb. 11 Sec. 16.5
Solve, write up the solutions, and turn in on M Feb.16: 16.5: 3,8,12
R Feb. 12 Review of polar coordinates.
F Feb. 13 Sec. 17.1
Solve, for practice: Sec. 17.1: 1,3,5,9,13 (part of it is done in class)
Solve, write up the solutions, and turn in on M Feb.16: 17.1: 2,4,6,8
The following write-ups are due Monday, Feb.23.
M Feb. 16 Sec. 17.3
Solve, for practice: Sec. 17.3: 1, 2ab, 3, 5cd
Solve, write up the solutions, and turn in on M Feb.23: Sec. 17.3: 2cd, 4, 5ab,6,8,9
T Feb. 17 Sec. 17.4
Solve, for practice: Sec. 17.4: 3,5,8
Solve, write up the solutions, and turn in on M Feb.23: Sec. 17.4: 2, 4,6,11
W Feb. 18 More from Sec. 17.3,4.
Solve, write up the solutions, and turn in on M Feb.23: Sec. 17.3: 7, 10, 11, 13
R Feb. 19 Finish Sec. 17.4.
Solve, write up the solutions, and turn in on M Feb.23: Sec. 17.4: 10,11,12; Sec.17.3: 14
You are strongly encouraged to participate in: 2009 Rasor-Bareis-Gordon MATH Competition - SATURDAY, February 21, 1-4 pm Cockins Hall 240
Second midterm test Monday Feb. 23. The following formula sheet will be provided with the exam. In addition, you are allowed to bring a cheat sheet (one page, written on both sides, with your own notes). Topics: Chapters 15, 16 and 17.1, 17.3,17.4.
The following write-ups are due Monday, March 2.
T Feb. 24 Sec. 17.5. See some osculating circles, circles of curvature.
Solve, for practice: Sec. 17.5: 1, 7
Solve, write up the solutions, and turn in on M Feb.23: Sec. 17.5: 2, 3,6,8
W Feb. 25 Sec. 17.5 and 17.6.
Solve, for practice: Sec. 17.6: Read Example 1 (I mean, try to solve the problem and use the textbook to check your work), 1,3,5,7,9,11
Solve, write up the solutions, and turn in on M Feb.23: Sec. 17.6: 2, 4, 6, 8,10,12
R Feb. 26 We solve problems from Sec. 17.5 and 17.6.
F Feb. 27 We finish Sec. 17.6 and go over 17.7. See the Geostationary Satellite Server at NASA. See the height of the orbit of geostationary satellites. Aphelion Away! A rare beautiful event this evening, see Headline stories. See comet trajectory.
Solve, write up the solutions, and turn in on M Feb.23: For the function y=ln x at x=1/sqrt(2) answer the questions of problem 6 Sec. 17.5 (page 615). Also, solve in Sec. 17.5 problem 13 (Hint: first discuss the cases a>1, a=1, a<1 then subdivide the case a<1.)
Some sites of interest:
Decartes' Discourse on the Method.