• SHE (Salas, Hille, Etgen), Sections 9.3,9.4
• H² (Hubbard and Hubbard), Section 0.6
• The beginning of the novel A Tale of Two Statisticians

• SHE, P. 535, #29-32, 41, 44, 59, 60 (for #41 on also use the "weird coordinates.")
• SHE, P. 542, #9-11, 27, 47
• H², P. 23, #0.6.3. Also, do the same problem but for the roots of i rather than the roots of 1.

• SHE, Sections 11.5,11.6,10.5,10.6,11.1

• SHE, P. 679, #5,6,17-20,38
• SHE, P. 683, #1-4,33,34
• SHE, P. 617, #5,9,11,23,33
• Recall that the hyperbolic functions are defined by
  cosh(x) := (exp(x) + exp(-x))/2, and
  sinh(x) := (exp(x) - exp(-x))/2.
  Find the Taylor series for these two functions. Use these series to discover interesting facts about cosh(i x), sinh(i x), cos(i x) and sin(i x). (Have fun!)
• Other than by a brutal calculation, can you explain why so many of the coefficients of the Taylor series for sin(tan(x)) - tan(sin(x)) are 0?
• This problem is to be done after you have seen l'Hôpital's rule (section 10.5): Consider the function defined as f(x) := exp(-1/x²) for x different from 0, and f(0) = 0. (This function and all of its derivatives are defined at all values of x, including x=0.) Using l'Hôpital's rule to calculate the derivatives of f(x) at x=0, write down the Taylor series for f(x), in powers of x.

Due Thursday, 7 September:

• Reading:
  • SHE, Sections 11.2, 11.2, 11.3 (ratio test only), 11.4,
  • The little crib sheet on series
• Exercises
• Take the test given last year (both versions). Afterwards, compare your solutions to Maple's (version 1 or version 2 - ignore problem 4).
• SHE, P. 645, #25,26,33,36,57,61
• SHE, P. 655, #1-4,20,23,48
• SHE, P. 660, #14,16,21,33
• SHE, P. 666, #9-12,29,30
• How far is the horizon from a mountain top? Use geometry to get an accurate formula for the distance to the horizon in terms of the height of the observer. (Assume that the earth is a perfect sphere of radius r=6360 km and the observer's eyes are at height h above sea level.) Then, supposing that you do not have a calculator when you are at the mountaintop, use Taylor's formula to estimate the distance to the horizon to second order in h. How bad is this estimate? Extra credit: What notable mountains far from the sea should be visible from the sea? Brasstown Bald? Aconcagua?
• Take Prof. Loss's old tests (Test 1 and problem I of test 2 are appropriate.)

• Reading:
  • SHE, Sections 11.7-11.9, 8.7
• Exercises
  • SHE, p. 690, #1,2,9-15
  • SHE, p. 701, #1,3,10,14,31,43
  • Express the following as a power series to the first 9 or so nonzero terms:

           _________1_________________
                     2      3     4      5
           (1-x) (1-x ) (1-x )(1-x ) (1-x ) ...
    
    

    Extra credit: cube the result and find a cool pattern.
  • Use power series to solve the following transcendental equation:
    exp(x y) = y-x
    for y(x) such that y(0) = 1.

• Reading:
  • SHE, Sections, 8.7,8.8
  • Prof. Harrell's derivation of Simpson's rule.
  • For those with Maple, look at the following worksheets from the pretty good basic on-line Maple tutorial from NC State (The worksheets are not strictly required, but they discuss material which is):
    1. Approximating definite integrals
    2. Simpson's rule
    3. Prof. Herod's Maple worksheet on using randomness to evaluate an integral numerically
    4. Taylor polynomials
• Exercises
  • SHE, P. 490, #3,8.13,14,25, Project 8.7
  • SHE, P. 499, #7-11,29
  • Use the "Monte-Carlo" method discussed in Prof. Herod's Maple worksheet mentioned above and in class to find the area of a quarter disk with radius . Suggestion: If you do not have Maple of a similar utility, most spreadsheets and programming languages have (pseudo)randum number commands. For example, you can produce a column of random numbers evenly distributed between 0 and 1 in Excel by pasting

    =RAND()
    =RAND()

    into the cells of the column.
  • Use power series to solve the following differential equations:
    1. y'(t) = (1+t-t²)y(t), such that y(0) = 1
    2. y''(t) = (2 - sin(t)) y(t), such that y(0) = y'(0) = -1.

• Reading:
  • SHE, Sections 8.9
  • H², Sections 1.1-1.2
  • SHE, Sections 12.1-12.3
  • The web page solving last week's quiz with power series
• Exercises
  • SHE, P. 505, #3-10, 12-14. Try to solve these by power series as well as by the methods of Sections 8.8-8.9
  • H², P. 127, exercises numbered 1.1.x and 1.2.x
  • Do the problems from Prof. Harrell's tests given last year
  • Do the problems in Prof. Carlen's practice tests (especially IIA and IIB) and Prof. Loss's practice tests (especially Old Test 2, #3-4 Old Preptest 2, #3 and #1 in Old Test 3 and Old Preptest 3.

• Reading:
  • H², Sections 1.2-1.4
  • Demko, Sections 1.1, 2.1-2.3
• Exercises
  • H², P. 127, #1.2.3, 1.2.5, 1.2.6, 1.2.10, 1.2.16, 1.2.22, 1.3.3, 1.3.12, 1.3.13
  • Demko, P. 6, #1,2,3
  • Demko, P. 37, #1,3,7
  • Demko, P.47, #1,2,5

• Reading:
  • H², Sections 1.4-1.5,2.1-2.2
  • Demko, Sections 1.2-1.3
• Exercises
  • H², P. 127, #1.4.4, 1.4.5, 1.4.8, 1.4.11, 1.4.13, 1.4.20

  ---

  NOTE: We made slower progress this week than I originally planned. Therefore, the homework problems having to do with
  row operations (H&H chapter 2)
  3 by 3 determinants
  have been extended to next week. On the other hand, for tomorrow you should be able to solve problems of the following form:

  
  Let P,Q, and R be the following position vectors:
  
        [-1]    [-2]   [ 3]
        [ 4]    [ 0]   [-4]
        [-3]    [ 2]   [ 1]
  
  (or any other three vectors you can name).
  
  1.  Find the area of the triangle with vertices P,Q,R
  2.  Find the formula for the plane through P and with normal vector Q
  3.  Find the formula for the plane containing P,Q,R
  4.  Find the lengths of the sides of the triangle and the angles between them.
  5.  Find the area of the parallelogram with vertices O (origin), P, Q, and
  P+Q
  
  Also, in the simple old plane:
  
  Find the inverse of any 2 by 2 matrix, such as
          [1  2]
    A =   [3  4]
  
  Solve
  
          [-1]
    A v = [ 2]
  
  
  Find the vector obtained by rotating v of the previous question by
  pi/3 radians counterclockwise.
  
  Find the matrix P which projects any given vector onto the line y = 2 x
  (hint:  the basic modeling theorem).
  

Due Thursday, 19 October:

Reading:
• H², Sections 1.4-1.5,2.1-2.3
• Demko, Sections 1.2-1.3, 3.6
• Exercises
  • H², P. 231, #2.1.1, 2.1.3, 2.1.7, 2.2.2,2.3.2,2.3.4
  • Demko, P. 16, # 1,2; P. 26, #1-4 (These problems involving solving systems of two equations in two unknowns and three equations in three unknowns. If you do not have Demko, make up some systems of your own or get some from H² or elsewhere. Check answers.)
  • Demko, P. 88, 1-3 (These problems are calculations of areas of parallelograms, volumes of parallelopipeds, and determinants. If you do not have Demko, make sure you can do such calculations with examples from H² or make up your own 2 by 2 or 3 by 3 examples. Check answers.)

• Reading:
  • H², Sections 1.1-1.5,2.1-2.4
  • Demko, Sections 1.2-1.3, 2.1-2.3, 3.1,3.6
• Exercises
  • H², P. 231, #2.3.2, 2.3.4, 2.3.10, 2.4.5,
  • Demko, P. 58, # 2-5 (These problems have to do with column spaces and planes. If you do not have Demko, you may wish to look at these problems in the library (ask at the reserve desk), as they are not close to problems in H².
  • Do the problems from Prof. Harrell's test given last year
  • Do the problems in Prof. Carlen's practice tests (especially IIIA and IIIB) and Prof. Loss's practice tests (especially Old Test 3 and Old Preptest 3

• Reading:
  • H², Sections 2.4-2.5
  • Demko, Sections 3.1, 3.3, 3.4
  • (optional) Watch a pi-minute long Quicktime © movie about an interesting number. (Click for free Quicktime reader if necessary, for Mac or Windows.
  • Maple's solutions to the recent test left-arrow version and right-arrow version). (left or right).
• Exercises
  • H², P. 231, #2.4.3,2.4.8,2.4.9,2.5.2, 2.5.7, 2.5.13
  • Demko, P. 75, # 1-4,6,7
  • Demko, P. 78, # 1,2

• Reading:
  • H², Sections 2.6
  • Demko, Sections 3.4-3.6
• Exercises
  • H², P. 231, #2.6.1,2.6,2,2.6.5
  • Demko, P. 65, # 1-6
  • Demko, P. 84, # 1-3

• Reading:
  • Demko, Chapter 4, all sections
• Exercises
  • Demko, P. 94, # 1-3
  • Demko, P. 103, # 4,5,8; also, find the matrices which
    1. scales perpendicularly to (1,1,1) by 1/2, stretches in the direction (1,-1,0) by 2 and stretches in the direction (1,1,-2) by 3; and
    2. projects onto the plane spanned by (4,9,2) and (3,5,7)
  • Demko, P. 108, # 1,3
  • Demko, P. 118, #1-3,9
  • Demko, P. 124, #2,3,5; also, find the functions which do the following:
    1. reflect through the plane x + y - 4z = 2
    2. rotate about the line x=t, y=2t, z=t+2 by angle theta.

• Reading:
  • Demko, Chapter 5, all sections (but 5.4 before 5.3)
• Exercises
  • Demko, P. 133, # 1,2,3,5,7,10
  • Demko, P. 140, # 1,2,4,6
  • Demko, P. 158, # 1,2,4,5
  • Correctly answer the Zen question: How can you rotate a (non-zero, two component) vector by 90 degrees, so that it still points in the original direction?

Past contests

1. The first contest was purely aesthetic: Use Maple or another graphing utility to graph the most creative or beautiful curve.
   Congratulations to Christos Tsonis for a graph which looked something like a hairy carneval mask:
   > plot[sin(260*t)^2 + cos(2*t),t,t=-Pi..Pi];
2. A professor had ten different pairs of socks, but a malicious dryer ate six (individual) socks. An optimist would say that with luck, there will still be 7 matching pairs, while a pessimist would say that there will more probably be only 4 matching pairs. Who is more likely to be right, and by how much? Give a quantitative comparison of the probabilities.
   Congratulations to Karthik Balakrishnan and Jay Underwood for correctly showing that the pessimist is 112 times more likely to be right than the optimist.
3. Over in Athens, Georgia, many people - far, far, too many - believe that (f g)' = f' g'.
   This is known as the Ugamath formula. Once in a while it is even right, for example when f and g are constants, or f(x) = g(x) = exp(2 x). How many examples can you find where (f g)' = f' g' but NOT = 0?
   Congratulations to Karthik Balakrishnan for the largest number of solutions.
4. As you know from a class in Euclidean geometry, two triangles are congruent if they share two angles and a side (ASA) or two sides and the angle between them (SAS), but not necessarily if, for example they share two sides and a different angle (SSA). The relative positions of the 6 possible elements (3 S's and 3 A's) need to be given for these theorems. This does not exclude the possibility that two triangles might share 5 of the 6 elements but fail to be congruent, if we don't say which angles are adjacent to which sides. How many examples can you find of such pairs of triangles with integer sides?
   Congratulations to Karthik, again, for many solutions.!
5. Define a sequence of polynomials by P₀ := 1, P₁ := x+1, and P_n+1 := P_n + x P_n-1. Show that all roots of P_n are real, for all n.
   Sadly for the polynomials, this one was not solved by the extended deadline. Keep trying for an exciting prize, though no points!
6. Arithmetic codes. There is a category of arithmetic puzzles, in which each digit is encoded as a letter, and the arithmetic message spells something meaningful. The classic one is a letter sent home from college, which reads

   Dear Father,
   
                    S E N D
                  + M O R E
                  _________
                  M O N E Y
   
   

   This is a good puzzle because there is exactly one solution.
   Our contest has two parts. The first is to solve for the arithmetic encoded by

                   AB*CD=EEE
                   E*CD-AB=CC
   

   in order to find A*B*D . This much may get you an exciting prize.
   The second part is more creative. Make up a puzzle, preferably something like the SEND + MORE = MONEY code by using concepts from this class.
   The winning puzzle will have one and only one solution, and will be judged on the basis of beauty.

   P.S., the father's response to the letter above was:

                      U S E
                  + L E S S
                  _________
                  S O N N Y
   

   Congratulations to Tim Lee for two sums, including

                          I
                    + AGREE
                    +   ITS
                  _________
                      TOUGH
   

7. Three-way duel, if "duel" is the proper word. Three duelists stand at the corners of an equilateral triangle. The rules of the duel are that they shoot in order, and they draw straws to see who begins. At each turn a given duelist gets one shot, and the shooting continues until there is only one left standing.

   Now, they all know that Dead-Eye Dana is a great sharpshooter who will hit the target 100% of the time, while Average Alex hits the target 80% of the target 80% of the time and So-So Sal hits the target 50% of the time. Although the duel began over some passionate irrationality, now their minds are focused, and they each act rationally to maximize the chances of their survival. The contest is to figure out the probability each of the three has to survive, and to explain your answer clearly.

   Congratulations to Steven Paul for showing (quantitatively) that the worst shot is the most likely to win!
8. The infidelity contest.

   Note: This logic puzzle is the creation of John Allen Paulos, and was recounted in his book Once upon a Number, which was chosen by the Los Angeles Times as one of the best nonfiction books of 1998. The innovation of alien genders is, however, my own.

   ---

   A nominally monogamous species on the planet Thrae consists of two genders, "wen" and "momen," the singulars being "wan" and "moman." Now, wen are very intelligent and logical, and each wan knows that all the other wen are equally so. They have the following well-known habits. 1. If a wan has proof that the moman to whom the wan is married has been unfaithful, the wan will kill the moman within one day. 2. Wen's intuition always allows them to know when another wan's spouse has been unfaithful. 3. Wen never tell any wan else about infidelity. 4. For some reason, wen's intuition fails them as to their own spouses. 5. Every wan takes as proved whatever is printed in the Equinchet, a local oracle.

   In the village of Chetag there are forty married couples, and all the momen have been unfaithful. Of course, every wan lives in the belief of being the one wan with a faithful spouse. Well, maybe. One day the Equinchet prints the statement that it has evidence that at least one moman in Chetag has been unfaithful.

   Does any wan learn anything not already known? Do any killings ensue? If so, how many and when? To win this one, you have to have the correct answer and the correct logical reasons for it. (Hint: One wins when one wonders well when one wan wonders whether one wan's one wanders.)

   Ah, well, no correct solutions to this one by the deadline
9. The literature contest. Who can find the best examples of realmath in real literature? I am aware of some examples in work by Poe, Calvino, Borges, Stephenson, and some others. If you find a good example our panel of judges was unaware of, and back it up with the evidence, you may win a few points.
   A split win this time, between Karthik Balakrishnan and Eric Rahm! Karthik provided a really nice website on math and fiction, while Eric came up with the movie ; check out the movie's web site. Honorable mention to Steven Paul for finding Flatland on the Web.
10. Three politicians are having a debate around a round table on the planet Thrae. To the left of A sits B, to the left of B sits C, and to the left of C sits A. Now, there are three political parties, the Truthtellers, who always tell the truth, the Liars, who always lie, and the Wafflers, who lie and tell the truth in strict alternation. In the course of the debate they report the following facts (in order) about their opponents:

    
    speaker   The person to my left is   The person to my right is     I am
    
    A            a Liar                    a Waffler                 a Truthteller
    
    B           a Waffler                  a Waffler                 a Truthteller
    
    C            a Liar                    a Truthteller             a Truthteller
    
    

    Congratulations to Tim Lee and, minutes later, Casey, Zach, Steven, and Karthik, for showing that two of the politicians are Wafflers and one a Truthteller.
11. Eigendoggerel. Write some rhymed couplets about eigenvectors and eigenvalues. Your couplet must contain at least one true mathematical statement (and no false mathematical statements), and must be original by you. I.e., not downloaded from the Web, where there is no doubt a vast literature of wonderful poetry about linear algebra. The winner will be judged on the basis of wit. Your entries are due on Monday, 27 November.
    Congratulations to Rebecca Fink, whose poem was voted the best, though there were also awe-inspiring eigenpoems by Tim Lee and Zach Eaton.

• The 1512 home page
• Evans Harrell's home page
• The School of Math on-line resources

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