MAT2705 07S homework and daily class log

Your homework will appear here each day as it is assigned, with occasional links to some MAPLE worksheets when helpful to illustrate some points where technology can be useful. [There are 56 class days in the semester, numbered consecutively below and  labeled by the (first initial of the) day of the week. Wednesday, January 18 thru May 3.] Usually it will be summarized on the white board in class, but if not, it is your responsibility to check it here. You are responsible for any hyperlinked material here as well as requesting any handouts or returned tests or quizzes from classes you missed. Homework is understood to be done by the  next class meeting (unless that class is a test, in which case the homework is due the following class meeting).

  1. W: GETTING STARTED STUFF. By Friday, January 19, e-mail me [robert.jantzen@villanova.edu] from your OFFICIAL Villanova e-mail account (which identifies you with your full name) with the subject heading "[MAT2705-XX]", where XX is your section number 01 or 04, telling about your last math courses, your comfort level with graphing calculators (can you do symbolic derivatives and integrals on your graphing calculator if not in Maple?) and computers and math itself, how much experience you have with MAPLE (and Mathcad if appropriate) so far, why you chose your major, etc. Tell me what your previous math course was named (Mat2500 = Calc 3 or Mat1505 = Calc 2). If you were my student, last semester, tell me if you thought you successfully learned something from the experience.
    [In ALL email to me, include the string "mat2705" somewhere in the subject heading if you want me to read it. I filter my email.]

    In class:
    1) log on     to your computer and open Internet Explorer. (IE allows you to open MAPLE files linked to web pages automatically if MAPLE is already open or if it is available through the Start Menu Program listing under Math Applications, in Netscape you must save the file locally and then open in it MAPLE using the File Open task.)
    2)     log on to the Villanova home page in Internet Explorer (click on the upper right "login" icon and use your standard VU email username and password) and check out our My Courses classroom site, and visit the link to my course homepage from it
    [ http://www.homepage.villanova.edu/robert.jantzen/courses/mat2705/ ],
    3) Open     Maple 10 Standard (red not yellow icon) from the Start Menu Program listing under Math Applications
    [or click on this maple file link: cmdlist4.mw]
    4)         bob will quickly show you the computer environment supporting our class. And chat up a bit the course.
    And Maple.
    [Using the File OPEN URL command in Citrix Maple, with this URL (right click, copy shortcut) will show you a Maple file of the exponential and sinusoidal example functions we played with in the latest version of Maple, correctly as opposed to what will happen in G86,87,88 where an older version of maple remains.]

    Afterclass:
    5) log on     to MyWebCT and look at the Grade book: you will find all your Quiz, Test and Maple grades here during the semester.
     [This is the only part of WebCT we will use this semester.]
    Check out the on-line links describing aspects of the course (no need yet to look at the MAPLE stuff). Fill out your paper schedule form (get  a copy in class to fill out or print it out back-to-back earlier to fill out in advance and bring to your first class already done) to return in class Wednesday.
    [You can     drop by my office St Aug 370 (third floor, Mendel side, by side stairwell) to talk with me about the course if you wish and to see where you can find me in the future when you need to.]
    6) Make sure you can access and open Maple 10 Standard in citrixweb on your laptop. If you have any trouble, email me with an explanation of the errors. If I receive no email, it means that you were successful.
    You are expected to be able to use Maple on your laptop or in class when needed.

    7) Read computer classroom etiquette. Then read the first paper handout: algebra/calc background sheet.

    8) Homework Problems: 1.1: 3, 5, 13, 33 (this links to a PDF scan of the HW problems from book if you have not yet purchased it);
    [ "A is proportional to B" means "A = k B" where k is some constant, independent of A and B]
    (short list so you can check out our class website and read about the course rules, advice, bob FAQ, etc, respond with your email).
    It is important that you read the section in the book from which homework problems have been selected before attempting them.
    Get acquainted with Maple10 Standard DE entry and "odetest" for problems 3,5,13 [even 33!]
    [use citrixweb Maple on your own computers]
     
  2. Th: return your schedule forms at the beginning of class [did you use the 3 letter dorm abbreviations?];
    1:1: 7a [i.e., only check y1(x)], 23;
    formulating DEs: 27, 29 [Hint: recall perp lines have slopes which are negative reciprocals; make a diagram of given point (0,1), a generic function graph curve, and a point (x,y) on the curve and the connecting line segment between points perpendicular to tangent line, then compute slope from two points, and from derivative, equate the two to get the DE], 35, 36.
  3. F: course information handout; Quiz 1 Not! (try it later without the book 20-30 minutes max: then check answer key [hand, maple]);
    1.2: 1, 5, 15, 19 [Hint: write piecewise linear function from graph: v1(t) for first expression, v2(t) for second expression, solve first IVP for x1(0) = 0, second IVP for x2(5) = x1(5)], 25 (like lunar landing problem), 39, 40 (ans: 2.4mi), 43.

    WEEK 2 [-1]
  4. M: deadline for sending me the email has passed, send it anyway (see first day assignment);
    1.3:  3: [hand draw in all the curves on the printout supplied by bob (rotated and scaled to fit option on Print), maple version],
    8* : our first Maple HW problem: read the Maple HW instruction page, then try the Maple direction field command in this linked template to reproduce the two given curves by using two appropriate initial data points [Hint: notice that the y-axis crossing points are integers!] Then include all the initial data points of the red dots (some half-integer values); this is also described on p.29  for both Maple and Matlab, but the above template already has the appropriate section copied into it for you; if you can find a partner to work with in time fine, otherwise try this yourself and find a partner for future problems, then merge your work.
    being careful: 11, 15 [note Dyf is just df/dy above Eq (9) in the text]; 27 [don't worry about making a diagram];
    a freebie: numerical problems caused by vertical tangent line.
  5. W: class list contact data sheet handout (not web published) to help form partnerships;
    1.4: 1, 5; 21, 25, 27, 29.
     
  6. Th: Post-Its, anyone? [for your textbook markers] ask bob, they are in his bag of stuff;
    handout on exponential behavior/ characteristic time [read this worksheet explicit plot example];
    1.4: 45, 47 [recall 1.1.35, use units of thousands of people, assume no one has heard the rumor at t=0: N(0)=0 (an approximation in units of thousands; really N(0)=.001 would be one person starting the rumor, or should it be two?)], 65;
    Optional challenge problems:
    68 (PHYS: parametric cycloid solution, obvious typo: theta = 2t in text) 
    or 69 (ME: suspension cable catenary: hyperbolic cosine)
    or Torricelli's law section and 64 (ChmE: ans r = 1/35 in).
     
  7. F: Quiz 1 thru 1.4; no-paper handout only on line: recipe for first order linear DE;
    1.5: 3, 7, 11, 21; 27;
    21*[check both the general solution and the initial value problem solution with the dsolve template].
    For future reference:
    > deq := y ' = x y      [space implies multiplication]
    > sol:=dsolve(deq, y(x))
    > solinit := dsolve({deq, y(0)=1}, y(x))
    [or enter DE plus IC separated by a comma, right click on output]
    > y ' = x y , y(0)=1
    > odetest(sol,deq)   [this returns 0 if LHS-RHS of DE is 0]

    WEEK 3[-1]:
    Weeks 3 thru 4:     come by and find me in my office, tell me how things are going.
    This is a required visit. Only takes 5 minutes or less. Come say hi!
     
  8. M: 1.5: 17, 26 [ans: x = y^(-2)/2 +Cy^(-4)], 29 [use the fact that the derivative of the function defined by the integral is the integrand including all multiplicative factors], 31 [show by differentiation and backsubstitution into the DE, not by rederiving it], 32 [find the constants by substituting the trial solution into the DE and choosing the values of A and B to make the DE satisfied: put all terms on LHS and set coefficients of the sine and cosine separately to zero],
    41 [Hint: Δt in years is the fraction of the year, so S(t) Δt is the approximate income during this time interval, 12 percent of which contributes to ΔA, while the interest contribution is the annual interest rate 6 percent prorated by multiplying by the fraction Δt for the interest contribution over this shorter time period, applied to the current value of the retirement amount A(t): 6 percent of A(t) Δt; adding these and dividing by Δt, taking limit to get dA/dt gives the linear DE [click here only if you have tried to get the DE and could not or its solution did not give the correct number].
    41* [use the dsolve template to solve this DE with its initial condition; evaluate answer to question of word problem to a floating point number and give the result in dollars and cents in a comment, then think about significant figures and give a response with fewer significant digits].

    Maple1.mw assignment complete. Due week between Wednesday to next Tuesday

    Test 1 thru chapter 1 next week Feb 5-9: you chose Thursday, February 8.
     MLRC problem session 5:30pm Tuesday or Wednesday??
     
  9. W: 1.5: 37 (Use Eq. 18 in the book; what is the final concentration of salt?);
    [this mixing problem is an example of developing and solving a differential equation that models a physical situation; for another word problem example see this]
    1.R(review): classify the odd problems 1-35 as: separable, linear in y (as unknown), linear in x (as unknown), some combination of these three, or NOTA (none of the above), for example dy/dx = y/x is all three and can be solved in three different ways;
    solve 25, express K in terms of C as given in the book supplied answers by combining the two terms in the K solution [identity for (x+1)^3 !] and then comparing with the C solution;
    solve 35 in two ways and compare the results    ;

    Optional Light Reading 1.5: Application: see how a slight generalization of the directionfield example 1.3.3 to include an initial time parameter and time and temperature scale parameters has a useful physical application on pp.56-58. No need to study it, but reading through it gives an idea of how one can use a simple linear DE to model an interesting problem.
  10. Th: REMIND BOB to match up people who need Maple partners;
    handout on solution of logistic DEQ (and similar eqns) [directionfield,  integral formula];
     [handout on DE's that don't involve the ind var explicitly];
    2.1: 5 [go thru the solution steps just this once with concrete numbers, remember the integral formula, follow the steps of the book or handout derivation of the solution, compare with the final logistic formula; use the formula for the remaining HW problems],
    9, 15 [note that by definition Bo = kMPo and  Do = -kPo2 , and these are easily solved for M and kM in terms of Bo, Do, and Po to get the other two logistic curve parameters needed to solve problem 16],
    16 [ans: P = .95 M after 27.69 months];
     
  11. F: Quiz 2 on linear DEs;
     2.1: 6 [ use handout integration formula with sign reversed, or use technology for integral and combine log terms;
    answer: > dsolve({x'(t) =3 x(t) (x(t)-5), x(0)=2},x(t)) (copy and paste into the input line);
    plot this solution with technology and choose a horizontal window (negative and positive t values) in which you see the reversed S-curve nicely, be ready to give the approximate time interval for an appropriate viewing window in class],
    11 ["inversely propto sqrt": use β = k1/P^(1/2), δ = k2/P^(1/2) in eq.(1)],
    23 [Note dx/dt = 0.004 x (200-x) so k = 0.004, M=200 for logistic curve solution],
    30, 31 [ans: find βo = 0.3 from condition dP/dt(0) = 3x10^5 using the DE at t =0, then use solution α=0.3915 from second condition (check numerically that it satisfies the 6 month condition) to find the limiting population as t goes to infinity (i.e. neglect decaying exponential)].

    WEEK 4[-1]:
  12. M: air resistance handout [comparison of linear, quadratic cases; numerical solution];
    2.3: 1, 2, 3, 9 [remember weight is mg, so mg = 32000 lb determines m = 1000 in USA units].

    T: 5:30pm voluntary Test 1 problem review session in MLRC
     
  13. W: Open Maple at class beginning;
    2.4: most basic numerical DEQ solution technique from calculus: maple2.mw; in class
    open MAPLE, then open IE and webmail, use email link to go to our HW page in a separate window and follow instructions for this maple assignment;
    hand work home work:
    2.4: 1 (do by hand and calculator, can check with euler.mw if you wish;
    optional read about improved Euler).
  14. Th: Test 1 thru chapter 1.
     
  15. F: handout: why LinAlg with DE?; [maple dsolve and DEplot for 2x2 systems of DEQs];
    Read 3.1 on linear systems (review of HS solving 2 or 3 linear equations in 2 or 3 variables),
    do 3.1: 1, 5, 7; 9, 15, 31;
    Maple2.mws due next week February 12-17.

    > {x - y = 2, x + y = 0}  Right-click and choose solve

    WEEK 5[-1]: office visit still to do??
  16. M: 3.2: 1, 3, 7; 11, 13, 15 (do a few by hand, then you may use step-by-step  row ops with MAPLE or a calculator;
    you must learn a technology method since this is insane to do by hand after the first few simple examples);
    23 [can your calculator handle this?]
     
  17. W:
    in class with neighbor as partner, read together and execute line by line this linked worksheet: rowredex0.mw;
    handout on RREF and solving linear systems example;
    3.3: 1, 3, 7, 11, 17,  23 = 3.2.13;
    19* [use the tutor and record your step by step reduction, annotating each step as in the rowredex0.mw worksheet; then check with ReducedRowEchelonForm;
    To solve a linear system, input augmented matrix with the palette or Matrix command:
    > with(Student[LinearAlgebra]):
       A:=Matrix([[3,8,7,20],[1,2,1,4],[2,7,9,23]]);
       ReducedRowEchelonForm(A);
       BackwardSubstitute(%);
    To step by step reduce a matrix and solve the system do (pick Gauss-Jordan):
    > LinearSolveTutor(A)
     
  18. Th: recall 3.2.23 above never discussed;
    3.2: 22      [solve using row ops on augmented matrix to rref form, then by hand backsubstitute, check answer with Maple]
     3.3: 29 = 3.2.19 [solve using row ops on augmented matrix to rref form, then by hand backsubstitute],
    29* [now check this same problem solution by solving with ReducedRowEchelonForm, BackwardSubstitute of augmented matrix];
    MAPLE chemical reaction problem*

    Don't forget: Maple2.mw is due this week.
    Not all Maple1.mw files have been received! These are easy points.
    Your grade starts down from 90 instead of 100 if you blow them off.
    If you got less than 80 on Test 1 (back today), you should come for another (or first!) office chat.
    Check your Q1,Q2,T1 grades on WebCT to be sure they correspond to what is written on your copies.
     
  19. F: word of the day: can you say "homogeneous"?;
    Quiz 3;
    3.4: 1, 5, 7, 11, 13; 17, 21; 27, 43.

    WEEK 6[-1]:
  20. M: 3.5: 1; 9, 11; 23 [this is a way of solving 3 linear systems with same coefficient matrix simultaneously];
     30 [just substitute for the inverse and then drop parentheses and multiply out: (ABC)-1 (ABC) = ...],
     32 [multiply on the left by A-1].
     Matrix multiplication, matrix inverse, determinant, transpose
     > A B [note space between symbols to imply multiplication]
     > A-1
     > |A|  [absolute value sign gives determinant]
         > A%T
     
  21. W: 3.6: (determinants abbreviated: forget about minors, cofactors, only need row reduction evaluation to understand)
    13, 17, 21 (plug in just so you've done it once);
    17* [record your Gaussian elimination steps in reducing this to triangular form and check det value against your result];
    3.6. Apps:  7* from textbook on p.197, using the inverse matrix as described on p.196,
    be sure to answer the word problem question with a text comment;
    are these sandwiches cheap or not?
    [if you know how to use MathCad, you might try it and compare with MAPLE]
     
  22. Th: maple3.mw assignment is complete; due next week; get it over with early, catch up with others if not done yet.
    [blowing off maple means subtracting 10 points from your cum for midterm grades...];
     4.1: 1; 5, 7; 9; 15, 17; 19, 23; 25 (use technology in evaluating the determinant or ReducedRowEchelonForm for these problems); [row reductions];
    hand out on determinants and area etc;
    hand out on the interpretation of solving linear homogeneous systems of equations: A x = 0 [visualize the vectors]
     
  23. F: Quiz 4;
     handout on nonstandard coordinates on R^2 and R^3 [goal: understand the jargon and how things work];
    1) From the old illustrated coordinate handout with the new basis {[2,1],[1,3]} of the plane: graphically find the new coordinates of the point [4,7], then confirm using the matrix multiplication by the inverse matrix of the coordinate transformation given there. Then do the reverse for the point whose new coordinates are  [2,2].
    2) With the blank graph paper handed out in class and a ruler and sharp pencil, make a new -2..2,-2..2 coordinate grid associated with the new basis vectors {[1,1],[-2,1]} and graphically represent the point [-5,1] and find its new coordinates, then confirm by matrix multiplication. Similarly read off the old coordinates of the point whose new coordinates are [2,-1] and then confirm their relationship using matrix multiplication. Write down the coordinate transformations between the old and new coordinates in both directions. Put your name on it and bring it to class Monday.

    WEEK 7[-1]: maple3.mws due this week
  24. M: study the handout on solving linear systems revisited;
     [note only a set of vectors resulting from a linear homogeneous condition A x = 0 can be a subspace of a vector space; interpretation: lines, planes, ..., hyperplanes through the origin];
    4.2: 1, 3, 11; 15, 19;
    look at the soln to the system 3.4.21 on p.182 (back of book p.712): what does it tell us about the 5 vectors (namely the columns of the coefficient matrix of the linear system) in R2? how many independent relationships are there among the 5 vectors? How many vectors are independent? Is this obvious if you look at the explicit components of these vectors? [no! but if you realize that no 2 vectors are proportional, and no more than 2 vectors can be independent in the plane, of course the answer is obvious, this is the power of reasoning].
  25. W: did you turn in your graph hw from the weekend by today?
    handout on linear combinations, forwards and backwards [maple to visualize];
    4.3:  1, 3, 5, 7; 9, 13; 17, 21; 23.
     
  26. Th: Quiz 5 thru 4.2: linear independence or dependence of a set or vectors, expressing a vector as a linear combination of a set of vectors (using technology for appropriate row reduction);
    handout summarizing linear vocabulary;
    4.4: 1, 3, 5, 7; 9 [find solution, pull apart to find coefficient vectors of parameters, repeat for remaining 4 problems],
    13; 15, 17, 25 [use technology for all HW row reductions and determinants (2x2 dets are easy by hand!)].
     
  27. F: Transition back to DEs:
    read p.274 on function spaces and examples 3, 6, 8, 9; equivalently a handout on the vector space of quadratic functions [.mw, 1.5MEG PDF];
    4.7: 15 [solve c1(1+x)+c2(1-x)+c3(1-x^2) = 0, for unknowns c1,c2,c3; are there nonzero solns? if not, these are lin. ind. polynomials, in terms of which any quadratic expression in x can be written; note that any two (nonzero!) functions of x that are not proportional are automatically linearly independent], 16 [same approach].
    Spring Break.     enjoy and be safe.

    [U calculate your own midterm Grade (check WebCT on-line grades)]: 4 quizzes (drop lowest), 1 test, 2 maple assignments; if you want better Maple grade, check how 4/4 =10 points of your cum makes a difference for you. Corrected worksheets must arrive by Monday after break for consideration for midterm grade.

    WEEK 8[-1]:
  28. Wed noon midterm grades, Maple upgrades for midterm grade possible until early Wed am;
    Did you miss Quiz 5? talk to bob;
     handout on sinusoidal example;
    M: 5.1:     1, 3, 5, 9; 13, 17;
    begin maple5.mw: 9* [use the 2nd order dsolve template to solve the 2nd order IVP as a check].
    > deq:= y'' + 3 y' + 2 y = 6 ex
    > inits:= y(0) = 4, y'(0) = 5
    > solgen:=dsolve(deq,y(x))
     > solivp:=dsolve({deq,inits},y(x))
     
  29. W: T: happy π Day! and Albert Einstein's 128th Birthday;
    5.1: 33, 35, 39,
    49 [find the IVP solution for y, set y ' = 0 and solve exactly for x, backsubstitute into y]
    5.2: 1, 11;
    5.1: 49*[check your IVP solution for the highest curve in Figure 5.1.6 (highest for x > 0) using the dsolve template and plot it together with the horizontal line y = 16/7 claimed by the back of book answer as the highest value of that function; recall how to plot multiple functions in the same plot:
    > plot([<expression_for_y(x)>,16/7], x=a..b, y=c..d, color=[red,blue]);
    the color option is useful in distinguishing two functions when it is not already clear which is which; is 16/7 the peak value of your solution?].
     
  30. Th: The Ides of March?
    Test 2 date next week: Th:22, MLRC 5:30 voluntary problem session W:21;
    handout on visualizing the initial value problem (IVP);
    handout on complex arithmetic, exponentials [maple commands; i is uppercase I  in Maple];
    5.2:     13, 17, 21, 26 [for fun now, later method].
     
  31. F: Quiz 6;
    [    optional web only handout for context and going beyond this course: power series and DEs (chapter 11)];
    handouts on the amplitude and phase shift of sinusoidal functions and exponentially modulated sinusoidal functions [maple videos showing different phase shifts and frequencies];
    5.3: complex roots:
    8 [ans: y =  exp(3x)(c1 cos(2x) + c2 sin(2x))] , 9; 17; 22, 23 (express in phase-shifted cosine form);
    use technology to factor or solve for roots of polynomial equations:
    > solve(r2+6 r+13 = 0)    (in general, finds all roots exactly up to fourth degree and sometimes higher degree if lucky)
    > factor(r2+6 r+13 = 0)    (in general, factors real polynomials into linear and irreducible quadratic factors)
    [Note: > fsolve(...) returns all numerical roots of a polynomial].

    WEEK 9[-1]:
  32. M: 5.3 higher order DEs [ignore instructions to factor by hand or polynomial long divide: use technology for all factoring]: 11, 13; 25, 33; 39; 49 ;
    49* [check your solution with the higher order dsolve template, and also check the IC linear system solution as in the template, edit away all stuff not directly relevant to your new problem].
  33. W:  happy spring! [equinox trivia];
    MLRC 5:30 voluntary problem session today;
     5.4: 1, 3, 13, 14 [b) the solution once put in the form x = A exp(-Kt) cos(ω t - δ) has the "envelope curves" x = +/- A exp(-Kt) ] , 17; 23 [pdf].
    handout on     linear homogeneous 2nd order DEQ with constant positive coefficients (damped harmonic oscillator) [examples in nature].
  34. Th: Test 2 thru 5.3 (lin alg up to only 2nd order DE stuff thru Friday); did you check the archived test?
  35. F: [for your amusement note 5.4.32,33 about relative extrema and the same period and decay time];
    metric system prefixes confusing?;
    handout on RLC circuits [plots for this example];
    5.4: 15, 21;
    underdamped RLC circuit parameters: R = 16 ohms, C = 1/40 = .025 = 25 millifarads, L = 8 henries, E0= 17 volts; evaluate τ0, ω0, T0, ω0 τ0,  τ, ω, T, ω τ, A0 = E0/(ω L), and I(t) from the handout formulas (either resolve the IVP with Maple or plug into the formulas for the solution);
    Maple: *[RLC Maple plot: Make a single plot showing the current and its amplitude envelope for 4 decay times t = 0.. 4τ for this new problem.]
    WEEK 10[-1]:
  36. M: word of the day: can you say "homogeneous"?;
    5.5: handout on driven (nonhomogeneous) constant coeff linear DEs
    (complete final exercise on sheet [solution], Maple: *check with dsolve);
    5.5: 1, 3, 9 (we will not cover "variation of parameters"; the book presentation of the method of undetermined coefficients is a recipe with little justification (educated guess), instead the handout shows exactly how and why one gets the particular solutions up to these coefficients).
  37. W: handout on damped harmonic oscillator driven by sinusoidal driving function [Maple specific example]
    [maple resonance plots: general , specific (frequency much higher than resonance)];
    5.5: 33, 38 [y = ( exp(-x)(176 cos(x)+197 sin(x)) )/85 - (6 cos(3x)+7 sin(3x))/85];
    repeat the undriven RLC problem of day 35 now with a voltage source E(t) = 4 sin(2 t) and initial conditions I(0) = 0 = I '(0), solving it ignoring the solution formulas derived on the handout; then check your result with these formulas.
    *RLC Maple plot: for this RLC problem, plot I, Iparticular (the steady state part of the solution) and E/(2L) on the same plot and see how long it takes for the transient (difference between these two currents) to essentially be zero to the pixel accuracy.
    [USA rms voltage 120 Volts means peak voltage 120 sqrt(2) = 169.71V]
     
  38. Th: W: handout on beating and resonance
    [Tacoma Narrows Bridge collapse: engineering explanation; Wikipedia bridge video; Google];
    maple5.mw due next week;
    5.6: 1 [rewrite as a product of sines using the cosine difference formula:
    cos(A) - cos(B) = {-2 sin((A-B)/2)} sin((A+B)/2), the expression in {} is the sinusoidal amplitude, try plotting x and +/- this amplitude function together],
    11 [convert steady state solution to phase shifted cosine at end of calculation], 17 [pdf], 23.

    maple5 (last maple assignment) is complete, due any time during next two weeks since Easter break takes away 3 days
     
  39. F: Quiz 7;
    recall coupled system of DEQs [now revisit it (read a preview of what we are about to embark on)];
    F: Watch the MIT Eigenvector 3 minute video [from their LinAlg course];
    if you like play with the Duke U applet [red is x, blue is A x, click on matrix entries to change];
    see the various possibilities of the video in this Maple DEPlot phaseplot template;
    6.1: 1, 9, 13 only 3 problems so you have time to do some of the above!
    [use determinant: |A-λI|, solve (solve if complex roots) to find characteristic equation and eigenvalues of a 3x3 or higher dim matrix].

    Su:

    April Fools Day!

    WEEK 11:
  40. M: 6.1: 19; 27, 25;
    For the matrix A = <<1|4>,<2|3>> entered by rows, A = <<1,2>|<4,3>> entered by columns,
    find a set of independent integer component eigenvectors {b1,b2} by hand and then check them with the MAPLE Student[LinearAlgebra] Eigenvector command, then make the basis changing matrix B = <b1|b2> whose columns are these vectors (so we can agree, choose their order so that one gets to the second from the first by moving counterclockwise in the plane by an angle less than 180°), and write out the two coordinate transformations X = BY and Y = B-1X explicitly in terms of the individual coordinates: old [x1,x2] and new [y1,y2]; use the either the matrix or scalar transformation equations to find the new coordinates of the point [-2,4] in the plane and to find the point whose new coordinates are [2,-1]; then make a hand plot [as in this MAPLE plot] of the new coordinate axes and unit grid for the range y1 = -2 .. 2, y2 = -2 .. 2 and mark these two vectors on the plot with the two eigenvectors and confirm visually the relationship with the coordinates you evaluated; finally evaluate AB = B-1AB.
    [bring this to next class to give to bob]

    > with(Student[LinearAlgebra]):
    > Eigenvectors(A)
     
  41. W: Questions on Thursday's HW (excluding 6.6.23)?
    Find an (smallest integer) eigenvector basis changing matrix B = augment(b1,b2,b3) for the matrix of 6.1.19 which is the result of your hand method yesterday and use it to find the new coordinates <y1,y2,y3> of the point with old coordinates <x1,x2,x3> = <2,1,2>; evaluate AB = B-1AB;
    6.2: 1, 9 ("defective"); 13, 17, 21 ("defective"); 34 [just write out the characteristic equation and solve with the quadratic formula, think about real distinct roots];
    13* [check with the Eigenvectors command, choose eigenvector basis matrix B, evaluate AB = B-1AB by matrix multiplication].

    Easter Recess: Th, F, M no class
     
  42. W: hand in eigenvector worksheet;
    handout on the geometry of diagonalization and 1st order linear homogeneous DE systems (real eigenvalues);
    repeat this exercise for the matrix of 6.2.1 (namely A = <<5|-4>,<2|-1>> input by rows) and the initial condition x(0) = <0,1>;
    *check your solution using the dsolve system template;
    repeat by hand for the matrix of problem 6.1.19, with initial condition x(0) = <2,1,2>.
  43. Th: handout on 1st order linear homogeneous DE systems (complex eigenvalues) [phaseplot];
    Find the general solution for the DE system x ' = A x for the matrix A = <<0|4>,<-4|0>> (input by rows) and then the solution satisfying the initial conditions x(0) = <1,0>; (solution on-line: .pdf)
    Repeat the process for problem 7.3.11.
     
  44. F: handout on transition from a complex to a real basis of a linear DE solution space;
    handout on     1st order linear homogeneous DE systems (purely imaginary eigenvalues);
    using the eigenvector technique, find the general solution for the DE system x ' = A x for the matrix A = <<1|-5>,<1|-1>> (input by rows) and then the solution satisfying the initial conditions x(0) = <1,2>; note the solutions are easily obtained with dsolve (*check your hand solution this way!);
    finally express the component functions for this IVP solution in phase-shifted cosine form by hand.

    Check out the archived test 3 over the weekend.

    WEEK 12: week long take home test 3 starts Monday
  45. M: homework discussion, last minute questions for review, archived test 3 questions;
    Test 3 out, due back next Monday. Make a BIG push on it over next two days.
    Think of the this take-home test as an exercise in "writing intensive" technical expression. Try to impress me as though it were material for a job interview (you're fired! or you're hired! ?). In a real world technical job, you need to be able to write coherent technical reports that other people can follow.
     
  46. W: Continue working on the test.
  47. Th: Read 7.3 [especially compartmental analysis example 4, which has complex eigenvectors (more examples)];
    do 7.3: 37 [closed 3 tank system with oscillations, plug into Eq(22), solve by eigenvector method].
     
  48. F: Today 2:30-4pm Iron Math Competition at the MLRC! light refreshments! fun with math!;
    Read 7.1 except for examples 5-7 (opposite of reduction of order, not needed);
    handouts on reduction of order (optional handout on phase spaces);
    7.1: 1, 8 [first let   = [x1,x2,x3,x4] = [x,y,x',y'], then re-express the 2 DEs replacing x1" by x3' and x2" by x4', adding the definitions x1' = x3, x2' = x4, then write the 4 DEs in matrix form x ' = A x + F],
    7.2: 5, 9.
    These are very short problems, no solving required, just rewrites, to make sure you understand matrix notation.

    WEEK 13:
  49. M: Take home test 3 due any time today, including night, but don't exaggerate, we need to move on;
    note the higher order system dsolve template;
    handout on extending eigenvalue decoupling (to nonhomogeneous case, and second order);
    7.4: 3, 9 (undriven/driven 3 spring system)
    [also solve the IVP with inits: x(0) = 0, x'(0) = 0 ; 7.4.9 continues 7.4.3; can use direct substitution of x1 = c5 cos(3t), x2 = c6 cos(3t) into DEs to determine undetermined coefficients c5,c6 since there is no damping term to mix cosines and sines].
     
  50. W: 7.4: 3,9 continued: explore resonance by replacing the frequency 3 by ω and recomputing the particular solution and evaluating the amplitude functions for each variable (amplitude = absolute value of the cosine coefficient for a pure cosine term); find the values of ω where Ai(ω) go infinite (since there is no damping); these are the resonant frequencies.
     
  51. Th: re-express 7.4.3-9 in first order form and use Maple to find the eigenvalues and eigenvectors of the 4x4 coefficient matrix. Divide each eigenvector by its second entry to compare with the 2x2 matrix and its eigenvectors, and write out the general homogeneous solution by taking real and imaginary parts of the products of the complex exponentials and eigenvectors which are the complex solutions, just like we have done before with complex eigenvectors.
     
  52. F: Quiz 8 [Problem 7.4.9 with numbers changed: quiz format]; catch up with Maple assignments. If you need more practice on this, try 7.4.8 [Maple will provide you with the solution of the IVP directly to check.]


    WEEK 14:
  53. M: in class eigenquiz [.mw, .pdf];
    handout answer key to ungraded quiz 8;
    7.4.Application.1: earthquake problem part 1 with only 2 floors: A = <<-20|10>,<10,-10>> (general matrix with first and last rows only). Find response (particular solution) to this 2 floor building to the earthquake oscillation of part 4 and calculate the amplitudes for the two floor position variables.
     
  54. T (F!): 06S Final out for practice; handout on 2 fl earthquake problem [.mw]
     
  55. W(M): Answer key for take home test 3 back;
    handout available on-line for coupled mass-spring problems.
     
  56. Th: Final Class, CATS evaluations;
    The final exam will be weighted as a 4th test: it will consist of 1) a driven damped harmonic oscillator problem, 2) a first order homogenous linear system of 3 deqs, and 3) an undriven coupled spring-mass system (2 masses) to solve, with a couple little questions about characteristic time, frequencies, periods and phase shifted cosines thrown in [similar but a bit easier than 06S].

    MLRC 5:30 Problem session for those who cannot make the

    M: MLRC 7pm problem session.
    T: exam-01 10:30 class + 3 transfers in G90 at 1:30-4:00 [3 delayed to 4:15-6 (conflict)]
    W: exam-04 12:30 class in G86 at 10:45-1:15

I try to keep a class or so ahead with the homework for those of you who might anticipate?

Weeks 3 thru 4: come by and find me in my office, tell me how things are going. This is a required visit. Only takes 5 minutes or less.

*MAPLE homework log and instructions [asterisk "*" marked homework problems]

Test 1: ; MLRC problem session .
Test 2:  ; MLRC problem session .
Test 3: Take home out , in  ; MLRC problem session .
Final Exam: MLRC problem sessions Thursday May 3, 5:30; Monday May 7, 7pm

FINAL EXAM: if convenient, you may get permission to change section (email bob)
     10:30MWF/12:00Th: Tues, May 8 1:30 - 4:00
     12:30MWF/02:30Th: Wed, May 9 10:45 - 1:15

                          MAPLE and G.Calc. CHECKING ALLOWED FOR QUIZZES, EXAMS
 

3-may-2007 [course homepage] [log from last time taught]

extras