MAT2705 11S 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, 4 each normal week, numbered consecutively below and  labeled by the (first initial   of the) day of the week
  BUT weekday 2 of 4 is T/W for sections 03/04 while weekday 3 of 4 is W/Th for sections 03/04;
Friday will usually be the quiz day. Monday quiz makeup day.]

It is your responsibility to check homework here. (Put a favorite in your browser to the class homepage.) 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. M: Monday January 10. GETTING STARTED STUFF. By Wednesday, January 12, 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 03 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 [sophomores only:  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 (for example, Mat2500 = Calc 3 or Mat1505 = Calc 2).
    [In ALL email to me, include the string "mat2705" (no spaces) somewhere in the subject heading if you want me to read it. I filter my email.]

    In class log in to the computer at your desk:
    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 a local application, while in citrix Maple, you must save the file locally and then open in it MAPLE using the File Open task or using the Open URL selection.)
    2)     log on to the Villanova home page MyNova in Internet Explorer (click on the upper right "login" icon and use your standard VU email username and password):
    In the Student Tab, click on the double person icon to the right of our class line to get to our photo class roster and click on my home page URL under my photo. Click on our class URL there [or just put "Jantzen" in the VU home page search window, it is the first hit].
    [ http://www.homepage.villanova.edu/robert.jantzen/courses/mat2705/ ],
    3) Open     Maple 14 Standard (red not yellow icon) from the Start Menu Program listing under Math Applications
    [or click on this maple file link:     Maple plotting example file, solving DE example file].
    [Using the File OPEN URL command in Citrix Maple, with this URL (right click, copy shortcut) will show you a Maple file of how to pick the viewing window for realistic exponential and sinusoidal example functions.] Clicking on the !!! icon on the tool bar with execute this worksheet and restore the output which has been removed to make the file smaller. It will also automatically open the two subsections, which can be closed by clicking on the triangle at the section title.]

    4) bob will quickly show you the computer environment supporting our class. And chat up a bit the course. [cell phones on vibrate or off: bob will give you his cell phone number in class, not on the web!] [maybe he will try to impress you with this gee whizz! Maple demo; naahh...we'll leave this only to the curious among us.]

    Afterclass:
    5) log on     to My Nova and go to WebCT and look at the Grade book for our course: you will find all your Quiz, Test and Maple grades here during the semester once there is something to post.
     [This is the only part of WebCT we will use this semester.]
    In the Student Tab, click on the double person icon to the right of our class line to get to our photo class roster [look at the photo class roster to identify your neighbors in class! here is a much better photo roster: click here]
    and click on my home page URL under my photo. Click on our class URL there. Check out the on-line links describing aspects of the course (no need yet to look at the MAPLE stuff).
    [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) Download Maple 14 if you don't already have it and install it on your laptop when you get a chance (it takes about 15 minutes, I will help you in my office if you wish, MAC USERS can install in my office). If you have any trouble, email me with an explanation of the errors.
    [    You can always access and open Maple 14 Standard in citrixweb on any Windows computer anywhere through Internet Explorer but the first time you use citrix on a computer, you must download and install the client from the link on that web page, but the local copy of Maple is preferable.]
    You are expected to be able to use Maple on your laptop or Windows Maple on class computers when needed. We will develop the experience as we go.

    7) Read computer classroom /laptop etiquette.

    8) Homework Problems: 1.1: 3, 5, 13, 33 (this links to a PDF scan of the HW problems from the book if you have not yet purchased it);
    [memorize!: "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.
    Optional: get acquainted with Maple Standard DE entry and "odetest" for problems 3,5,13 [even 33!]

    9) Fill out the paper schedule form bob handed out in class. [see handouts];  use the 3 letter dorm abbreviations     to return in class the next class day.
    Proportionality statements must be converted to equalities with a constant of proportionality introduced:
    y is proportional to x:  y ∝ x  means y = k x . [y is a multiple of x]
    y     is inversely proportional to x:  y ∝ 1/x  means y = k/x .
    y     is inversely proportional to the square of x:  y ∝ 1/x2  means y = k/x2
     
  2. T/W: return your schedule forms at the beginning of class;
    1:1: 7a [i.e., only check y1(x)], 23 [see Maple plot];
    formulating DEs: 27,
    29 [Hint: recall perp lines have slopes which are negative reciprocals, the normal line is perpendicular to the tangent line and passes through the same point on the curve; make a diagram of the given point (0,1), a "generic" function graph curve whose normal at a point (x,y) on the curve passes through (0,1), and draw in the connecting normal line segment between these two points and the perpendicular tangent line, then compute the slope of the normal line from the two points, and then from the negative reciprocal of the derivative value, finally equate the two to get the DE],
    35, 36;
    read the first paper handout: algebra/calc background sheet [online only: more rules of algebra NOT!];
    on-line handout: initial data: what's the deal?
     
  3. W/Th:
    1.2  (antiderivatives as DEs): 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, see example), 39, 40 (ans: 2.4mi), 43 (pdf) [convert to appropriate units!].
     
  4. F: Bring your laptop for Friday class so I can introduce you to Maple, have it on after the quiz, go to our HW page
    {which means you should try to install Maple by class time if you can, otherwise you will have to look on with a neighbor.];
    Quiz 1 (checking a solution of a differential equation, imposing initial conditions,  formulating a DE from a rate of change condition; consult quiz archive;);
    Read the course info handout.
    Right click menu driven Maple intro for DEs explained step by step in class.

    1.3:
    3: [hand draw in all the curves on the full page paper printout supplied by bob]
    11, 15-16 [note Dyf is just df/dy above Eq (9) in the text]; 27 [don't worry about making a diagram: .mw];
    Maple HW not due till the week after we finish chapter 1, there is time next week to get a partner and ask bob questions about how to use Maple;
    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 [will not be due until we finish chapter 1].

    WEEK 2[-1]:
    M: Martin Luther King Day
     
  5. T/W:  1.4 ( separable DEs ) 1, 5; 21, 25, 27, 29.
    [old business: see 1.1.23 Maple plot above on day 2]

    Class list paper handout for forming partnerships.
     
  6. W/Th: handout on exponential behavior/ characteristic time [read this worksheet explicit plot example];
    1.4: 45 (if your cell phone were waterproof: "Can you hear me now?" attentuation of signal--characteristic length),
    47 [recall answer to 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 [CSI problem  (Newton's law of cooling): pdf, .mw please read worksheet to see the power of mathematics to solve all such problems at once;
    please read the PDF to see how to convert the word problem line by line to a mathematical problem];
    Optional challenge problems (more fun/interesting but safe to ignore):
    68 (PHYS: parametric cycloid solution, obvious typo in text: θ = 2 a should instead be  θ = 2 t) 
    or 69 (ME: suspension cable catenary: hyperbolic cosine)
    or Torricelli's law section and 64 (ChmE: ans r = 1/35 in).
     
  7. F: Quiz 2 on separable DEs and directionfields (see last semester quiz); bring your laptops to use to check your solution of the IVP;
    check WebCT/Blackboard gradebook for correctly entered quiz grade;
    online handout: 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, choose Solve DE, for y(x), or Solve DE interactively;
    rightclick "Simplify, Simplify" may be necessary to simplify the result    ]
    > y ' = x y , y(0)=1

    WEEK 3[-1]:  Mandatory visit to bob's office weeks 3,4; soon if you are finding difficulties!
    [Test 4 in week 4: W or F or M week5? give me feedback about preferred date]
  8. M: try to network to find 1 or 2 Maple partners (you can change after this assignment if you wish); maple 1 will be due week 4;
    how to understand the DE machine (example of switching variables);
    1.5: 17, 26 [ans: x = ½ y -2 +Cy -4],
    29 [use the fact that the derivative of the function defined by the integral is the integrand including all multiplicative factors (explanation of error function)],
    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].
    When you are satisfied, compare with retirement.pdf.

    TODAY 4:30: VU alumnus Sean Carroll talk The Origin of the Universe and the Arrow of Time at Nursing School Driscoll Hall 132.
     
  9. T/W->Th (2nd snow day)
    1.5    : 37 (Use Eq. 18 in the book; extra question: what is the final concentration of salt? How does it compare to the initial concentration and the incoming concentration?);
    [this mixing problem (online handout) 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, i.e., from skipped section 1.6), for example dy/dx = y/x is all three and can be solved in three different ways; don't solve them but:
    solve 25 [it can be done in two ways by expanding out the square on the RHS before integrating with integration constant C or by using a u-substitution as the book did with integration constant K; express K in terms of C as given in the book supplied answers by combining the two terms in the K solution [identity for expanding out (x+1)3 !] and then comparing with the C solution];
    solve 35 in two ways and compare the results    .
     
  10. W-Th: SNOW DAY!
    Print this sheet for problem 1.5.37 from the book and neatly solve the problem and answer the questions at the end with complete English sentences. Hand in during class Friday. This will help prepare you for the quiz, modeled after the last quiz in the archive (check it out).

    Quiz 2 answer key is on line now in the archive.
    Maple 1 is due anytime next week. Read submission instructions.
    Do you have a partner? If not ask bob in class to match you up.
     
  11. F: Quiz 3 on linear DEs; problem 1.5.37 is due today, or if not today for whatever reason, MONDAY.
    handout on solution of logistic DEQ [directionfield,  integral formula];
    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 [READ CAREFULLY: 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].

    WEEK 4[-1]: maple1.mw due with one or two partners, no exceptions, anytime this week
    any objection to test 1 next monday?
  12. M:  HW discussion for past week, come prepared to ask specific questions;
    handout on DE's that don't involve the ind var explicitly;
     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 formula],
    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 (solve this yourself or at least check numerically that it satisfies the 6 month condition) to find the limiting population as t goes to infinity (i.e., neglect decaying exponential)].

    Did you stop by bob's office yet?  TEST 1 Monday, February 7?
    5:30pm MLRC voluntary Test 1 problem session this Thursday?

    MAPLE:     To quickly plot some functions (and constants) together, use this template:
    >     plot( [f(t),g(t),2], t=-2..2, color=[red,blue,green])
     
  13. T/W:  air resistance handout;
    [optional reading to show what is possible: comparison of linear, quadratic cases; numerical solution for any power];
    2.3: 1, 2, 3,
    9 [remember weight is mg, so mg = 32000 lb determines m = 1000 in USA units, convert final speed to mph for interpretation!],
    22 [this is the tanh case: vterminal = 20.7 ft/s, t = 8 min 5 s][solution, .mw].
     
  14. W/Th: 2.4: 1 (do by hand and calculator, can check with euler.mw if you wish);
    optional read about improved Euler;
    maple2.mw: instructions for this second maple assignment. [due the week after Test 1 roughly; make sure you have at least one partner and follow the submission instructions].
     
  15. F: Quiz 4 on chapter 2 word problems;
    5:00pm voluntary MLRC problem session for Test 1;
    Read this: written homework returned;

    handout: why LinAlg with DE?; [maple dsolve and DEplot for 2x2 systems of DEQs];
    Read 3.1 on linear systems (review of high school topic solving 2 or 3 linear equations in 2 or 3 variables),
    do 3.1: 1, 5, 7; 9, 15, 31, 33, 34 [due after Test].


    WEEK 5[-1]:
  16. M: Test 1 on chapter 1 (sep DE, linear DE, exponential decay). You may use Maple and/or GCs, but you will need technology.

    Test 1 Answer key online now.
     
  17. T/W: handout on RREF (Reduced Row Echelon Form, section 3.3) [see bob's first 2x2 example];
    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 for the rest;
    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?]
     
  18. W/Th:
    maple2 is due by end of this weekend;
    handout on      solving linear systems example; we always want to do a full reduction, not the partial Gauss elimination reduction also described by the textbook;
    in class open this Maple file with a partner and use the LinearSolveTutor and enter the matrix for the system 3.2.15 given there and reduce step by step, then solve the system (if possible), then switch positions and enter the matrix for the system 3.2.18 given there and repeat using instead the Reduction command template there;
     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 row ops with MAPLE worksheet (here are some more examples rowredex0.mw); then check with ReducedRowEchelonForm or by right-clicking and choosing from the menu (solvers and forms)].

    To solve a linear system directly in Maple, input augmented matrix with the palette or Matrix command and these 4 lines will do it:
    > with(Student[LinearAlgebra]):   #[you can load this from Tools, Load Package,...]
       A:=Matrix    ([[3,8,7,20],[1,2,1,4],[2,7,9,23]]);  #[row by row entry, you enter matrix using palette instead]
       ReducedRowEchelonForm(A);  #    [you can right click reduced this: solvers and forms]
       BackwardSubstitute(%);
    To step by step reduce a matrix (max 5x6) and solve the system do (pick Gauss-Jordan):
    > LinearSolveTutor(A)
    Matrices for doing tonight's homework with Maple

    Control SpaceBar turns on command autocomplete on an input line. If you type
    > Back
    and hit this key combination it will give you a menu of choices to pick BackwardSubstitute (even I cannot remember this command past Back) which is the key command to solve a system once an augmented matrix is completely reduced, which you can do by a right click menu choice. "Red" will autocomplete to ReducedRowEchelonForm.

    You can bypass all of this by useing the linear solve tutor from the Tools Menu:
    Maple Tools Menu: Tutors, Linear Algebra, Linear System Solving: Gauss-Jordan elimination
     
  19. F: No quiz because test week.
    Quiz 5 not! [try this for HW without checking the answer key for good practice];
     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 after inputting the Student Linear Algebra package from the Tools menu or by using the line from the Maple template on the previous day 18];
    MAPLE chemical reaction problem* (what can you find out on the web about interpreting this chemical reaction?);
    word of the day: can you say "homogeneous"? [online handout only]

    WEEK 6[-1]: maple2 is past due.
  20. M: finally matrix multiplication!; [and a special day:   ]
    3.4: 1, 5, 7, 11, 13; 17, 21; 27, 43.
    Matrix algebra is easy in Maple.
     
  21. T/W: 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 and matrix inverse, determinant, transpose [or right-click and use Standard Operations menu]
     > A B [note space between symbols to imply multiplication]
     > A-1
     > |A|   [absolute value sign gives determinant]
         > A%T       [transpose, we don't need it]
     
  22. W/Th:  read on-line handout: determinants and area etc
    (to combine vectors we arrange them into rows of matrix, transpose maps columns to rows, rows back to columns);
    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];
    3.5. Application:  7* from textbook on p.201, using the inverse matrix as described on p.200,
    be sure to answer the word problem question with a text comment;
    And also the question: are these sandwiches cheap or not?
     
  23. F: all quiz answer keys now on-line---be sure to look at these after getting back your quizzes (remind me to post if not up a few class days later!);
    Quiz 5 on vector form of solution of a linear system, using inverse matrix to solve a square linear system (step by step technology row reduction required);
    maple3.mw assignment is complete; due next week or at least during the week after break if you have too much midterm stuff going on; 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: maple1 and maple2 only for midterm grades];
    try to meet to finish maple3 this weekend, make sure maple1 upgraded to 2/2 (I will try to return maple2 by Monday).

    WEEK 7[-1]:
  24. M: now we look at coefficient matrices as collections of columns, matrix multiplication on the right by a column of coefficients as taking linear combinations of those columns, and solving linear systems of equations as looking for relationships among them;
     4.1: 1; 5, 7; 9; 15, 17; 19, 23; 25 [use technology in both 1) evaluating the determinant  and 2) rightclick menu to the ReducedRowEchelonForm needed for these problems];
    [HWready row reductions for these problems];
    hand out on the interpretation of solving linear homogeneous systems of equations: A x = 0 [optional read: visualize the vectors from the handout]
     
  25. T/W: study the handout on solving linear systems revisited [remember this one: solving linear systems example];
     [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 solution to the system 3.4.21 on p.186 (back of book p.724): 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].
     
  26. W/Th: handout on linear combinations, forwards and backwards [maple to visualize];
    [optional worksheet to read: what does the solution of a nonhomogeneous system of equations say about the column vectors in the augmented matrix?]
    4.3:  1, 3, 5, 7; 9, 13; 17, 21; 23.

    Midterm grades up through Quiz 4 and Maple 2 [total of 4 for midterm Maple grade] now online.
    Check grade entries. Hope to get Quizzes 5 and 6 by Monday after break for midterm grades.
     
  27. F: Quiz 6;
     4.4: 1, 3, 5, 7;
    9 [this is already reduced, find solution as in class example, pull apart to find coefficient vectors of parameters, repeat for next 4 problems reducing first],
    13; 15, 17, 25 [use technology for all HW row reductions and determinants (but 2x2 dets are easy by hand!)].

    handout on nonstandard coordinates on R2 and R3 [goal: understand the jargon and how things work];
    1) Using the completed first page coordinate handout made for 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 find the old coordinates for the point whose new coordinates are  [2,2].
    2) Using the previous handout as a guide, first fill in the blanks on the sheet handed out in class for the basis transformation matrix and its inverse, and then for the change of coordinates, and then use either the matrix or scalar equations of the change of variables to calculate the other coordinates for the two given points.
    Then following the example, using 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 [x1,x2] = [-5,1] and find its new coordinates  [y1,y2] using the grid and confirm that this agrees with your previous calculation. Similarly read off the old coordinates [x1,x2] of the point whose new coordinates are [y1,y2] = [2,-1]. Confirm that these agree with your calculations. Put your name on it and bring it to class to hand in  next time. [You could also edit the Maple version of the example to see that your hand work is correct.]

    [Here is a solution scan.]

    Spring Break.      :-) Enjoy. Be safe in your travels. 

    WEEK 8[-1]:
  28. M: handout summarizing linear vocabulary;
    [maple illustration of how to describe a plane in space using a subspace basis].
    [optional on-line handout: linear system vocabulary]
     Part 2 of course begins: transition back to DEs:
    read the 4.7 subsection (p.279) on function spaces and then examples 3, 6, 7, 8;
    if you are curious, example 5 illustrates the partial fraction decomposition needed in engineering integration;
    read the worksheet on the vector space of quadratic functions [quadratics.mw, pdf];
    4.7: 15 [Hint: solve c1(1 + x) + c2(1 - x) + c3(1 - x2) = 0, for unknowns c1,c2,c3; are there nonzero solutions? if not, these are linearly independent 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, 17 [same approach].
     
  29. T/W: Memorize: y ' = k y  < -- > y = C e k x
       y '' + ω2 y  = 0 < -- >  y = C1 cos(ω x) + C2 sin(ω x)
    ["omega" ω is the angular frequency "radians per time unit" as opposed to just "frequency" f = ω/(2π ) or "revolutions or cycles per time unit" as in physics, but in our class we will just say "frequency" for ω, assumed to be expressed in radians per time unit or converted to revolutions per time unit as convenient    ; see also damped harmonic oscillators and RLC circuits]
    5.1[up thru page 295, plus Example 7]:     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))
    or right click on output of (use solve DE interactive, Solve Symbolically, Solve, Quit):
    > y'' + 3 y' + 2 y = 6 ex , y(0)  = 4, y'(0)  = 5
    or if x is not the independent variable, then use function notation:
    > y''(t) + 3 y'(t)  + 2 y(t) = 6 et , y(0) = 4, y'(0)  = 5
     
  30. W/Th: handout on sinusoidal example;
     5.1[after page 295]: 33, 35, 39,
    49 [find the IVP solution for y, set y ' = 0 and solve exactly for x, backsubstitute into y and simplify using rules of exponents]
    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 2nd order 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?].
     
  31. F: Quiz 7;
    read handout on complex arithmetic, exponentials [maple commands; the complex number i is uppercase I  in Maple];
    5.2:     13, 17, 21, 26 [this problem is for "fun" now, but later we learn a method to attack it];

    [optional: read the on-line handout on visualizing the initial value problem (IVP) if you wish to see the interpretation of solving the IVP]

    WEEK 9 [-1]: Test 2 Wednesday, voluntary MLRC problem session Monday 5:30pm TODAY!
  32. M: 3/14  Pi Day:  π!  [and Einstein's birthday]
     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, see handout on the amplitude and phase shift of sinusoidal functions [example, DE], 4 quadrant examples: mw, pdf).

    use technology to solve for roots of polynomial equations (or right-click on eqn output, Solve, Solve) :
    > solve(r2+6 r+13 = 0)    (in general, finds all roots exactly up to fourth degree and sometimes higher degree if lucky)
    [Note: > fsolve(...) returns all numerical roots of a polynomial].

     Managing Mental Health, Stress and Well Being on Campus: 3/14
    The Patricia B. and Gary M. Holloway University Counseling Center present "Managing Mental Health, Stress and Well Being on Campus: Challenges and Opportunities," by Richard Kadison, MD, chief, Mental Health Service, Harvard University Health Services, on March 14 at 3:30 p.m. in the Radnor-St. David's Room, Connelly Center. If you have questions, please contact: Joan.Whitney@Villanova.edu  For more info: click here.

     voluntary MLRC problem session Monday 5:30pm TODAY!
     
  33. T/W-> T/Th: 5.3: 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].
    handouts on     exponentially modulated sinusoidal functions [maple videos showing different phase shifts and frequencies];
    do the short problem at the end of the handout (ignore C and W) after reading it. [soln]

          Maple3.mw is long overdue for 13 students!
     
  34. W/Th-> W: Test 2 [chapters 3, 4]. Wednesday March 16.    
     
  35. F: No quiz;
     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)
    [on-line only: examples in nature].

    WEEK 10 [-1]:
  36. M: 5.4: 21 (solve by  hand),
    21* [solve the two DEs with the same initial conditions using Maple and plot the two solution curves together, but also plot the corresponding third solution with pure damping but no oscillator as in the template];
    handout on driven (nonhomogeneous) constant coeff linear DEs [IVP problem completion]
    [complete final exercise on handout sheet, Maple: *check this with dsolve;  pdf solution];
    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. T/W: handout on damped harmonic oscillator driven by sinusoidal driving function [in class example (pdf), maple resonance plots: general, specific];
    5.5: 33, 38 [y = ( exp(-x)(176 cos(x)+197 sin(x)) )/85 - (6 cos(3x)+7 sin(3x))/85]; [not many of the book driving functions are physically interesting here]
    5.6: 8. Ignore the book instructions and solve this problem by hand for the initial conditions x(0) = 0 = x'(0). Read the Maple worksheet about the interpretation of the solution.

    Check quiz, test grade entries in WebCT gradebook, report discrepencies, see current cum
    [U calculate your own Grade (check WebCT on-line grades)];
     Discuss Test 2 with bob if you did not do well.
     
  38. W/Th: [   > x, -x, x sin(x)    Enter and right click on output (choose Plot Builder, 2D Plot) to plot together to see oscillation and its envelope. or use bob's templates in each problem.]
    handout on beating and resonance;
    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 envelope function [and its period is 2(2 π)/(A-B)], look at the plot of x and +/- this amplitude function together, then repeat for the HW problem * at the end of the beating example worksheet (be sure to plot one full period of the envelope function to see 2 beats) and include only that HW problem section in your maple5 worksheet],
    11 [then convert steady state solution to phase shifted cosine at end of calculation], 17 [pdf], 23 [pdf].
     
  39. F: Quiz 8; catch up on HW (please take it seriously);
    [Tacoma Narrows Bridge collapse (resonance NOT!): engineering explanation; Wikipedia; Google (choose a You-Tube video, lowest bidder!)];
    Transition back to linear algebra:
    Fun.
    Watch the MIT Eigenvector 4 minute video [there are 5 frames which then repeat, so stop when you see it beginning again---from their LinAlg course]; see the 5 possibilities of the video in this Maple DEPlot directionfield phaseplot template;
    then play a computer game with the Duke U applet lining up the vectors [red is x, blue is A x, click on matrix entries to change, click on tip of red vector and drag around an approximate circle to see corresponding blue vector]; try first with the default values, then try for a11=0, a12=1, a21=1, a22=0, then try  for the matrices of 6.1: 1,2. See if you can guess the eigenvector directions and the corresponding eigenvalues (all integer triangles locating the vectors and integer eigenvalues); write down a simple representative eigenvector (with the smallest integer components, say) and its eigenvalue that you can read off from the applet as explained within the introductory webpage.

    maple5.mw due next week (5 problems now complete, last maple assignment)

    WEEK 11[-1]: maple5.mw due this week
  40. M:  recall previous handout coupled system of DEQs [motivation: direction fields for Maple (DEPlot directionfield) help visualize eigendirections of a 2x2 matrix]
    examples: A = <<0,1>,<1,0>>, <<2,0>,<1,3>>, <<1,2>,<2,4>>
    6.1: 1, 9     , 13, 19; 25, 27;
    [do everything by hand for a 2x2 matrix; go thru process: use Maple determinant: |A-λI| = 0, then solve to find characteristic equation and eigenvalues of a 3x3 or higher dim matrix, NO POLYNOMIAL DIVISION!].
     
  41. T/W: [now revisit coupled system of DEQs in a handout giving a preview of what we are about to embark on: why diagonalization?];
    For the matrix A = <<1|4>,<2|3>> entered by rows,
    from the two independent integer component eigenvectors {b1 = <1,1>,b2 = <-2,1>}, make the basis changing matrix B = <b1|b2> whose columns are these vectors, use the two coordinate transformations X = B Y and Y = B-1X to find the new coordinates of the point [x1, x2] = [-2,4] in the plane and to find the point whose new coordinates are [y1, y2] = [2,-1]; finally evaluate the matrix product by hand AB = B-1AB to see that it is diagonal and has the corresponding eigenvalues in order along the diagonal.[multiply b1 and b2 by A to see what multiple you get in each case---the eigenvalue];
    [solution: remember this? (now it is the geometry of diagonalization)];
    6.2: 1, 9 ("defective", see direction field plot); 13 (upper triangular!, det easy), 21 ("defective"); [use Maple for det and solve for eigenvalues, then by hand find eigenvectors]
    34 [just write out the quadratic characteristic eqn and solve with the quadratic formula, think about real distinct roots; Delta is the discriminant; what if b = c when the matrix is "symmetric"?].
     
  42. W/Th: handout on the geometry of diagonalization and first order linear homogeneous DE systems (2-d example: real eigenvalues [phaseplot]);
    1) Repeat this handout 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>;
    2)     solve the 3x3 system x' = A x for the matrix of problem 6.1.19, with initial condition x(0) = <2,1,2>;
    3) find the eigenvalues and matrix B of eigenvectors for the matrix  A = <<0|4>,<-4|0>> (input by rows) and verify that AD = B-1A B is diagonal.
     
  43. F: Quiz 9; catch up on HW, complete Maple5.mw by this weekend. Look at Quiz 8 answer key and 5.6. 17 [pdf].

    WEEK 12[-1]:
  44. M: handout on 1st order linear homogeneous DE systems (complex eigenvalues) (2-d example: [phaseplot]);
    handout on transition from a complex to a real basis of a linear DE solution space;
    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:  x ' = A x for the matrix A = <<1|-2>,<2|1>>, initial conditions x(0) = <0,4>
    [if you like you can quickly edit the DEPlot phaseplot template to reproduce the back of the book diagram for that problem].
     
  45. T/W: online handout on 1st order linear homogeneous DE systems (2d example:  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!);
    Read 7.3 [especially compartmental analysis example 4, which has complex eigenvectors (more examples)];
    do 7.3: 37 [closed 3 tank system with oscillations (explanation, text example .mw), plug into Eq(22), solve by eigenvector method: pdf, .mw].
     
  46. W/Th: Read 7.1 excluding examples 5-7 (these are the opposite of reduction of order, not needed);
    we should already understand the matrix approach to solving first order homogeneous linear systems, now we begin thinking about nonhomogeneous and then 2nd order systems;
    read handout on reduction of order with exercise and do exercise begun in class; [pdf solution, .mw];
    (optional on-line handout on phase spaces if you want to understand how this is connected to velocity and momentum spaces);
    the rest of the homework consists of very short problems, no solving required, just rewrites, to make sure you understand matrix notation:
    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],
    Note: substituting x = B y , leads to B y ' = A B y +F hence y ' = B-1A B y + B-1F so we can also solve nonhomogeneous linear DE systems in the same way, adding the new components B-1F of the driving vector function to the new decoupled DEs;
    7.2: 5, 9.
     
  47. F: summary handout on eigenvalue decoupling so far;
    handout on extending eigenvalue decoupling (to nonhomogeneous case, and second order) [example];
    solve x''(t) = A x(t) + F(t), where x(t) = <x1,x2>,
    for  A = <<-5/2,3/2>,<3/2,-5/2>>,  F(t) = 0, x(0) = <1,0>, x'(0) = <0,1>, make a hand plot of the eigenvectors and new coordinates y1,y2, the initial data position vector x(0)  and the velocity vector x'(0) with tail at the tip of the initial position vector; express the mode variables  y1(t), y2(t) in phase shifted cosine form so you can compare the amplitudes and phaseshifts of these two oscillations.
    Check your solutions with Maple by solving the corresponding scalar differential equations for x1(t), x2(t).
    The following two commands will show you the two relevant plots of your results if you are curious:
    > plot([x1(t), x2(t),t = 0..2π])  a nice figure 8 ish design!!
    > plot([x1(t), x2(t)],t = 0..2π, color = [red,blue])

    TODAY 5:30pm MLRC voluntary problem session for test 3.
    Quiz answer keys on line are now up to date.

    WEEK 13[-1]:
  48. M: HW solution .mw;
     In class start "take home" Test 3. Read the take home test instructions as part of the pledge signing step.
    You will need the DEPlot directionfield phaseplot template:
    here is a stripped down example for the direction field and solution plot versus t, all you need for problem 3.
     
  49. T/W: 2 mass 3 spring systems: 2mass-3spring-figure8curves.mw [pdf solution of day 47 hw];
    solve x''(t) = A x(t) + F(t), where x(t) = <x1,x2>,
    for A = <<0,4>,<4,0>>,  F(t) = <0,cos(t)>, x(0) = <1,0>, x'(0) = <0,0>.
     
  50. W/Th:  In class in teams of 2 or 3 start together the exercise with 2 spring, 2 mass system and gravity, easy numbers; complete for homework.
    Read 7.4 on coupled mass spring systems if you have time. Otherwise keep working on test 3;

    solution:      [.pdf, .mw] but since b1 is twice Maple's choice, my first components are half Maple's.
     
  51. F: T/W new exercise with driven 3 spring, 2 mass system, easy numbers, resonance. [pdf,mw]

    WEEK 14[+1]:
  52. M: Take Home Test 3 in.
    We explore resonance for the last class system [pdf, mw]
     
  53. Tuesday class for section 03 will be an optional office hour in the usual classroom,
    Final Thursday class for section 04 (April 28) will be an optional office hour in the usual classroom.
    You can ask me anything you wish about past final exams.
     
  54. W: in class 2 equal masses, 2 or 3 equal springs; solve and plot both the curve x(t) = <x1,x2> (with equal units on the axes) and x1 and x2 versus time; solutions are not periodic; show larger and larger time intervals: 50, 500, 5000
    > plot( [x1(t),x2(t)], t=0..50)   x1 and x2 together versus t
    > plot( [x1(t),x2(t), t=0..50])   x2 versus x1
    See worksheet with plots.

    The new default turn in date for test 2 is before you leave campus Wednesday, unless things are really hectic and you need till Tuesday after break
    Easter Recess: 
     
  55.  T[F]: Maple Amnesty Day: bring your laptops to help each other with upgrades of past due Maple worksheets;
    Know what you are heading for by hand calculations:
    > x1''(t) = -2 x1(t) + x2(t), x2''(t) = x1(t) -2 x2(t), x1(0) = 2, x2(0)=0, x1'(0) = 0, x2'(0)=0
    right click on output, Solve DE interactively, Solve Symbolically, Solve,...<wait till solution appears>, Quit.
     
  56. W[M]: Earthquake problem example [earthquake links];
    CATS evaluation forms.

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.
If you are at all confused, try to do this way before Test 1 in week 4.

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

Test 1: week 4
Test 2: week 9
Test 3: Take home out , in  week 13
Final Exam: MLRC problem session

FINAL EXAM:  [switching between these slots with permission]
     2705-03: MWF/T  11:30/12:00 class:   Wed, May 4 10:45 - 1:15
     2705-04: MWF/Th 12:30/1:00 class:     Wed, May 4 1:30 - 4:00

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

26-apr-2011 [course homepage] [log from last time taught]

extras, unused stuff