MAT2705 15S homework and daily class log

Your homework will appear here each day as it is assigned, including some PDF and/or Maple problem solution files or PDF notes, 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. 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 (January 12): GETTING STARTED STUFF. By Wednesday, January 14, 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]", telling about your last math courses, your comfort level with graphing calculators and computers and math itself, [for sophomores only] how much experience you have with Maple (and Mathcad if appropriate) so far, why you chose your major, etc, anything you want to let me know about yourself. Tell me what your previous math course was named (if at VU: Mat1500 = Calc 1, Mat1505 = Calc 2, Mat2500 = Calc3).
   HINT: Just reply to the welcome email I sent you before classes started.
   [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.]
   
   DURING CLASS (THIS IS THE FIRST DAY PLAY PART)
   On your laptop if you brought it:
   1) Open Internet Explorer. (IE allows you to open Maple files linked to web pages automatically if Maple is installed on your computer.)
   2) Log in to MyNova on 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 class photo roster, and visit the link to my course homepage from it by clicking on my home page URL under my photo and then on our class homepage, directly:
   [ http://www.homepage.villanova.edu/robert.jantzen/courses/mat2705/ ],
   3) Open Maple 17 or 18 if you already have it
   [or click on this maple file link: firstdayplay.mw] bob will discuss a few things to remember the rest of the course: firstday2015.pdf [firstday2015.mw]
   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] [maybe he will try to impress you with this gee whizz! Maple demo; naahh...we'll leave this only to the curious among us.]
   
   AFTER CLASS (THIS IS THE HOMEWORK)
   5) log on to My Nova, choose the student tab, and go to BlackBoard 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 BlackBoard 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!]
   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 18 for Windows or Mac if you haven't already done so 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). If you have any trouble, email me with an explanation of the errors.
   [You can always access and open Maple 18 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, especially for printing purposes.]
   You are expected to be able to use Maple on your laptop when needed. We will develop the experience as we go. Maple 17 lacks some key right click menu choices for matrix manipulation.
   
   7) Read computer classroom /laptop etiquette and check out the university academic integrity site.
   
   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 clickable calculus 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: return your schedule forms at the beginning of class;
   1:1: 7a [i.e., only check y₁(x)], 23 [see Maple plot (execute worksheet by clicking on the !!! icon on the toolbar)];
   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:  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!].
   
   Thursday Inquirer TechLife column: wireless power transmission relies on resonance (chapter 5).
    
4. F: Read the course info handout. Office hours are set;
   
   Quiz 1 through 1.1 (see quiz archive  for an idea of what to expect);
   1.3: 3: [hand draw in all the curves on the full page paper printout supplied by bob: projectable image]
   11, 15-16 [note D_yf is just df/dy above Eq (9) in the text]; 27 [don't worry about making a diagram, just view this worksheet: .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 [more][his own words]
    
5. T: Kahn Academy link added to our class home page;
   classlist contact information will be distributed in paper handout not reproduced on the web (privacy concerns). Find one or two Maple partners for the first assignment, you can change partners later if you wish.
   1.4:  1.4 ( separable DEs ) 1, 5; 21, 25, 27, 29
   [ignorable unless you are curious: example 2: technology hint,  other complications visualized].
   
   
6. W: handout on exponential behavior/ characteristic time [read this worksheet explicit plot example, execute worksheet first with !!! icon on toolbar];
    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): after you solve this problem yourself, consult both [pdf, .mw];
     please read Maple 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 problem 64 (ChmE: ans r = 1/35 in). <<<<< ChE majors!
   
   
7. F: Check BlackBoard grades for Quiz 1; read answer key;
   Quiz 2 [you must use Maple to check your separable DE IVP solution, see archived online 14F quiz 2 or: check your IVP solution];
   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" or "Simplify, Symbolic" (with radicals) may be necessary to simplify the result]
   > y ' = x y , y(0)=1
   Use function notation to change independent variable:
   > y '(t) = t y(t) , y(0) = 1
   
   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.
   If you are at all confused, try to do this way before Test 1 in week 4. Time is running out.
8. M:  online handout: 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 (antiderivatives as definite integral functions, more 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 the hand written solution retirement.pdf (don't submit a Maple worksheet without consulting this to be sure your text comments are correct).
   
   Maple1.mw hw due any time next week including the weekend, but preferably be familiar with using Maple to check your work before the Tues Test (read Maple HW page).
   
   
9. T: 1.5 (tank mixing problems): 37 (Use Eq. 18 in the book; extra questions: 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.
   
   Quiz 2 answer key online.
   
   
10. W: handout on solution of logistic DEQ [directionfield,  integral formula, characteristic time];
    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 B_o = kMP_o and  D_o = kP_o² , and these are easily solved for M and kM in terms of B_o, D_o, and P_o to get the other two logistic curve parameters  M and kM needed to use the solution formula to solve problem 16],
    16 [ans: P = .95 M after 27.69 -> 27.7 -> 28 months].
    
    
11. F: Quiz 3 on linear soln technique; check BlackBoard grades against paper grades;
    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 β_o = 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)].
    
    WEEK 4[-1]: Maple1.mw due any time this week through the weekend. [you must have a partner!]
12. M: voluntary Test 1 problem session 5:30 in MLRC;
    air resistance handout (example of a piecewise defined solution and the importance of dimensionless variables);
    [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: v_terminal = 20.7 ft/s, t = 8 min 5 s]. [soln, .mw].
    
    
13. T: Test 1
    
    
14. W: 2.4: 1 (do by hand and calculator, can check with euler.mw if you wish: soln);
    optional: read about improved Euler in textbook;
    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].
    
    read this handout on DE's that don't involve the ind var explicitly.
    
    
15. F: no quiz on test weeks;
    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.
    
    WEEK 5[-1]:   maple2 is due by end of this weekend
16. M: check Blackboard to match your grades with those on your quiz/tests;
    check out the quiz 3, test 1 answer keys online; come talk to bob if you did not do well;
    handout on RREF (Reduced Row Echelon Form, section 3.3) [see bob's first 2x2 example,example 3];
    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) ;
    [Matrices for doing some of tonight's homework with Maple preloaded, will be explained next time];
    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?]
    
    To be explained next class:
    Maple 18 (not 17): Right click on augmented matrix entered with the Matrix palette, choose Solvers and Forms, then Row Echelon Form, then Reduced to get the RREF form of the matrix, then right click on new matrix and choose Solvers and Forms, then Linear Solve, and you get the RHS of the solution with no LHS variables assigned to these expressions because Maple has no clue what variable names you wish to use.
    See Example 3: row ops with MAPLE.
    
    
17. T: Bring laptops to class for hands on row reduction tutor experience;
    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 bob as 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;
    
    You can get to the Linear Solve Tutor from the Tools Menu:
    Maple Tools Menu: Tutors, Linear Algebra, Linear System Solving: Gauss-Jordan elimination
    
    3.3: 1, 3, 7, 11, 17,  23 = 3.2.13 (refers back to system in 3.2 HW) [Matrices for doing tonight's homework with Maple preloaded]
    19* [use the tutor and record your step by step reduction, annotating each step as in the row ops with MAPLE worksheet; then check with ReducedRowEchelonForm or by right-clicking and choosing from the menu (solvers and forms)].
    
    
18. W: 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 right click RREF of the augmented matrix, then right click Linear Solve;
    MAPLE chemical reaction problem* [pdf] (what can you find out on the web about interpreting this chemical reaction? chem site reaction balancer gives balanced reactions but no description, this one of our exercise turns out to be interesting);
    
    word of the day (semester really): can you say "homogeneous"? [online handout only]
    Get used to this word, it will be used the rest of the semester. It refers to any linear equation not containing any terms which do not have either the unknowns or any of their derivatives present, i.e., if in standard form with all the terms involving only the unknowns on the left hand side of the linear equation, the right hand side is ZERO. [nonhomogeneous means the RHS is nonzero]
    
    
    
19. F:  Quiz 4 on row reduction and solving a linear system [archived Quiz];
    finally matrix multiplication! [handout];
    3.4: 1, 5, 7, 11, 13; 17, 21; 27, 43.
    Matrix algebra is easy in Maple [see here for how to do Matrix stuff in Maple]. <<<<<<<<<<<
    
    WEEK 6[-1]: still not too late to come visit bob in his office! [especially those who did not do well on test 1]
20. M: handout example for matrix inverse algorithm;
    3.5: 1; 9, 11; 23 [this is a way of solving 3 linear systems with same coefficient matrix simultaneously (see matrix multiplication handout)];
     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, or just right click]
    Memorize: . Switch diagonal entries. Change sign off-diagonal entries. Divide by determinant.
    Always check inverse in Maple if you are not good at remembering this.
    
    
21. T: Snow Day!
    
    optional on-line handout: determinants and area etc ;
    [optional: why the transpose? (to combine vectors we arrange them into rows of matrix, the transpose maps columns to rows, rows back to columns)];
    Read explanation of why we need determinants;
    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?
    [this is the final problem of maple3.mw. make sure you find a partner or join a 2 partner group for this assignment]
    
    
    
22. W: linear independence of a set of vectors;
    now we look at linear system coefficient matrices A as collections of columns A = < C₁| ... |C_n >,
    matrix multiplication on the right by a column of coefficients as taking linear combinations of those columns,
    A x = x₁ C₁ + ... + x_n C_n
    and solving homogeneous linear systems of equations  A x = 0  as looking for linear relationships among those columns (nonzero solns=linear rels) ;
    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].
    
    
    
23. F: Quiz 5;
    vector spaces and subspaces;
    study the handout on solving linear systems revisited [remember the original one: solving linear systems example];
    [note only a set of vectors x resulting from the solution of 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.
    
    WEEK 7[-1]: maple3.mw due this week; make sure you have 2/2 on first two assignments;
24. M: book catches up with us: linear independence (A x = 0), express a vector as a linear combination (A x = b); span of a set of vectors;
    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.
    
    
    
25. T: bases of vector spaces, subspaces [optional: pdf,  calculate  and visualize];
    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]
    [in class practice on blank grid [ .mw, .pdf] with b₁= <3,-1>,  b₂= <2,2> ; what are new coords of <-2,6>? what point has new coords <2,-1>?] ;
    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 corresponding blank worksheet handed out in class for a new basis transformation matrix and its inverse given on that sheet, 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. [Let the larger grid squares of a centimeter determine your unit tickmarks!]
    Then following the completed 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 class. [If you are at all confused, here is a completed worksheet for the in class practice example above.]
    [If you are feeling ambitious, you could also edit the Maple version of the example to see that your hand work is correct.]
    
    
26. W: handout summarizing linear vocabulary for sets of vectors;  
    [optional on-line handout: linear system vocabulary for linear systems of equations ]
    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 (at most) 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].
    
    
27. F: Quiz 6;
    
    Spring Break. enjoy and be safe.
    
    WEEK 8[-1]: midterm grades Maple thru Maple2.mw (see Blackboard); upgrades possible thru Monday; spreadsheet calculation of letter grade average;
    late quiz 6 takers see me today? don't forget to submit Maple3.mw
28. M: 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, and Hertz, not just a rental car company but "cycles per second"]
    5.1[up thru page 295, plus Example 7]: 1, 3, 5, 9; 13, 17.
    
    
29. T: 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]);     (let a = 0 here to plot only x > 0)
    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. W: Th: 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]
    required: look at the Quiz 6 answer key online.
    
    
31. F: Quiz 7; complex roots: handout on the amplitude and phase shift of sinusoidal functions and DE example [.mw];
          extras: [shorter presentation: pdf], 4 quadrant amplitude-phase shift examples: [mw, pdf].
    5.3: 8 [ans: y =  exp(3x) (c1 cos(2x) + c2 sin(2x))] , 9; 17; 22,
    23 (express in phase-shifted cosine form, see handout).
    
    Saturday is a very special Pi Day! 3.14.15 [also Einstein's birthday!]
    
    WEEK 9[-1]:
32. M: MLRC 6:00pm voluntary problem session for Test 2 [note delayed time!];
    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].
    handout on exponentially modulated sinusoidal functions [maple videos showing different phase shifts and frequencies];
    do the short problem at the end of the handout (ignore complex C and W unless you are EE or physics oriented) after reading it. [soln: pdf, mw].
    
    
33. T: Test 2 on chapters 3,4.
    
    
34. W: 5.4: 1, 3 [use meter units!], 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] (this is a useful application problem, please study the solution carefully after trying to solve it yourself);
    handout on linear homogeneous 2nd order DEQ with constant positive coefficients (damped harmonic oscillator)
    [on-line only: examples in nature].
    
    
35. F: no quiz; check answer key for Quiz 7 (use Maple to check solution even if not requested!);
    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
    [complete final exercise on this handout sheet, Maple: *check this with dsolve, (pdf solution; the algebra works if you are careful!)];
    5.5: 1, 3, 9 [see PDF solution after you try this yourself:  version 1, version 2]
    [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].
    
    ignore this: EE/Physics majors if you are interested: RLC circuit application: RLC circuits [maple plots]
    
    WEEK 10[-1]: Check Blackboard grades (and answer keys) thru Test 3, Quiz 7, Maple 3;
36. M: catchup day. in class: 5.4,5.5 HW discussion. After class, go back and do those HW problems that you did not get to.
    For second order DEs, if a RHS driving function term satisfies the homogeneous DE, multiply the corresponding trial function (the most general solution of the homogeneous DE for which that term is a particular solution) by the independent variable (repeat if the new trial function still satisfies the homogeneous DE).
     [not many of the book driving functions are physically interesting here in this exercise set]
    Do a new version of 5.5.9 : a slight change to this problem to make it physically interesting describing an overdamped spring system at rest to which an impulse force is applied, from which one can calculate the maximum displacement which occurs: y''+3y'+2y = x exp(-x), y(0) = 0 = y'(0). Find the maximum of y(x) for y≥0.
    [solution: part1, part2]
    5.5: 33, 38 [y = ( exp(-x)(176 cos(x)+197 sin(x)) )/85 - (6 cos(3x)+7 sin(3x))/85].
    
    
37. T: handout on damped harmonic oscillator driven by sinusoidal driving function [1 sheet printable version]
    [in class example.mw pdf: resonance calculation side by side], maple resonance plots: general, specific];
    For the curious, Wiki: driven harmonic motion, amplitude plots: they introduce ζ = 1/(2Q).
    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; hand solution];
    15, 17. Follow the handout procedure as in this particular case, find the amplitude function of the frequency and plot it to compare with the back of the book. Evaluate the natural frequency and Q value for each. [soln: pdf, mw; 17more-pdf]
    
    Oscar Romero Solidarity Lecture-Immigration and Solidarity 7pm
    
    
38. W: handout on beating and resonance [includes resonance calculation for previous day example];
    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 T = 2(2 π)/(A-B)], look at the plot of x and +/- this amplitude function (the envelope) together in this beating example worksheet, then do:
    5.6.1*: as explained here, repeat for the HW problem at the end of this 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];
     23 (earthquake!): consistent units: sec, ft, lbs; first evaluate Hookes' law constant, then find x''+10 x =  A₀ ω² sin(ω t), convert frequency to ω = 2 π /2.25 radians/s, set A₀  (=3in) = 1/4 ft. Solve. Find amplitude of response oscillation in inches. [soln]
    
    Maple plotting of multiple functions
    > 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. We need this for envelopes!
    
    
    
39. F: Quiz 8:  linear homogeneous constant coefficient 2nd order DEs: complex root solutions, identifying natural decay rate/time, natural frequency and period, Q factor,  solve IVP (like HW problem 5.4.14);
     catch up on Maple5.mw homework due any time next week. Read solution for 5.6.23;
    optional Quiz 8b due Monday: if you want it to be graded to take the place of your second lowest quiz grade, write "Grade this" in the upper right corner. If not I will just check it over to make sure you understand what you are doing.
    
    If you are curious watch the Google linked video of resonance NOT! (but see the engineering explanation linked PDF for the detailed explanation):
    [Tacoma Narrows Bridge collapse (resonance NOT!): Wikipedia; Google (choose a You-Tube video)].
    
    WEEK 11[+1]: maple5.mw due anytime this week (which extends to after Easter break)
40. M: 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 worksheet 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: 1,2,9. See if you can guess the eigenvector directions and the corresponding eigenvalues (all integer triangles locating the vectors and integer eigenvalues)  for these matrices; 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 to compare with your matrix calculations for these textbook HW problems . Then find them by the eigenvector process.
    
    6.1: 1, 2, 9 [1, 9 mw].  Ignore polynomial division discussion, use technology for roots of polynomials!
    
    Recall old handout coupled system of DEQs and its directionfield [motivation: direction fields for Maple help visualize eigendirections of a 2x2 matrix]
    matrix examples: A = <<0,1>,<1,0>> [done]; <<2,0>,<1,3>>; <<1,2>,<2,4>>.
    
    
41. T: eigenvector analysis: 3x3 example (repeated eigenvalues:  <<9, -6, 6> | <4, -1, 4> | <0, 0, 3>>);
    we begin HW in class:
    6.1: 13, 19; 21, 25 (upper triangular so diagonal values are eigenvalues!),
    27 (complex!);
    [do everything by hand for 2x2 matrices; for 3x3 or higher, go thru process: use Maple determinant: |A-λI| = 0 (right click on matrix, Standard Operations), then solve to find characteristic equation and its solutions, the eigenvalues, back sub them into the matrix equations (A-λI) x = 0 and solve by rref, backsub, read off eigenvector basis of soln space, NO POLYNOMIAL DIVISION!].
    
    
42. W: diagonalization [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^-1 x to find the new coordinates of the point <x₁, x_2> = <-2,4> in the plane and to find the point whose new coordinates are <y₁, y_2> = <2,-1>; finally evaluate the matrix product by hand A_B = 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" and what kind of special matrix is this if Delta is zero?].
    
    Quiz 8, 8b answer keys online: check them out.
    
    I also explained that for complex conjugate eigenvalues which always come in complex conjugate pairs, you may always choose the eigenvector corresponding to the complex conjugage eigenvalue to be the complex conjugate of the original eigenvector. And why the set collecting all of the bases of eigenvectors for each individual eigenvalue is a linearly independent set.
    
    Easter Break.   Maple5.mw due as soon as possible
43. T: 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>; [make a rough drawing of the new axes and the initial data decomposition along the eigenvector directions, or use the phaseplot worksheet to make the diagram];
    2) solve the 3x3 system x' = A x for the matrix of problem 6.1.19, with initial condition x(0) = <2,1,2> (this worksheet shows how the Eigenvectors Tutor works);
    3) find the eigenvalues and matrix B of eigenvectors for the matrix  A = <<0|4>,<-4|0>> (input by rows) and verify that A_D = B^-1A B is diagonal.
    
    
44. W: 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 [4 page double handout print];
    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. F: Quiz 9 [solve DE IVP with real eigenvalues, starting from Maple Eigenvalues];
    Quiz 10-NOT! for HW [solve DE IVP with complex eigenvalues, plot solution, print out and attach to quiz; same deal as last time, if you decide to have this graded, it can take the place of the next lowest quiz grade that has not yet been deleted].
    handout on 1st order linear homogeneous DE systems (2d example:  purely imaginary eigenvalues);
    optional:
    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.
    
    WEEK 12[+1]:
46. M: Read 7.3
    optional read with many extras, only for the curious:
    [compartmental analysis: real eigenvalues: example 2 (open case+comparison with closed case: real/complex eigenvalues) ,
     complex eigenvalues: example 4 (closed case);(more examples 7.3.35,36)];
    do 7.3: 37 [closed 3 tank system with oscillations (explanation, text example4.mw), plug into Eq(22), solve by eigenvector method] [soln: .mw, pdf];
    choose one problem from 7.3: 27-30. [soln: mw]
    
    for complex eigenvector matrices that do not come out in "rationalized" form but involve a quotient of complex numbers, right click on matrix, select "map command onto" and insert the command "evalc" to evaluate to a complex number, expressing each entry its real part plus i times its imaginary part.
    
    
47. T: summary handout on eigenvalue decoupling so far;
    handout on extending eigenvalue decoupling (to nonhomogeneous case, and second order);
    online examples handout [.mw] (lower rightmost diagram incorrect!); go over this handout to see how we can easily extend our present tool to cover the nonhomogeneous case, and when we have a second derivative instead of a first derivative;
    solve x''(t) = A x(t) + F(t), where x(t) = <x₁(t),x₂(t)>,
    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).
    Optional. The following two commands will show you the two relevant plots of your results if you are curious (replace unknown variables by their expressions that you find):
    > 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]) .
    
    MLRC voluntary test problem session 5:30
    
    Quiz 8b, Quiz 9, Quiz 10 answer keys online now.
    
    
48. W:  Take home Test 3 will be linked here; please read the short instructions on the test and the long instructions online; this is not a collaborative effort; start in class;
    here is a stripped down example for the direction field and solution plot versus t, all you need for problem 3.
    
49. F: Work on Test 3 again.
    
    WEEK 13[+1]:
50. M: in class: do  day 47 HW exercise (following this handout example), then repeat together with driving function: F(t) = < 0,50 cos(3t) > ;
    [interpretation coming soon---this represents an equivalent 2 mass 3 spring system: beautiful Maple animations,   and the JPG movie.]
    HW: continue working on Test 3.
    
    
51. T: 2 mass spring systems: theory plus worked examples; 
       [days 47, 50 exercise description: drivenDEsystemExercise.pdf, solution: figure8curve12.pdf [included in previous line PDF]
         plot: figure8curvemodesplot.mw, figure8curvemodesplot.pdf ];
    7.4: 3, 9 [same system, with/without driving term, same procedure as above, namely: driven 3 spring, 2 mass system, easy numbers, resonance (parts a-f).].
    
    Maple grades posted in Blackboard, let me know if your total does not agree: 8/8 full credit for 4x2 assignments.
    
    
    
52. W: HW 7.4:3,9 do parts a)-c) then skip homogeneous soln and do only g) in class without initial conditions to find only the response solution by the method of undetermined coefficients: but consider resonance by replacing the driving frequency 3 by omega and only find the particular (response) solution; check solution for omega = 3 with back of book formula.
    [pdf resonance only hand solution, maple plot]
    complete problem solution: pdf (no resonance), maple (resonance too)
    
    
53. F: Tentative Test 3 due back date (extension by email request possible);
    back to work:  handout on reduction of order with exercise [mixed order DE system example: pdf, mw]
    [in book you can read 7.1 First Order Systems, examples 3,4 (examples 5-7 are the opposite of reduction of order, ignore)];
    no homework.
    
    WEEK 14:
54. M: damped coupled oscillators without reduction of order exercise: we derive by hand in class the solution to the decoupled 2nd order equations [pdf exercise];
     Solution: [pdf, mw; note natural frequencies 1, 1.73 slow down to 0.87, 1.66, while transient time constant is 2 so a decay window is 0..10 for the transient]
    
    
55. T (=F): archived final exam.
    
    
56. W(=M):  MLRC voluntary problem session for final exam: Thursday, Friday? preferences?
    animations of damped figure8 system: figure8curve-reduction.mw; aperiodic example: 2mass2springaperiodic.mw
    CATS evaluation forms.
    
    Friday May 1, 3pm voluntary MLRC problem session.