MAT1505 21F homework and daily class log

Jump to current date! [where @ is located]

Your homework will appear here each day as it is assigned, with occasional links to some MAPLE worksheets when helpful to illustrate some points where technology can be useful. [There are 56 class days in the semester, numbered consecutively below and  labeled by the (first initial of the) day of the week.]

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). WebAssign deadlines are at 11:59pm of the day they are due, allowing you to complete problems you have trouble with after class discussion. [If Friday is quiz day, HW due Friday has a WebAssign deadline of midnight Monday.] Red numbered problems are a relic of the old online HW hints that are now superceded by the e-book addons.

" *" asterisk marked problems are to be done with MAPLE as explained in the separate but still tentative MAPLE homework log, which will be edited as we go.

Textbook technology: WebAssign homework management/grading is required, giving you access to an incredible wealth of multimedia tools together with the online e-book textbook you can access from any internet connection. WebAssign deadlines are suggested to keep students on track, but extensions in time and number of tries are always granted.
Textbook problems
are labeled by chapter and section and problem number, and identify each WebAssign problem. Those problems which are not available in WebAssign will be square bracketed and should be done outside of WebAssign. Check this page for hints and some linked Maple worksheet solutions. Use the Ask Your Instructor tool to get help on any problem for which you cannot get the correct answer. There is no reason for anyone not to get 100 percent credit for the homework assignments. This is the most important component of learning in this class, doing problems to digest the ideas.

If you have any questions, drop by my office St Aug 370 (third floor, Mendel side, by side stairwell) or just come to see where you can find me in the future when you need to; I welcome visitors.

  1. GETTING STARTED STUFF Monday August 23.
    In class open your favorite browser and log in to MyNova at http://mynova.villanova.edu/ (use your standard VU email username and password) and check out our class photo roster (search for Class Schedule with Photos), and visit my course homepage
    [ http://www.homepage.villanova.edu/robert.jantzen/courses/mat1505/ ],
    and bob will give you and overview of the class materials.
    (No need yet to look at the MAPLE examples and tips yet: we will get around to Maple soon.

    Lecture Notes 5.3 on reviewing diff/int notation; [Exploreplot.mw]

    Homework (light first day assignment):
    Make sure you read my welcoming email sent on the weekend, and register with WebAssign.
    [If not already done: download Maple here. Then install it.]
    Watch this short video about New Users of Maple.
    Explore the on-line resources. Browse the pages linked to our class home page.
    Read this online handout on functions and differentiation to see what background is assumed.

    Read 5.3. Do 5.3: 7, 9 using WebAssign, and the "Getting Started with WebAssign" assignment. Notice the "Ask Your Instructor" button!

    Read what Villanova students say about the most important things you can do to succeed here.

    By Wednesday August 25, reply to my welcoming email with the subject heading "[MAT1505-00?] <cell#> <nickname> <dormabbreviation>
    where <cell#> is your cell phone number (I will give you mine in class),
    where <nickname> is the first name other than your legal first name by which you wish to be addressed (leave blank if legal name ok),
    where <dormabbreviation> is from the list 3 letter dorm abbreviations (OFF for  offcampus).

     [always include the string [MAT1505] in your subject heading if don't want your email to be lost in my overflowing mailbox]
    telling about your last math courses and how far you reached in your high school calc class (freshmen), your comfort level with graphing calculators and computers and math itself, why you chose your major, and anything about yourself for me to get to know you better, etc.
    Please attach your class schedule from MyNova saved to PDF, or as an image file, or perhaps a screen shot saved to JPG, and name the file LastName-FirstName.extension so I can easily extract them to a folder on my laptop.

  2. W: Lecture Notes 5.4 indefinite integration;
    Think about this online handout on algebra and differentiation/integration rules (and: expanded diff rules);
    [Note Maple does not add the constant of integration when evaluating an indefinite integral, leaving you to choose its symbol.]
    Velocity example of net change and the fundamental theorem of calculus: s5-4-60.mw;
    5.4: 1, 5, 31, 33, 45[hint: separate into two integrals], 50, 52, 59,
    tabular data: 65 (units! hrs, secs);

    if you have a moment for reflection read:
    differentiation versus integration: notation and associated pictures [just a summary of day 1 lecture]

  3. Th: bring your laptops today, with Maple installed; bob will help you adjust your palettes and show you the interface.
    Here is the worksheet bob used in class to illustrate document versus worksheet mode in Maple:
    document mode, worksheet mode;
    Then open up this file: datafitint.mw and read it with a partner, then do the final section exercise with the date from Stewart 5.4.74* on a shuttle liftoff height calculation (convert to mile units at the end). Ask bob when you are confused about anything. Finish this as part of a first Maple assignment. Find a partner or two eventually to join with you in consolidating your work into a single group worksheet and do other problems from chapter 5 (class contact info will be available after drop and add);


  4. F: Quiz 1 on 5.3-4 (see archive); use Adobe scan to upload your work;
    Lecture Notes 5.5a change variable in integral;
    handout: "u-substitution";
    chain rule for scaled integration variable in a function;
    5.5 ("u-substitution"): 1, 3, 7, 13, 19, 21, 25, 45 [hint: separate into two integrals], 59 [express as a definite integral in u];

    WEEK 2: Read Purchase WebAssign Access choices!
  5. M: Lecture Notes 5.5b symmetry and transforming definite integrals;
     5.5:  81, 83, dimensionless variable discussion: 85 (read!), 86, 87, 91;
    square bracketed problems on this HW page are not on WebAssign;
    [WebAssign HW problem numerical parameters online change so you can't just enter the answer from the back of the book for odd numbered problems!]


  6. W: class syllabus with office hours;
    paper only classlist contact information handout (you don't want your cell numbers on the web!);
    5.R. Review problems [no review problems are on WebAssign]: 4, 7, 15, 23, 31, 39, 47, 53, 58, 59, 61, 63,
    66 [Maple gives you the value of the limit, your task is to derive it using the fundamental theorem of calculus; the worksheet explains why this limit is interesting for an optional read];

    64* [Maple problem, use template; forget parts a,b, just graph C, then C, S together on interval -10..10, solve the condition c) for the first positive solution showing the graph of C together with the constant function 0.7 on the interval 0..1 which contains it (click on gridlines) since there are an infinite number,The Maple5.mw assignment is due anytime next week through the weekend. You can use this template: lastname-lastname-maple5.mw to get started ]; 18* [Use Maple to evaluate this definite integral in a second section in your worksheet, just to prove you can enter a definite integral and evaluate it.]


  7. Th: Lecture Notes 6.1 on Areas between Curves;
     6.1: 1. 3. 9, 11, 15, 27, 31, 33, 49; 53 [see this approach]; numerical root necessary?;
    Income inequality is a serious problem today, made painfully clear by COVID.
    Read Applied Project "The Gini Index", the applied project following section 6.1 (inserted as a result of the lobbying of our very own Math&Stat Dept prof. Klaus Volpert) which is important in quantifying income and wealth inequality.
    [If curious checkout this Gini index calculus application only using calculus you already know.]


  8. F: Lecture Notes 6.2 on Volumes (of Revolution, etc); [elliptical version of example 7: 55]
    Quiz 2  on u-substitution;
    6.2: 1, 7, 9, 11, 33, 41, 47, 49. 


    WEEK 3[-1]: Monday is Labor Day!
  9. W: in class group work:
    6.2:  48 [with bob hint], we then work on these WebAssign setups:
    HW:
     53 diagram: [this called a trirectangular tetrahedron; use horizontal plane cross-sections] ,
    55 [recall sides of isosceles right triangle are hypotenuse divided by sqrt(2), Maple solution and plots]
    ,
    57, 61.
    online handout on regions of the plane: relationships between variables [areas, volumes of revolution];
    optional fun: apply the volume formula to the Great Pyramid of Egypt using the data you find online.

  10. Th:  Lecture Notes 6.3 on Work in 1-d motion;
    6.4 [skip 6.3]:   3, 7,
    13
    [hint: this is tricky and only works for lifting halfway, no other fraction, see the Maple worksheet explanation],
    15 [hint: also tricky,the extra single weight is moved a constant distance, so the product work value can be incorporated into the integral for the cable, see the Maple worksheet explanation]
    , 23,
    30 [this problem is all about units! you need to convert inches to feet to get ft-lbs in the final answer],
    34 [keep maximum significant figures in part a) to obtain correct integer for part b).]

  11. F: Please read remarks on Quiz 2 answer key (page 2);
    Quiz 3 on volumes;
     Lecture Notes 6.5 on Average Value of a Function; [temperature example][23]
     6.5:  1, 7, 9,  in class: 11,[13], 17, 18 [blood flow as an example of dimensionless variables].


    WEEK 4[-1]:
  12. M:  6.R: 11 (in class), 15, 25, 26 [part b gives you the answer!],
    33 [l'Hopital's rule];
    6.Plus.2Hint: solve for the point x = a of intersection in terms of the slope m of the line ym x (easy!) , and express m as a function of a.  Then use this value to set the area over x = 0..a equal to half the area over the entire interval. This condition is easily solved by hand if you are careful! Plot the resulting line with the curve.  [soln].

    Maple6.mw assignment: 6.AppliedProject: Where to Sit at the Movies* (after 6.5) (p 465). Assume the result of part 1 (just trigonometry), then do parts 2-4 to get a real life example of an average viewing angle. Here is a worksheet maple6.mw to accomplish this.

  13. W: Chapter 7:1-4 "Methods of Integration" is obsolete in the age of computer algebra systems; there is no general approach to finding antiderivatives.
    Lecture Notes 7.1a on Integration by Parts;
    7.1:  (integration by parts is useful in physics/math variational problems, not so much as an integration technique that we need to worry about, but we need to be aware of the process):
    3, 5, 9, 17, 39 [example of using Tutor], 45 [this can also be done by change of variable, but try IbP]; 
    just for fun: [iteration example]; [change of variable example].

  14. Th: Lecture Notes 7.1b on Integration by Parts etc;
     7.1: 57, [59, 65], 66, [67], 68 [dimensional discussion and HW template], [69], 71; (boldface problems worked in class?);
    optional minor challenge: 73 [this shows how one can manipulate integral formulas with u-sub and int by parts in combination]

  15. F: Quiz 4 on integration by parts due Tuesday;
     Lecture Notes 7.7a on Approximate Integration;
    7.7 (approximate integration; Simpson):  2, 3, 7 [ApproxInTutor!], 30, 35, 42;
     [You can use the ApproximateIntTutor, and the "Compare" button which gives all the approximations simultaneously];
    [Hint: Maple approach to these problems, with explanations.]

    Parents Weekend. potato head family

    WEEK 5[-1]: Test 1 Thursday in class through chapter 6.
  16. M:  Lecture Notes on Approximate Integration;
    7.7: (numerical error?): 22, 37 (tabular data with Simpson);
    [in class: 23* use template problem 24 here]

    Note that Simpson 3/8 uses a 3rd degree polynomial, while Newton Cotes formulas (see the Approximate Integration Tutor window menu) generalize the trapezoidal and these Simpson rules to even higher order polynomials. [wiki]
    Amusingly enough the proof of the error estimate for the trapezoidal rule involves two clever integration by parts and once verified, the whole proof becomes relatively straightforward. So as I mentioned, integration by parts is essential for advanced mathematics and applications. The Simpson's rule error estimate proof  is more of the same.

  17. W: Lecture Notes 7.8a on  Improper Integrals (infinite limits);
    7.8 [Maple examples]: 1, 3, 5, 13, 21, 23, 25 [Hint: u-sub!].

  18. Th: Test 1 through chapter 6 in class [no work problems.] [no lined paper needed, bob will bring it]

  19. F:  Lecture Notes 7.8b on Improper Integrals (infinite integrands);
    7.8: 28, 29, 31, 32, 33 [ab-surd!], 35, 40 [Hint: u-sub for antiderivative.] 

    WEEK 6[-1]:  Quiz 4 answer key online
  20. M:  Lecture Notes 7.8c on Improper Integral (comparison and applications); [62: mw, wiki];
    49, 51, 53, 54, 55 [u-sub!];
    applications:  68, 70 [in class, first do rescaling u-sub to dimensionless variable, then handle the improper integral in the dimensionless integral].


  21. W:   Lecture Notes 5.5c on change of variable in Definite Integrals; [5.5: 64 example, 65 by students]
    What? A Nobel prize for an improper integral?
    7.8: 66 workshop with Maple (voluntary completion with partner(s), submit if you wish feedback).

    Please study Test 1 Answer key

    5.5: 62, 63, 67, 69,71 [in each case, 1) re-express integrand in terms of u AND  re-express differential in terms of u and du, and 2) change to new u-limits of integration. Then evaluate this completely new integral expressed entirely in terms of the new variable.]


  22. Th: This evening special Zoom panel  on feeling off as a student: how to get help;
    7.R: see Maple worksheet collection of problems:  70 [liters!], 72 [hyperbolic function!],
    74
    (area: let u = sqrt(x) and transform the integral to a simpler form, then compare with Maple result: hyperbolic arctanh!),
    78 [games with limits];
    79 [this is just a change of variable problem, we don't have to actually do the integral),  80.
    [boldface done together in class]


  23. F: Quiz 5 on improper integrals;
    Lecture Notes 8.1a on arclength;  [fun: powers]
    8.1(arclength Tutor):  1, [3*] (use Maple), 7,
    9 (simple u-sub), 13 (perfect square!),
    15 (use tan^2 +1 =  sec^2 to simplify perfect square, then use technology for antiderivative),
    23 (use Maple evalf of inert form of integral or wait 25 seconds to be amazed: estimate),
    33 (how to plot; solve for y, perfect square is a power function!, do one quadrant, multiply by 4).

    WEEK 7[-1]:
  24. M: Lecture Notes 8.1b on arclength functions;  [revisit 13];
    8.1: 35, [try this for fun: 36*],
    37 [we have to simplify the integrand by combining fractions],
    39, 43,
     44 (remember definitions of cosh and sinh; you need to numerically solve for a in the interval a = 40..100,  Solve, Numerically Solve From Point, choose a = 40);
    44* [if curious, plot the final profile over the given interval together with the ground line y = 0 and the line y = h where h is the height you found to hang the wire, and set equal units on both axes so that it looks like you would see it; also show your plot of the condition determining a and show the numerical solution].

    We skip 8.2 on surface area since this is not even required in the syllabus of multivariable calculus MAT2500.
    We skip 8.3 since moments will be  dealt with in MAT2500, while hydrostatic pressure seems to far afield for most of us.
    |
  25. W: Lecture Notes 8.4a on business calc [demand curve, etc]
    8.4: 5, 6, 8, 12 [9: total surplus: maximized?? bob finds minimized!]

  26. Th: Lecture Notes 8.4b on blood flow, cardiac output;
    8.4 (dye dilution method example worksheet to measure flow rate out of heart, cardiac output summary): 
    not enough problems on WebAssign!
     [optional: basic financial literacy! 15 (future value), 16 (present value) , 17 (incnme distribution)],
    18 (net change integral),
    19 (plug in numbers, blood flow),
    21 [dye problem with expression for c(t)],
    AE2 [cardiac dye, like 22 cardiac dye].

  27. F: Quiz 6 on arclength, due Tuesday after break.

    Fall Break.  

    WEEK 8[-1]:
  28. M: Lecture Notes 8.5a on probability;
    8.5: 1, 2, 5, 6,
    Poisson (decaying exponential) distribution applications (tables!):
    10 (median value of x: probability half below x, half above x), 11.

  29. W: Lecture Notes 8.5b on probability: normal distributions etc;
    [gas tank filling example and changing variable: pdf, mw]; [circular target];
     8.5: normal distribution: 15, 16 [template: adjusting the mean to achieve a goal],
    17, 19
    [21* Use Maple to answer each part of the problem, keeping  a0 symbolic except for the graph in part c)].

    WebAssign has a smart phone Cengage Mobile app!

    And today a post with 3 sigma, 5 sigma  mentioned!

  30. Th: 8.R: [9 also find s(x), 21, 22, 23 recall Poisson mean is 0.69 mu!];
    Quicky in class exercise Maple 8.5.21.

  31. F: Short Quiz 7 on probability distributions
    Lecture Notes 11.1 on infinite sequences;
    Read 11.0 ultraviolet catastrophe;
    Watch 11.1 video on Fibonacci sequence; (read this motivating example: planck, sequence worksheet):
    11.1:  1, 5, 11, [17], 23, 25, 29, 33, 41 (l'Hopital's rule: ∞/∞), 49, [59* (use the template in this worksheet)], 65.

    > seq(f(n), n = 1..N)    this is how you generate sequences in Maple; use evalf(f(n)) if you want the decimal values.

    WEEK 9[-1]: Facinated by math? Check this out [YouTube]
  32. M: Lecture Notes 11.2 on infinite series;
     (short problems, I avoided tedious entry of numerical values of first n terms):
    11.2 (series): 1, 3, 15, 17, 18, 22, 27, 31, 41, 43 [Hint: convert(1/(n^2 - 1), parfrac)],
    49 [d: all rational numbers that have a terminating decimal representation except for 0],
    51, 57, 69 [wording misleading, limiting amount after each injestion, not the minimal level always remaining in body],
    75.

  33. W:  Lecture Notes 11.3 on the integral test [mw, pseries];
    11.3  [1], [2], 7, 8, 11, 14, 15, 27, 29, 35, 37 (use Maple integral test template), 39 [template].

  34. Th: Test 2 through chapter 8. arclength, improper integrals, probability.

  35. F: Lecture Notes 11.4 on comparison tests;
    11.4: 1, 2, 5, 7, 23, 27, 28, 29,  31, [35*];

    2pm Mendel Award Public Talk in the Villanova Room. Don't miss it!

    WEEK 10[-1]:
  36. M: Lecture Notes 11.5 on alternating series; [limit comparison easier than direct comparison];
    (alternating series mw);
    11.5: 1, 2, 3, 5, 7, 11, 17, 23, 32.

    Check blackboard grades against test grades. Test 2 answer key online. Test 2 back Thursday (oops, forgot them at home)

  37. W: Lecture Notes 11.6 on Absolute Convergence, Ratio [& Root] Tests;
    [video example mw];
    11.6: 1, 2, 5, 7, 13, 17, 23, 25, 31, 37, 43;
    [optional fun reading 50] (this problem mentions William Gosper calculating 17 million digits of Pi! and shows dramatically how using an efficient series representation pays off; > Digits:=20 is needed for part b) which looks at accuracy in the teens of digits; Digits:=10: returns to the default, use Maple to evaluate the limit in the ratio test; if you want to be amazed, read this worksheet).

    We are skipping 11.7 since general series convergence tests are not as important for Taylor series, where "absolute convergence rules" and the absolute convergence ratio test plus p-series comparison is all that matters; now we start the important stuff.

  38. Th:  Lecture Notes 11.8 on power series (almost Taylor, finally!);  [Bessel]
    11.8: 5, 7, 9, 15, 17, 19, 21, 27, 37.

  39. F: Quiz 8 on series so far;
    Lecture Notes 11.9 on power series tricks;
    11.9 (tricks with power series): 3, 5,
     [9, to make this less painful, use the fact that (x-1)/(x+2) = 1 -3/(x+2)],
    13 [see this worked example pdf, .mw], 15, 25, 32 [use Maple worksheet].

    WEEK 11[-1]:
  40. M: Lecture Notes 11.10a on Taylor series;
    11.10 (Taylor series!): [2], 3, 4, [5, 7], 11, 12,
    17 [cosh' = sinh, sinh' = cosh, cosh(0) = 1,sinh(0) = 0 follow immediately from their definitions in terms of the exponential function],
    21, 25, 39, 55.
    > taylor(exp(x),x = 1, 6) ;convert(%,polynom)        # up to 5th power, centered at 1

  41. W: Lecture Notes 11.10b on  binomial expansion series;
    [wiki: arcsin.mw; binomial thm];
    11.10: 31, 33, 37, 51,
    59 [Hint: integrated series is alternating after the first term, so next term in series is error estimate],
    63, 73.

  42. Th:Lecture Notes 11.10c on Taylor series maybe not: grabbag day?;
     (error: theory, practice not for testing);
     (products and quotients of series? not for us): [tan(x):  video, wiki; Bernoulli numbers? tantaylor, tan.mw];
    limits with power series;
    [exp,ln,trig,binom series (all we can handle)];
    11.10: 60, 62, 78 [for fun look at 84].


  43. F: Quiz 9 on Taylor series;
    Lecture Notes 11.11 on Taylor series apps
    11.11 (applications): [kinetic energy, black body radiation]
    [use Maple > taylor(f(x),x=0,6) to get Taylor series you need in these problems].
    35, 37 [Hint: the angle from the center in radians is so small that it can be approximated by its tangent L/R; lowest terms of secant series from Maple! For part c) you need Digits:=20; forget about parts a), b) they are worth 0 pts and no one will ever see what you enter.]

    optional: [3336 Hint: factor out d from the radical to use binomial series in small ratio R/d, as in 33 for d/D].

    WEEK 12[-1]:
  44. M: 11.R: in 11-22 recognize how to quickly identify the large n behavior [22 taylor expand in x = 1/n after dividing through)], some diverge, consider absolute value when not a positive series etc;
    37 [remainder series less than geo series without 2, use its sum to place constraint on error];
    look at some of the following problems that seem reasonable: 40-44 you should be able to answer these (41);
    47-54 use tricks by manipulating simple series;
    5659;
    57 (just read s11-R-57.mw);
    60
    is a great application we will do together in class;
    |
    Taylor series summary: the big picture.

  45. W: Lecture Notes 10.1 on parametrized curves;
    parametrized curves can be fun (tease)!; [hearts!];
     10.1 (execute worksheet and look at many other examples):
    (use Maple to visualize curves [set parameters =1 to plot]! Google named curves if interested):
    [1, evaluate start and stop points to see direction],
    7 [where is t = 0? where is start, stop?],
    9, 11, 13,  [optional for fun only: 28], [useful: 31],
    33, 41,
    [43: use identity cot^2+1=csc^2 to eliminate parameter, what continuous range of angle makes it a continuous curve?],
    45 [display both animatecurves!], 46 [see how you can Explore the trajectories for all forward angles].

    fun: Venus in EarthCentric coordinates?

  46. Th: Lecture Notes 10.2 on calculus of parametrized curves; (semicircle video);
    (tangents, areas, arclengths)
    10.2: 7, 13, 17, 31, 48, [53, 55];
     [many Maple worked HW problems];
    [optional viewing: Project 10.1/2 running circles around circles: fun with cycloids].

  47. F:  no quiz! [fun: Earth centric venus orbit!];
    Lecture Notes 10.3 polar coordinates;
    (trig refresher online: trig,  inverse trig; handout: polar coords);
    10.3: 3, 5, 11, 17, 19, 21, 25, 27, 30, 32, 37, [49, 62]. [cardioidfamily.gif]

    WEEK 13[+1]:
  48. M: Lecture Notes 9.0 on series soln of DEs and Maple sol (for fun);
    [Zoom recording  in email] [soln]
    ODE 101 with Maple, a preview,
    plus how power series solution of DEs defines special functions like Bessel.

    Thanksgiving Break.

    F: Take home Test 3 out on BlackBoard. First read the test rules carefully please.

  49. M: Hard copy of Test 3 distributed in class; continue working on this in class.

  50. W:  Lecture Notes 10.4a on polar curve angular ranges; [mw]
    after this preliminary issue with polar curves, work on Test 3 will continue in class.

  51. Th: Lecture Notes 10.4b on polar curve areas; [video example] [circle and cardioid]
    10.4:  7, 21,
    31 (> plot([cos(2*t), sin(2*t)], t = 0 .. 2*Pi, coords = polar) find intersection point, find area of one half a single region between the two curves, multiply by twice the number of such regions; what percent of area of the circle passing through the intersection points does this represent? Is that reasonable?),
    44 (use numerical solution for angle of intersection).

  52. F: Lecture Notes 10.4c on polar curve arclength; [circle] [zoom]
    in class arclength of r = sin(theta/4), what is period over which you have to integrate?
    10.4: 47 (easy u-sub!), 50 (perfect square!), 51 (exact via Maple),
    517.XP [convert(L,ln) to force arcsinh to ln expression for WebAssign?).

    WEEK 14:
  53. M: Test 3 due in class today stapled;
    Optional Lecture Notes 10.5 on ellipses;
    in class group exercise: area and circumference of a specific ellipse (as part of review for both parametrized curves and polar coordinate integration).
    ellipses in action [even after Einstein]


  54. T [F]: 10.R (plots): 1, 11, 19 (what happens at theta = 0?),
    23 (m = -1), 25,
    27, 28 [areas of loops, not discussed in the text! A=4.05];
    29
    (hor/vert tangents: plot with a = 1),
    30
    (6 Pi a^2); 31 (18), 40 (perfect square! 0.9).

  55. W: checking a solution of a differential equation;
    history:  hbar? [ellipse area]

    [optional glance: Spaghetti curves? How an innocent math problem can  lead to crazy stuff.]

  56. @ Th: Last Class; CATS at beginning of class (instructions: Blackboard);
    discussion of final exam on Chapter 10:
    areas enclosed by parametrized curves or polar curves, arclengths in both contexts, vertical and horizontal tangents, tangent lines of parametric curves or polar curves.

    We made it! party cats (party cats!)

end of list! scroll up for current day assignment.


Weeks 3 and 4 or 5 or earlier: come by and find me in my office, spend 5 minutes, tell me how things are going. This is a required visit.
If you are having troubles, don't wait to see me.

FINAL EXAM: switching times allowed with instructor permission
                       1505-001 MWF 10:30:  Fri, Dec 17 08:30 AM - 11:00 AM
                       1505-002 MWF 11:30: Sat, Dec 11  10:45 AM - 01:15 PM

                          MAPLE CHECKING ALLOWED FOR all Quizzes, EXAMS

log from last time 20-may-2021 [course homepage]
 

here is a sample worksheet report, with exported animated GIF. we will return here later for volumes of revolution).
and optional bell curve example of change of integration variable [pdf, mw];