MAT2500 19S [Jantzen] homework and daily class log

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

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 Red numbered problems are a relic of the old online HW hints that are now superceded by the e-book addons.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.
Problems which are not available in WebAssign will be square bracketed and must be done outside of WebAssign. Check this page for hints and some linked Maple worksheet solutions.

M (January 14, 2019):
GETTING STARTED STUFF. By Friday, January 18, reply to my welcome e-mail [robert.jantzen@villanova.edu] sent to your OFFICIAL Villanova e-mail account (which identifies you with your full name), telling me 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(s) was(were) named (Mat1500 = Calc 1, Mat1505 = Calc 2, Mat2705 = DEwLA).
HINT: Just reply to the welcome email I sent you before classes started.
[In ALL email to me, include the string "mat2500" somewhere in the subject heading if you want me to read it. I filter my email.]

On your laptop/tablet if you brought it:
1) Open Internet Explorer or your favorite browser. (You can 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 a browser (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, or you can Google me with the string "dr bob jantzen" to find my home page and then this class home page:
http://www.homepage.villanova.edu/robert.jantzen/courses/mat2500/ ].
3) bob will quickly show you the computer environment supporting our class
[like clicking here to register with WebAssign; class keys 10:30: villanova 3227 8678, 12:30: villanova 1288 7300 ].
4) Open Maple if you already have it by clicking on this maple file link: bloodflow.mw
And then bob will then set the stage for 12.1, showing how reading a example Maple worksheet can illuminate all the steps in a hand solution of a problem involving the distance formula in space without dwelling on the details [Stewart 12.1.45: s12-1-45.mw].
5) Here instead is an example of a problem [Stewart 12.1.46] embedded in 3d space but quickly reduced to a Calc 2 volume of revolution problem:
hand solution [ s12-1-46.pdf], [notice how different the hand written and Maple typeset formatting of the solution]
corresponding Maple worksheet solution [ s12-1-46.mw] and illustration of the problem. All you have to do for such solution worksheets is read them if you are interested. Don't worry, we will take it slow with Maple.]

During class in the first part of the semester, a signup sheet will be passed around for your signature. Make sure you sign at the end if it bypasses you. Today please put your nickname or your first name to be used in class, and include your cell phone number and your 3 letter dorm abbreviation listed on the short list side of the signup sheet.

Log on to My Nova, choose 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.]

Homework (light first day assignment):
Make sure you read my welcoming email sent on the weekend (and reply to it within a few days), and register with WebAssign (immediately, if not already done).
Explore the on-line resources. Read the pages linked to our class home page. [Read computer classroom /laptop etiquette.]
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.

WebAssign Problems: WebAssign101 is a very quick intro to WebAssign due Wednesday midnight;
read 12.1 reviewing 3d Cartesian coordinate systems, distance formula and equations of spheres;
instead due in class Thursday but in WebAssign by midnight Thursday so you can ask questions in class if necessary:
12.1: 23, 45 (WebAssign has random numbers in your problem, solved above for the textbook numbers);
This short list is so you can check out our class website and read about the course rules, advice, bob FAQ, etc, respond with your email; those who do not yet have the book should be using the e-book through WebAssign. It is important that you read the section in the book from which homework problems have been selected before attempting them.

Download Maple 2018 if you haven't already done so and install it on your laptop when you get a chance (it takes about 15 minutes or less total), I will help you in my office if you ;wish.). If you have any trouble, email me with an explanation of the errors.
You are expected to be able to use Maple on your laptop when needed. We will develop the experience as we go. No problem if you never used it before.
W: return your schedule forms at the beginning of class;
look over the class paper handout on diff/int/algebra;
check cell phone number, dorm info on daily signup sheet from first day entry;
12.1: 11, 15, 17, [21a hint: show the distance from P₁ to M is the same as from P₂ to M and equal to half the total distance; this is the hard way with points and not vectors],
12.2: 1, [2], 3, 5, 7, 13, 15, 19, 21, 25, 29.
Th: Office hours and course info document handout;
12.2: vector diagram problems [numbers for example 7];
34 (draw a picture, express the components of each vector, add them exactly (symbolically), evaluate to decimal numbers, think significant digits),
37 [same as example 7, different numbers],
45 [grid].
F: Quiz 0 on 12.1 distance formula, sphere analysis (completing the square);
12.3 handout on dot product: [example 3];
1, 2, 5, 9, 11, 15, 21, 23, 33;
optional fun problems if you like math: [55] (geometry [pdf,.mw]),
[57] (chemical geometry [ soln.mw, plot: methane.mw]).

WEEK 2[-1]:
M: MLK DAY.
Problems which are not available in WebAssign will be square bracketed and must be done outside of WebAssign.
W: handout on resolving a vector
and read this Maple worksheet: [using Maple (for dot and cross products and projection)];
12.3 projection: 39, 41, [45], 46, 49.

7pm today in the Villanova Room: MLK address and Peace Prize talk. [don't miss it!]
Th: 12.4 cross product: handout on geometric definition;
[why component and geometric definitions agree: crossprodetails.pdf];
1, 7, 11, 13, 16, 19, 27 (find 2 edge vectors from a mutual corner first, use 3 vectors and cross product),
31 (Maple example: trianglearea.mw),
33, 35, 37 (zero triple scalar product => zero volume => coplanar),
39 (first redo diagram with same initial points for F and r).
[54, why are a) and b) obvious? visualization]

Note:
> <2,1,1> · (<1,-1,2> x <0,-2,3>)
[boldface "times" sign and boldface centered "dot" from Common Symbols palette parentheses required]
> <2,1,1> · <2,1,1> then take sqrt (Expressions palette) to get length [example worksheet: babyvectorops.mw]
F: Quiz 1 on projection (easy! identical to 18S with 3 new points);
[animation , .mw];
detour: handout on parametrized curves:
[textbook example curves: s10-1.mw (wow!)][parametrized curve tutorial];
open these worksheets and execute them by hitting the !!! icon on the toolbar (then read them!);
it is not very useful to try to draw parametrized curves based on what the graphs of x and y look like: technology is meant for visualizing math!;
BUT REMEMBER, WE ONLY NEED TO WORK WITH SIMPLE CURVES.
This detour now is because we want to describe equations of simple straight lines.
10.1: 4, 9, 13, 17 [hyperbolic functions, Stewart 8e section 3.11: cosh² x- sinh² x = 1, recognition is enough],
21, just for fun: [28 : jpg; it does not hurt to use technology if you cannot guess them all];
33, [37].

> plot([cos(t), sin(t)], t =0..2)           square bracket after last function, plots functions versus t on same axis
> plot([cos(t), sin(t), t =0..2])           square bracket after parameter range, plots parametrized curve in plane
or
> cos(t), sin(t)   right margin context menu, cursor on output, choose Plots, PlotBuilder, 2d parametrized curve (3d for curves in space); NEVER SMART PLOT!

WEEK 3[-1]:
M: Everybody is politely requested to stop by my office for a 5 minute chat during the next few weeks just to see where it is an report on how things are going; [Test 1 week 5 thursday?];
handout on lines and planes [.mw];
never use the symmetric equations of a line: they are useless for all practical purposes!;
12.5: 1 (draw a quick sketch to understand each statement),
3, 5, 7 (parametric only),
13, 16, 19; 24, 31, 41, 45, 51 , 57[.mw].
W: class roster handout (paper only)---let's try to find one or two (preferable) partners for Maple assignments;
handout on geometry of points, lines and planes (distances between);
in these problems do not just plug into a formula: this is practice in vector projection geometry, we really don't care about the distance!:
12.5: 69 (DO NOT PLUG into FORMULA, find point on line, project their difference vector perpedicular to the line as in handout),
71 (find point on plane, project their difference along the normal) ,
73 (find pt on each plane, project their difference vector along the normal);
76 (Don't use distance formula; draw a figure, find a point on the plane and move from it along the normal in both directions to get a point in the two desired parallel planes),
78 [.mw finding the closest points!] (find pt on each line (set parameters to zero!), project the 2 point difference vector along the normal to the parallel planes that contain them; ans: D = 2);
[optional challenge problem: 81].
Th: 12:R (Review problems are not in WebAssign): 24, 25, 26, 38 [what is the distance of a point from any coordinate plane---we did this problem in 12.1.23, so what is the distance of the point (x,y,z) from the plane y = 1? [soln is ellipsoid]];
read section 12.6 only for fun, since we will be using quadric surfaces during the course so why not skim through this material quickly?
in class we work together on 26.
[Hint on 25: you get the normal to the desired plane from the cross product of the orientation vector of the line of intersection, and the normal to the third plane, since the third plane is orthogonal to the desired plane hence its normal is parallel to the plane].
F: Quiz 2 [see archived quiz 2];
Maple assignments start (read these instructions): note asterisks *;
13.1: [cubic, cutcylinder] [1], 3,[5], 7, 13, 21, 25
[21-26, for fun only, do quickly by thinking, note technology is not necessary here to distinguish the different formulas: .mw],
27, 33, [39*: refer back to similar problem 27: note that z² = (x²+y²)! plot the spacecurve and the surface together as in the template, adjust the ranges for the surface so it is just contains the curve and it not a lot bigger; clue: look at boxed axes ranges for the curve to chose your window for the surface; HINT: the template example is a paraboloid, this exercise surface is a cone];
43 [eliminate z first by setting: z² (for cone) = z² (for plane) and solve for y in terms of x, and then express z in terms of x and finally let x be t; Maple can solve the pair of equations for {y,z} in terms of x],
[12.5.57*: using the answer in the back of the book and this template, plot the two planes and the line of intersection and confirm that visually it looks right. Adjust your plot to be pleasing, i.e., so the line segment is roughly a bit bigger than the intersecting planes (choosing the range of values for t)].

Did you know about Villanova TV on You Tube?

WEEK 4[-1]:
M: 13.2 : [in class: 1: pdf, 2], 5 [recall: exp(2t) = (exp(t))², what kind of curve is this?], 7, 10, 15, 19, 21, 31, 29a (by hand),
29b* [graph your results using this tangent line template; make a comment about how it looks using text mode].

on-line handout: key idea of vector-valued functions and the tangent vector
(see Maple video: secantlinevideo.mw; vectorfunctionlinearapprox.mw);
Maple stuff:
> <1,2,-3> × <1,1,1>
                 5 e_x - 4 e_y - e_z
[this is just new notation for the unit vectors i, j, k; > BasisFormat(false): returns to column notation]
> F := t → <t, t² , t³> : F(t) or even F(t):=<t, t² , t³> creates a Maple function
> F '(t)
> ∫ F(t) dt
> with(plots):
> spacecurve(F(t), t=0..1, axes=boxed)
or just
> t, t², t³
then Right Click, Plots, Plot Builder, 3d parametric curve (also for 2d parametric curves with 2 expressions);
and recall for 2D plots:

> plot([cos(t), sin(t)], t =0..2)           square bracket after last function, plots functions versus t on same axis
> plot([cos(t), sin(t), t =0..2])           square bracket after parameter range, plots parametrized curve in plane
W: Maple worksheet vectoravg.mw to see how the integral of a vector-valued function can be interpreted visually; rules of differentiation extend from scalar-functions to vector-valued functions easily (example below).
13.2: 22 [check your answers against Maple worksheet],
29 [another template for plotting curve and tan line together),
33 [angle between tangent vectors],
39 [use technology to do the integrals],
[43], 48, 49,[54, use combined dot/cross product differentiation rules 4,5 from this section:
(a · (b x c)) ' = a' · (b x c)) + a · (b' x c)) + a · (b x c')) ].
[product rule holds for all the products involving vector factors as long as you keep the order of factors the same in each resulting term if cross products are involved; usual sum rules always apply]
Th: 13.3 (arclength) arclength toy problems require squared length of tangent vector to be a perfect square to be integrable usually! [or a factorization that makes a u-sub work!]
online handout on arclength and arclength parametrization;
[integral of speed with respect to time is distance traveled! so the length of a curve is the integral of the magnitude of the tangent vector! easy.]
3 [note the input of the sqrt in the integrand is a perfect square in this problem];
5, [note the factorization to make an obvious u-sub];
10 [use numerical integration either with your graphing calculator or if you use Maple, right-click on output of integral, choose "Approximate";
for the curious: oops! what a mess!];
11 [hint: to parametrize the curve, first express y and then z in terms of x, then let x = t; another perfect square],
12 [hint: let x = cos(t), y = 2 sin(t) for the ellipse, then solve the plane eqn for z to get the parametrization, for one revolution of this ellipse]
F: Quiz 3;
13.3 (curvature): handouts on geometry of spacecurves (page 1 for 13.3) and space curve curvature and acceleration (pages 2-3 for 13.4 later, 4 for both, print together);
[example Maple worksheets on these handouts: rescaled twisted cubic (page 1), helix (page4)]
13.3: 17, 25,
27 [do not use formula 11: instead use the parametrized curve form r = <t,t⁴,0> of the curve y = x⁴, then let t = x to compare with back of book or to enter in WebAssign];
47 [twisted cubic: perfect square!], 49;
51 [standard eqn: x²/9+y²/4 = 1, so r = <3 cos(t),2 sin(t),0>, Hint: for osculating circle, normals at axis intercepts are pointing inward along axes, just go distance ρ along axis to get center, write equation of circle with radius or curvature there.]

> with(Student[VectorCalculus]):
SpaceCurveTutor(<t,t²,0>,t=-1..1) from the Tools Menu, Tutors, Vector Calculus, Space Curves [choose animate osculating circles]
2d parabola osculating circle zoom.
[quick show and tell 2d curves: osculate-parabola.mw, osculate-ellipse.mw]

WEEK 5[-1]:
M: 13.3: Taylor approximation of a spacecurve: spacecurvetaylorapprox.mw
optional on-line only: osculating circle, how to describe mathematically using vector algebra [.mw].
online only: projections revisited just for those who like vector geometry;
on-line reminder of dot and cross products and length, area, volume.

In class exercise: calculate all the quantities for a tilted parabola: r = <2t,t²,2t>,t=-1..1 in general and at t = 1.

Work on Maple problems, look over archived tests.
W: 13.4 (splitting the acceleration vector, ignore Kepler's laws):
1, [2 avg velocity = vector displacement / time interval],
5, 11 [recall v = exp(t) + exp(-t) since v² is a perfect square],
17a, 17b*[graph your spacecurve using the template; pick the time interval t = -n π..n π, where n is a small integer, and by trial and error, reproduce the figure in the back of the book with 6 peaks, rotating the curve around and comparing with the back of the book sketch (note the horizontal axis tickmarks); if you wish, then animate the curve with the template provided],
19 (minimize a function when its derivative is zero (critical point)! confirm minimum by plotting function);
[37 note that v² = 3²(1 + t²)² is a perfect square],
39,
[41? also perfect square, see 11] , 45 [visualize it!]

Maple13.mw HW due anytime next week, through the weekend. Please read the instructions carefully.

5:30pm MLRC voluntary problem session for Test 1.
Th: Test 1 through 13.3, evaluation of formulas for various projections along curve.
F: 13.R: (p.882-883): [4, 6, 10, 16, 24];
in class we play with 24 and use the worksheet (without continuous curvature but just continuous value and slope, such piecewise curves joining points are called cubic splines, important in engineering and design);
[14a: use parametrized curve r = [t, t⁴- t²,0] to evaluate the curvature, by symmetry N(0) easily must point down find osc circle: x² + (y+1/2)² = 1/4],
14b*: edit the template with your hand results including comments and also do the zoom plot to see the close match of the circle to the curve].

WEEK 6[-1]: Maple13.mw assignment due anytime this week thru the weekend
M: 14.1 [text examples]: 1, 3, 11, 15, 25, 31, [33], 35, 41, 49;
|maple14.mw problems begin:
55*, using this template just do a single appropriate plot3d and contourplot after loading plots and defining the maple function f (x,y)];
81a (read only b,c; if you are interested to see how the data is fit see example 3 among the interesting examples from 14.1 shown in Maple);
after finishing the preceding, for fun only if you have extra time look at 61-66 (maple plots reveal relationships, try to see correlations between formulas and 3d plots, then the contourplots).
W: Snow Day;
work on maple13.mw.
Th: Test 1 back, answer keys for quizzes; check Blackboard grades;
14.3(partial derivatives, finally): [1], 3, 5, 11, 15, 17, 21, 31, 33, 41, 51 [in class if time: 22, 24, 28, 30].
F: New plan: class will meet.
Take home Quiz 4 on 14.1 [tabular data like 14.1: 2 or 7, be able to evaluate a tabular derivative of a single row of data; due Monday (open resource);
discussion framed by this worksheet 14.2-explore.mw:
14.2: [1], 2, [4, 5], 9, 13, [15], 17,
23* [toolbar plot option: contour, or "style=surfacecontour" or right-click style "surface with contour", explain in comment],
25, 31, 37* [does a 3d plot of the expression support your conclusion? that is, your conclusion drawn before looking at the back of the book obviously, plot and explain], 39.

WEEK 7[-1]:
M: hand in take home quiz 4;
14.3: second and higher derivatives (and implicit differentiation!; partials commute) 47, 49; 53, 56, 59, 63,
65 [use this example for higher derivatives];
73 [just average the adjacent secant line slopes on either side of the point where the partial derivative is to be evaluated, as in the opening example: pdf, this is not a testing problem! tedious so I show you how to work through it],
[81], 83, [84, 88 ideal gas law], 90.
W: 14.4 (linear approximation and tangent planes: differentiability illustrated; moving tangent plane):
1, 3, [7,] 11, [15], 17, 21, 22 [.mw];
7*[calculate by hand, then do two plots: first > plot3d( [f(x,y),L(x,y)],x = a..b, y = c..d); choose appropriate ranges centered about the point of tangency to show a good part of the surface behavior together with the tangent plane, then zoom in by choosing a smaller window about the point of tangency as instructed by the textbook, check that they agree, make a comment that it looks right confirming differentiability].
online handout on linear approximation.
Th: 14.4: handout on differential approximation and error estimates; [diff approx example]
25, 27, 31, 33, 35, 38, 39 [remember partials of this function from 14.3.83]; extra:
In the USA (inch units), the 4x6 photo prints have dimensions 4 in by 6 in. In Europe (cm units) the 10x15 prints have dimensions 10cm by 15 cm. Unit conversion: 1 inch = 2.54 cm. Use the differential approximation to estimate the absolute change and percentage change in the (computed) area of the USA format (new) compared to the European format (old): A = x y . Then compare your linear estimates for both to the corresponding exact changes. [HINT: apply the differential approximation using the x and y values of the European format, with the differentials dx and dy given by the differences USA format dimensions minus the European dimensions.] [Solution: 4x6prints.mw]
F: Snow Day;
online Quiz 5 [linear approximation, be able to evaluate 2nd derivatives].

FALL BREAK :-) Enjoy. Be safe in your travels.

WEEK 8[-1]: submit corrected Maple for midterm grade by Tuesday;
M: 14.5: chain rule: 1, 11, 15, 17 (I never use tree diagrams), 21, 31 31 (no need to remember these equations, just implicit differentiate and solve in practice), 35, 39, [wave equation: 49].
W: 14.5: 41;
14.6 (directional derivatives; stop at tangent planes): 1, 3, 5, 7, 9, 11, 15,
19 [first find a unit vector in the given direction! sketch the two points],
23[just split the gradient into its length and unit vector direction information], 29.
Th: Pi Day: π! [and Einstein's birthday] [Vu Math Pi Day Events];
handout on derivatives of 2d and 2d functions [maple 2d-gradient and directional derivative example][3d: Stewart Example 14.6.7];
14.6 (level surface tangent planes; note z = f(x,y) corresponds to F(x,y,z) = z - f(x,y) = 0):
14.6: 27, 31, [in class: 36, 38,] 41, 45, [47 (derive equations of plane and line by hand)], 49, 52, [this one is fun: _ 61 [soln]], 65 [soln];
47* [plot your results in an appropriate window (using the 1-1 toggle or the option "scaling=constrained" to see the right angle correctly), i.e., adjust windows of function, plane, line to be compatible, after doing problem by hand];
head start on problem 52 in class with any partner?
F: VILLANOVA CLIMATE STRIKE/WALK-OUT 12pm @ the Oreo;
https://www.fridaysforfuture.org/ and https://www.youthclimatestrikeus.org/.
12:30 class slightly delayed but Quiz is in manila envelope at instructor desk: honor system;
short Quiz 6 linked here as well on tangent plane, normal line, directional derivative for 2d graphs and 3d level surfaces;
Work on Maple14.mw this weekend; check out the archived tests 2.

WEEK 9[-1]:
M: handout on 2D max-min 2nd derivative test [derivation; with ignorable bonus handout on m ultivariable derivative and differential notation];
14.7: [1], 3, 5, 11, 12, 23 (do by hand, including second derivative test and evaluation of f at critical points); 23* [next time do 25, template shows how to narrow down your search to find extrema by trial and error, record your tweaked image or images confirming your hand results, include commentary, see additional comments on Maple HW page summary (inconclusive saddle point?), .mw];
optional: if you are interested in the more realistic case of example 4 where numerical root finding is required, read this worksheet.
W: 14.7: [21] (a warning that extrema are not always isolated points);
boundaries: 31; 37 [plot3d: > plot3d(y^4+2*x^3, x = -1 .. 1, y = -sqrt(-x^2+1) .. sqrt(-x^2+1)), express circular boundary in terms of polar angle to extremize there > plot(2*cos(t)^3+sin(t)^2, t = 0 .. 2*Pi)],
word problems (box): 41 [minimize square of distance],
49 (similar to 45 only with different coefficients in the constraint equation),
53,
[58: word problem with triangular boundary, use constraint to eliminate r, maximize resulting function of 2 variables on triangular region, consider triangular boundary extrema; plot3d: (try first, then sol: pdf, mw)];
read 59 [this explains least squares fitting of lines to data, and perhaps the most important application of this technique to practical problems].
Th: 14.R. (review problems; note some of the highest numbered problems refer to 14.8, which we did not do):
some in class if time: 1, 7, 15, 18, 21, 25, 29, 31, 33, 34a, 39, 53,
14.7: 52 [ans: the height is 2.5 times the square base; obviously cost of materials is not the design factor for normal aquariums, no?].

Maple 14 is due any time during next week thru weekend. Test 2 next Thursday.
F: Quiz 7 (max, min);
15.1 (Riemann on rectangles):
5:30pm voluntary problem session for Test 2.
Maple Tools Menu, Select Calculus Multi-Variable, Approximate Integration Tutor (midpoint evaluation usually best)
for Wednesday:
15.1: 1, [3 do by hand first], 3 [after doing this by hand, before next class: repeat this problem using the Maple Approximate Integration Tutor (with midpoint evaluation for (m,n) = (2,2), then (20,20), then (200,200), comparing it with the exact value given by the Tutor],
[5], 6 [midpoint sampling: (m,n)=(2,3), x along 20 ft side, y along 30 ft side: answer = 3600], 7, 11;
[15* ; note: (m,n) = (1,1), (2,2), (4,4), (8,8), (16,16), (32,32) = (2^p,2^p) for p = 0..5 is what the problem is asking for (see 3 line Maple template); what can you conclude about the probable approximate value of the exact integral to 4 decimal places?] .

WEEK 10[-1]: Test 2 Thurs, MLRC problem session Wed 5:30pm
M: 15.1 (iterated integrals on rectangles): 15, 21, 29, [33, which order avoids integration by parts?], 35, 43, [45* (use boxed axes!)], 47, 49;
[iterated integrals in Maple (how to enter, how to check step by step)]

step by step checking of multiple integration (Shift-Enter allows multiple line inputs):
> x + y
> ∫ % dx
> eval(%,x=b) - eval(%,x=a)
> ∫ % dy
> eval(%,y=d) - eval(%,y=c)
> etc... if triple integral (and simplify may help along the way)
W: 15.2: handout on double integrals;
5, 13, 15, 17, 23, 25, 27; 35, 39, 49, 51, [57, 59], [70*, follow the instructions in the template].

Keep in mind multivariable integration is really about parametrizing the bounding curves of regions in the plane or the bounding surfaces of regions in space, to set up iterated integrals, whose evaluation is just a succession of calc2 integrations, easily done by Maple. Setting up the integrals Maple cannot do. This is your job.

5:30 voluntary MLRC problem session. But I will only show up if at least one person confirms they will show.
Th: Test 2 on differential and linear approximations, equations of tangent planes and normal lines to 2d graphs and 3d level surfaces, directional derivatives and gradient, chain rule. [no max-min]
F: Maple 14 should be done by this weekend; DETOUR:
online only: review polar coordinate trig; inverse trig; [angles in all 4 quadrants!];
handout on polar coordinates and polar coordinate integration (the integration is next time);
review from MAT1505:
10.3 pp.658-663 (stop midpage: tangents in polar coords unnecessary for us), pp.665-666 (read graphing in polar coords [more polar fun]);
10.3: 3, 5, 8, 11, 12, 17, 19, 21, 25, 30, 33, 37 (all short review problems);
part of Maple15.mw:
67* [Nephroid of Freeth: starting at θ = 0 how far does theta have to go for the sine to undergo one full cycle? i.e., stop at θ/2 = 2 π ; this is the plotting interval];
keep in mind that our most important curves for later use are circles centered at the origin or passing through the origin with a center on one of the coordinate axes, and vertical and horizontal lines, and lines passing through the origin, as in the handout examples.

WEEK 11[-1]: Maple14 due [try x=1/2..2,y=1/2..2 to avoid infinity on contour plot in last problem! and option view=0..10 on the plot3d because of the infinite value at the origin]
M: W: 15.3: [1, 4, 5], 7, 11, 14, 15, 17, 21,
23 [twice the volume under the hemisphere z = sqrt(a²- x²- y²) above the circle x²+ y² ≤ a²or the volume between the upper and lower hemispheres],
25 [integrand is difference of z values from cone (below) to sphere (above) expressed as graphs of functions in polar coordinates].
35, 36.
W: handout on distributions of stuff;
15.4 (center of mass, "centroids" when constant density; skip moment of inertia---of course who cares about centers of mass or geometric centers of regions?--- but this is typical of many "distribution" problems, including probability, and we have some intution about where these points should lie):
[2], 5, 7, 11 [see example 3], 25* [integrals, visualize etc use one of these sections as a template: 15-4-5.mw].
Th: 15.4: read this worksheet on probability (Stewart section 8.5); [standard deviation?, Poisson distribution?];
15.4: (probability): 27, 29, 30 (30a: P(x<=1000,y<=1000) =??, 30b: P(x+y<=1000) = ??] [Maple is the right tool for evaluating probability integrals!],
[31*, use this template for the normal distribution; do a 3d plot as in the example with boxed axes to estimate the volume to compare with the numerical value of the integral to see if it makes sense, roughly].
F: Quiz 9 polar coordinate integral setup [see 17S quiz 8];
handout: example of iterating triple integral 6 different ways [tripleintexample.mw];
changing iteration to adapt to rotational symmetry 15.6.ex3 [pdf];
15.6 (triple integrals) : 4, 5, 13 [optional visualization], 17, 21,
("deconstructing a triple integral"): 27.

WEEK 12 [-1]:
M: handout exercise: exercise in setting up triple integrals in Cartesian coordinates (please take this seriously, hand in at our next class stapled to your work with name for review, not a grade) [soln];
try to sketch 31 using the same technique: solid enclosed by y = x², y = x, y+2z = 4;
[then see 3d Maple Maple plot; note two projections of the solid onto coordinate planes are actually faces of the solid, the third face has a border obtained by eliminating y from the two equations given in the figure to describe the condition on x and z for that edge curve];
use the standard maple expression palette icon for the definite triple integral of the constant function 1 to check the agreement of two different iterations with two different variables for the innermost integration step.
we go over previous day handout exercise.
W: 15.6: in class we try to sketch this solid (3d centroid example): 23; find the volume of the wedge in the first octant that is cut from the cylinder y²+ z²= 1 by the planes y = x, x = 1 then use Maple to evaluate triple integral you set up for its volume; how many ways can this be done as a single triple integral? Find the centroid; [centroid of hemisphere];
15.6: 31 [careful explanation from scratch], 33, 35, 40.
Th: read handouts on cylindrical and spherical coordinates and cylindrical and spherical regions of space and their bounding surfaces: examples (cylindrical part only until tomorrow, bob example);
15.7 (cylindrical coords): 1, 3, [5, 7], 9 (in addition give ranges of cylindrical coordinates describing interior of this sphere), 12, [15], 17, 20, 21, 29, 30.
F: Take Home Quiz 9 [Like 15.6.34 but deconstruct integral to see how diagram is given, including supporting diagrams, then reorder in another given iteration order to re-evaluate the same integral, then use both to evaluate the centroid];
handout on cylindrical and spherical triple integrals: examples;
handout on radial integration diagrams for simple circles and lines (cylinders, spheres, planes);
[bob example again];
15.8 (spherical): 1, 3, 5, 7, 9 (use double angle formula!), 11, 13 , 15, 17, 25, 29, 35, 41.

12 noon brief Climate Action "Fridays for Future" at the Oreo.

The logical extension of multivariable calculus and linear algebra leads to this:
simulated black hole image.

WEEK 13[+1]:
M: 15.R (be able to use arctrig functions as in this polar coordinate integration example);
review online: integration over 2d and 3d regions;
15.R: 12, 18, 19, 25, 27, 38, 43a, 47, 48, 53.

Tuesday 5:15-6:15pm Voluntary MLRC problem session if at least one person commits to attending.
W: Take home Test 3 out: four HW-like problems on Chapter 15 (no tricks, open resource,no collaboration).

Easter Recess:
W: 16.1: 1, 5, 11; 21, 23, 29, 31, 33;
comparison shopping (think of this as matching game, to see how to distinguish some feature of the formula by its graphical representation):
11-14: < x,-y>, <y, x - y>, <y, y + 2>, <cos(x+y), x> ;
15-18, <1, 2, 3>, <1, 2, z>, <x, y, 3>, <x, y, z> ;
29-32: x²+y², x(x+y), (x+y)², sin(x²+y²)^1/2;
19*; just try the template for this problem, no need to submit, or just read the worksheet and then the result, with bonus problem 25 done as well.

For full Maple credit you only need 4 total points, which you can get from Maple13, 14 or 15 combined.

Armenian Genocide Day.
Th: handout on line integrals (ignore text discussion of "scalar" integrals with respect to dx, dy, dz separately);
16.2 ( ∫ f ds scalar line integrals): 2, 3, 11 [write vector eq of line, t = 0..1];
31, 33, 36 [if radius 1, ans: <4.60,0.14,-0.44>, worksheet compares with centroid: obvious midpoint, also bonus: 33 solution].
F: Takehome Test back?
16.2: handout with exercise on vector line integrals (solution of exercise);
( ∫ F · dr = ∫ F · (dr/dt) dt = ∫ F · T ds = ∫ F₁ dx +F₂ dy + ... ; always use vector notation!):
7 [ ∫_C <x+2y, x²> · <dx, dy>], 15, 17, [read: 27 Maple template for vector line integrals], 32, 39;
45, 51 (notice projection along line constant on each line segment, so can multiply it by the length, add two separate results);
scalar line int: 48;
inverse square force line integral example from 16.3 example 1.

WEEK 14:
M: handout on "antidifferentiation" in multivariable calculus: potential function for conservative vector field;
16.3: 3, 7, 11 (b: find potential function and take difference, or do straight line segment line integral), 15 (potential function); 23, 25, 33, 35.
Optional note: the final section of 16.3 on conservation of energy is really important for physical applications and so it is worth reading even if it is not required for this course. Enough said.

Optional glimpse at surface integrals for the curious.

Take home Test 3 due in class if you did not already turn it in Friday or ask for further extension.
T[W]: 16.4 (Green's theorem): 3, 7, 9, 17, "WA:501"
[polar coord problem not in WebAssign: 18, convert double integral to polar coordinates];
[optional: the line integral technique for integrating areas of regions of the plane is cute but we just don't have time for it so you can ignore it.]
W: 16.5: handout on divergence and curl, Gauss and Stokes versions of Green's Theorem;
handout on interpretation of circulation and flux densities for curl and div [.mw: visualize];
[magnetic field lines; electric field lines];
16.5 (curl, div): 1, 3, 9-11,
12 [easier to interpret vectorially if convert to "grad = del, div = del dot, curl = del cross" form],
13, 19 [a magnetic field has div H = 0, so H = curl A is a way of representing it in terms of a vector potential so that it automatically has zero divergence, see problem 38; a static electric field has curl E = 0 so a scalar potential E = - grad φ is relevant---the minus sign is another story!],
31 [but read 37, 38 and look at identities 23-29], 32 [pdf].

Test 3 answer key online.
Th: CATS 20 minutes; short story about the rest of the chapter; final exam chat: not cumulative.
[Final exam will involve line integrals, potential functions, Green's theorem, double integrals in Cartesian and polar coordinates of regions of the plane, divergence and curl evaluation; no triple integrals in non-Cartesian coords]
archived exams

F: 4:30-6pm MLRC voluntary problem session for those taking final exam Saturday.

Th: exam week, 3:00pm MLRC voluntary problem session for those taking final exam Friday.

∞ scroll up for current day

FINAL EXAM: (switching days allowed but notify bob)
MWF 10:30 Sat, May 4 10:45 - 1:15 JB 211
MWF 12:30 Fri, May 10 2:30 - 5:00 Mendel G92!

MAPLE HW files:
maple13.mw due: Week 6
maple14.mw due: Week 10
maple15.mw due: Week 12? nah!

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

Test 1: Week 5: ; MLRC 5:30 problem session .
Test 2: Week 9-10: ; MLRC 5:30 problem session .
Test 3: WEEK 13: Take home out ; in ; MLRC problem session

FINAL EXAM: MWF 10:30 Sat May 6 10:45--1:15; MWF 12:30 Fri May 12 2:30 - 5:00; (switching days allowed but notify bob)

Graphing Calculator / Maple Checking ALLOWED FOR ALL QUIZZES/EXAMS

30-nov-2018 [course homepage]
[log16s, log17s (18s lost!) from last times taught with Stewart Calculus 8e]

does anyone ever scroll down to the end?