MAT2500 14S [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. [Sorry, no amusing You
Tube links.] [There are 56 class days in the semester, numbered consecutively below and labeled
by the (first initial of the) day of the week.] Usually it will be summarized on the white board in class, but if
not, it is your responsibility to check it here. You are responsible for any hyperlinked
material here as well as requesting any handouts or returned tests or quizzes from classes
you missed. Homework is understood to be done by the
next class meeting (unless that class is a test).
Textbook technology: Red numbered homework
problems have hints on the
textbook [7e Early
Transcendentals] website
TEC site, which also has tutorials and on-line quizzes and web extras. You just
need to do a short Java software install on your laptop first to use it.
- M (January 13, 2014):
GETTING STARTED STUFF. By Wednesday, January
15,
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 "[mat2500]",
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 (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 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 (better yet, right click on
the link and open it in another tab to get rid of all that MyNova crap at
the top of the window!):
[
http://www.homepage.villanova.edu/robert.jantzen/courses/mat2500/ ],
3)
Open Maple if you already
have it by clicking on this maple file link:
equianglespiralshell.mw]
[Duke
website]
And then bob will open the first Maple file in the handouts folder
linked to our course home page.
4) bob will quickly show you the computer environment supporting
our class.
[maybe he will try to impress you with this gee whizz!
Maple video; naahh...we'll leave this
only to the curious among us.]
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.
Afterclass:
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
17
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.
7) Read computer classroom
/laptop etiquette.
8)
Homework Problems:
12.1: 1, 2, 3, 5 (short list 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: handouts/stewart6e_12-1.pdf). It is
important that you read the section in the book from which homework problems
have been selected before attempting them. Here is an example of a PDF
problem solution: Stewart 12.1.42 [Okay,
I cheated and looked at the solution manual to see how to get started. Then I
made a nice Maple worksheet of the
problem, just to have an example of a Maple worksheet to show you.
Don't worry, we will take it slow with Maple.]
It is
important that you read the section in the book from which homework problems
have been selected before attempting them.
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.
Block out your time slots, identify by name of course like MAT2500.
- T: handout on diff/int/algebra [ok,
tomorrow, I forgot the copies were in the same folder as my schedule sheets];
return your schedule forms at the beginning of
class;
check phone number, dorm info on daily signup sheet from first day entry;
check out the
textbook website
homework hints (Calculus, Early Trancendentals) and extras;
12.1: 11, 13, 15,
19a
[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],
21a, 23, 33,
37;
12.2: 1, 2, 3, 5, 7, 9, 13, 15, 19,
21,
25.
- W:
handout on course rules, syllabus;
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 (tension result
given in units of force; vertical component balances downward gravitational
force F = mg, g = 9.8 N/kg, where m = 0.8kg) [pdf,.mw],
45.
- F: No Quiz first week;
Review ends, now we start new material: vector geometry
beginning with the dot product;
12.3: [example 3];
1, 3, 5, 9, 11, 15, 21 (it is enough practice
to find just one angle, say between PQ and PR), 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: no class MLK Day.
- T:
SNOW DAY. Read handout on resolving a vector
and this Maple worksheet: [using
Maple (for dot and cross products and projection)];
12.3: 39, 45, 46 [ans: b⊥ = ortha
b = <1.18,-0.29>], 49.
[Here I can put vectors
in boldface, but by hand we always need an overarrow to remember the symbol
represents a vector!]
- W: SNOW DAY; cross product: handout on
geometric definition;
12.4: [why component and geometric definitions agree: crossprodetails.pdf];
1, 5, 8, 11, 15 (move u right so initial points coincide),
16,
19, 21,
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).
> <2,1,1>
· (<1,-1,2>
x <0,-2,3>)
[boldface "times" sign and boldface
centered "dot" from Common Symbols palette]
> <2,1,1> · <2,1,1>
then take sqrt (Expressions palette) to get length [example
worksheet: babyvectorops.mw]
ignore this:
[triplecrossproduct identity?]
- F: 12.4.38: Use Maple to enter the difference vectors AB,
AC, AD
and find their triple scalar product.
32a. Use Maple to find a vector as
requested, then find the unit vector giving its direction.
Study 13S Quiz
1 [for practice, redo the projection diagram sketch for projecting P2P3
along P1P2]. Catch up on HW.
WEEK 3[-1]:
- M: Quiz 1
; it is helpful to bring a straight edge to draw lines;
[look at quiz archive
2500 13S to
get an idea what I expect, given any two vectors in the plane, one can
project one along and perpendicular to the other];
we can even make an animation of this with a little extra care showing both
the acute and obtuse included angle case for the projection:
handouts/triangleprojectionvideo.gif
[.mw];
detour: 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 JUST NEED TO WORK WITH SIMPLE
CURVES.
10.1: 1,
9,
13, 17 [hyperbolic
functions, Stewart 7e section 3.11: cosh2 x- sinh2
x = 1, recognition is enough],
19, 21,
just for fun:
28
[eqns; 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 click on output, choose Plots,
PlotBuilder, 2d parametrized curve.
- T: 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 [ans: a) x = 2+ t, y = 4 - t, z =
6 + 3 t ; b) (0,6,0), (6,0,18)],
17,
19; 23, 25, 31,
41, 45, 51, 55.
- W: handout on geometry of 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: 57, 69 (find point on line, project their difference vector
perpedicular to the
line),
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 (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 (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).
come visit me
5 minutes in my office during weeks 3, 4, 5,
the sooner the
better if you are having any troubles [test 1 in week 5: tues?]
- F:
Quiz 1 answer key online, make sure you read it carefully;
13.1:class roster handout; let's try to find one
or two partners for Maple assignments (groups of 2 or 3);
Maple assignments start (read these
instructions): note asterisks;
13.1: [cubic,
cutcylinder] 1, 3, 5, 7,
11, 13, 21-26
(do quickly, note technology is not necessary here to distinguish the
different formulas: .mw),
25,
37*
[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],
41
[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],
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).
WEEK 4[-1]:
- M: SNOW DAY;
13.2: 1[pdf], 2,
5, 7 [recall: exp(2t) = (exp(t))2], 9, 13,
15, 19, 21,
31, 29a (by hand),
29b* [graph your results using this
template; make a comment about how it
looks].
> with(Student[VectorCalculus]):
or Menu: Tools, Load Package, Student Vector Calculus
> <1,2,-3> × <1,1,1>
5 ex - 4 ey - ez
[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)
> 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,
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
- T: Quiz 2.
eqns of lines, planes, projection, area of parallelogram, volume of
parallelopiped;
on-line handout:
key idea of vector-valued
functions and the tangent vector (see Maple video:
handouts/secantlinevideo.mw);
13.2: 22 [check
your answers against Maple worksheet],
33 [angle between
tangent vectors],
37 [use
technology to do the integrals,
then repeat using u-substitution on each simple integral and compare
results],
43, 51, 55 [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]
- W: SNOW DAY.
13.3: arclength toy problems require
squared length of tangent vector to be a perfect square to be integrable
usually!
on-line handout:
arclength and arclength parametrization;
3 [note the input of the sqrt in the integrand is a perfect square in
this problem];
7 (or 9) [use numerical integration either with your graphing calculator
or if you use Maple, right-click on output of integral, choose
"Approximate"; 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],
13.
- F: SNOW DAY.
13.3: 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: rescaled
twisted cubic, helix]
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];
47 [perfect square!], 49;
>
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.
show and tell 2d curves:
osculate-parabola.mw,
osculate-ellipse.mw
WEEK
5[-1]: make up quiz 2 today, tomorrow if you missed it, talk to bob.
- M: catch up. Friday materials postponed to today.
- T:
13.4 (no 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],
37 [note that v2 = 32(1 + t2)2
is a perfect square],
41 [also perfect square, see 11] .
optional on-line only:
osculating circle how to
describe mathematically using vector algebra.
- W: online only: projections revisited
just for those who like vector geometry;
on-line reminder of dot and cross products and
length, area, volume;
Except
for the maple osculating circle plot with center C(0,-1/2) and radius 1/2,
IGNORE THIS first block of the ASSIGNMENT; SNOW DAY SHORTENING OF
EXTRA DAY ON CHAPTER 13,
but read solution below for physical
application of what we have learned to banking of curves on a highway.
13.4: 19 (minimize a function when its derivative is zero
(critical point)! confirm minimum by plotting function);
13.R. (p.874-875):
14a [use parametrized curve r
= [t, t4 - t2,0], evaluate T '(0) before simplifying derivative
(i.e., set t = 0 before simplifying the expressions after
differentiating) to find
N(0)
easily, 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];
This is the most interesting HW problem:
24. [Note b) has
answer 52 ft/sec = 36 mph] READ THIS: [solution].
Maple 13
is due any time next week;
did you do your 5 minute office visit?
INSTEAD DO THIS:
14.1: 1, 3, 11, 13,
15, 21, 27, 31, 33, 39,
47;
|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)],
79a (read only b,c; if you are interested to see how the data is fit
see example 3);
after finishing the
preceding, for fun look at 59-62 (maple plots
reveal relationships, try to see correlations between formulas and 3d plots,
then the contourplots).
Monday
5:30-6:30pm voluntary problem session for
Test 1 at the MLRC (Falvey, Floor 2, right side).
- F: Review archived
13S Test 1:
14.2: 1, 2,
5, 13, 15,
23* [toolbar
plot option: contour, or "style=patchcontour" 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].
Valentine's Day:
but
better!
[Maple: heart.mw]
WEEK 6[-1]? Maple 13.mw
due this week any time through the weekend, read Maple HW page, see collected problems there;
- M: 5:30-6:30 pm voluntary problem session for
Test 1 at the MLRC (Falvey, Floor 2, right side).
finally partial derivatives! 14.3:
1, 3,
5,
11, 15, 17,
21, 31, 33, 41,
51
[in class if time: 22, 24, 28, 30].
- T: Test 2 (thru 13.4). Come a bit early if you can!
- W: 14.3: second and higher derivatives
(and
implicit differentiation!)
47, 49; 53,
55, 59, 63, 67;
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],
83, 84, 88.
- F: 14.4: (linear approximation and tangent planes:
differentiability illustrated): 1, 3, 7,
11, 15, 17,
21, 23.
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 and differentials
WEEK 7[-1]:
- M:
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.
Make your response self-contained. [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]
14.4: (differentials):
25, 27, 29, 33, 35, 39 [remember partials of this function from 14.3.83].
Note:
order of partial derivatives does not matter if they are all continuous!
14.3: 97 (note: continuity of the first partial derivatives of a function
imply the function itself is continuous).
- T: We skipped Quiz 3 so Quiz 4; 14.5 chain
rule: 1, 11, 13, 15,
17 (I never use tree diagrams), 21,
35,
39a.
- W: 14.5: 31 (no need to remember these equations,
just implicit differentiate and solve in practice), 41 [units?], 43 [in
degrees per second?], 45 [pdf],
49.
optional: 53 [just read this to see how it works if you are
interested: maple,
pdf; note this "coordinate transformation"
of this second order derivative expression is extremely important for
gravitational, electromagnetic, quantum mechanical and heat transfer
problems, among many others].
- F:
14.6 (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.
Test 1 and Quiz 4
answer keys now online.
Spring Break.
enjoy and be safe.
- M: handout on derivatives of 2d and 2d
functions [maple gradient and
directional derivative example][Stewart
Example 14.6.7];
14.6: 27b, 31,
36, 38,
41, 47 (derive
equations of plane and line by hand), 49, 55,
this one is fun: 61 [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 in class with any partner? f(x,y,z) = x y + y z + z x
= 3 at (1,1,1) -> tan plane
math procrastination links:
cool
surface graphics link [umbilic
torus mirrored surface]
[more math art]
more procrastination: if you missed the first Cosmos show last night,
see it tonight 10pm on the History Channel:
http://www.cosmosontv.com/?gclid=COTWzfKBiL0CFewDOgodhnQA5g [
http://en.wikipedia.org/wiki/Cosmos:_A_Spacetime_Odyssey ] [National
Geographic Channel]
- T: Quiz 5 on implicit differentiation, linear and
differential approximations;
handout on 2D 2nd derivative test
[with bonus handout on multivariable
derivative and differential notation];
14.7: 1, 3, 5,
7, 13, 23 (do by hand,
including second derivative test and evaluation of f at critical points);
23* [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?)];
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: 19 (a warning that extrema are not always isolated
points);
boundaries: 31;
word problems: 39
[minimize square of distance],
47 (similar to 43 only with different
coefficients in the constraint equation),
51, 54
[use constraint to eliminate r, maximize resulting function of 2
variables on triangular region, consider triangular boundary (sol:
pdf, mw)];
read
55 [this explains least squares fitting of lines to data, and perhaps
the most important application of this technique to practical problems].
- F: Today is
Pi Day:
π!
[and Einstein's birthday];
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: 50 [ans: the height is 2.5 times the square
base; obviously cost of materials is not the design factor for normal
aquariums,
no?].
Maple14.mw is due
anytime next week through the weekend.
WEEK 9[-1]:
- M: St Pat's Day! [Hoops
and Yoyo]
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, 9;
15* [note: (m,n) = (1,1), (2,2),
(4,4), (8,8), (16,16), (32,32) = (2p,2p)
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?] .
5:30-6:30pm voluntary problem session for
Test 2 at the MLRC (Falvey, Floor 2, right side).
Note archived quiz 7 on max-min
problems.
- T: Test 2 chapter 14, including a simple
max-min word problem, which was part a of an assigned HW problem, and
totally equivalent to another one: minimizing the distance of a point to a
plane.
- W: 15.2: 1,
3, 7, 11, 21 [which order avoids integration by
parts?], 23,
31,
33* (used boxed axes!);
[iterated integrals in Maple (how to enter)]
step by step checking of multiple integration (worksheet):
> 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)
- F: 15.3: handout on double
integrals;
5, 15,
17, 25, 27;
39* [just use Maple to evaluate the integral once you set it up],
47, 49,
57, optional: 1, 9, 53.
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.
Confirm Test 2 and
recent quiz grades on Blackboard agree with your paper copy grade. Check out
answer key to test 2.
WEEK 10[-1]:
- M: jump to 15.7: 1, 3, 5,
13 [optional
visualization],
15 [make a diagram, do x or y integration
first, note that the tilted plane faces are described by the equations of
lines in the xz or yz planes],
21 [write 6 integrals for it], 27 [write 4 integrals
for it];
handout: example of iterating triple
integral 6 different ways.
- T: Quiz 8 (exchanging order of
integration in double integral: you need Maple to do
integrals you cannot do by hand);
first do the handout exercise:
exercise in setting up triple integrals in Cartesian
coordinates (please take this seriously, hand in
Wednesday stapled to your work with name); then try unaided the book problems:
15.7:
29,
31 [see 3d Maple plot: 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,
careful explanation pdf];
if you cannot get both 29 and 31, try 33 where the diagram is made for you.
31* 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.
CONTEXT: While a few of you may learn how to illustrate triple
integrals (my hope), I will only test all of you on being able to iterate
triple integrals given the 3d figure already drawn for you as in problems
33, 34, and the handout problem once you have the 3d figure given to you.
- W: turn in triple integral worksheet; catch up on 3d
integrals. [return to instruction page
for links to solution]
15.R. (review problems at end of chapter): just
draw the rough sketch of the regions of integration for 41, 42 which we will
revisit later with polar and spherical coordinates to evaluate (identify the
circle and sphere by their standard equations); 47, 48.
- F:
handout: review polar
coordinate trig;
handout on
polar coordinates and polar coordinate integration
(the integration is next time);
10.3 pp.654-659 (stop midpage: tangents in polar coords unnecessary
for us),
pp.661-662 (read graphing in polar coords [more
polar fun]);
10.3:
1, 3, 5, 7, 9, 11, 15,
17, 21,
25, 29, 31,
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., top 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]:
- M: 15.4:
(areas but no lengths): use 1, 4, 5, 7, 21,
23 [twice the volume under the hemisphere
z = sqrt(a2 - x2 - y2) above
the circle x2 + y2
≤ a2],
25
[integrand is difference of z values from cone (below) to sphere
(above) expressed as graphs of functions in polar coordinates].
If you like are intellectually motivated by the pursuit of knowledge as
opposed to only doing college as a way to increase your economic standing in the
world, go see the film Particle Fever playing this week in
Bryn Mawr.
- T: 15.4, 15.7 revisited briefly with polar
coordinates
recall 15.7.ex3 and re-express the outer
double integral, inner limits of integration and integrand in
polar coordinates in the x-z plane (pretend z is y,
or simply interchange them since they are dummy variables of integration):
∫ ∫ r<=2
[∫y= x2+z2..4 (x2+z2)1/2 dy]
dA = ∫ ∫ r<=2
[∫z= x2+y2..4
(x2+y2)1/2 dz]
dA (see book discussion on page 1020);
15.4: 15 [at what acute angular negative or
positive values of theta
does cos(3θ) =0? these are the starting and stopping values of
θ
for one loop],
29, 31, 35 [what is
the average depth? (integral of depth divided by area of region)],
36 [ans:
a): 2π(1-(1+R) e-R)],
39
(make a diagram, assembling the 3 integration regions into one simple region,
then it is easy!);
Optional: Read 40 [this enables one to sum the probability under a
normal curve in statistics, grade curving, it's important!]
Mandatory: read solution to triple integral worksheet exercise
[.mw each integral with integrand 1
should give volume 8/15, but so do many incorrect iterations!] Quz answer
key posted.
- W: Read 15.5 carefully;
15.5 (center of mass, "centroids" when constant density; skip moment of
inertia): 5, 7, 11 [see example 3], [integrals, visualize etc: .mw].
- F: Quiz 9 on polar coordinate integration (see
13S c-e Quiz 8 [10S, 06S,
recall relevant handout]);
15.5:
(read this worksheet: probability):
27, 29,
30a [P(x<=1000,y<=1000) = .3996], 30b
[P(x+y<=1000) = .2642]. [Maple is
the right tool for evaluating probability integrals!],
31* [handout:
distributions of stuff]
WEEK 12[-1]:
- M:
handouts on
cylindrical and spherical coordinates and cylindrical and spherical
regions of space and their bounding surfaces: examples;
15.8 (cylindrical): 1, 3, 5, 7, 9 (in addition
give ranges of cylindrical coordinates describing interior of this sphere), 13, 15, 17,
21, 29.
- T:
handout on cylindrical and
spherical triple integrals: examples;
15.9 (spherical): 1, 3, 5, 7, 9, 11, 13 , 15,
17, 21,
25, 39.
Quiz 9 answer key
online.
- W: handout on
radial integration diagrams for simple circles
and lines (cylinders, spheres, planes);
in class review problems,
homework:
15.8: 27;
15.9: 23, 29, 35;
15.R (p.1050): 19, 20, 27, 37a, 41, 47;
review online: integration over 2d and 3d
regions.
Th: 5:40pm MLCR problem review session.
- F: Take home Test 3 out in class after HW discussion.
READ TEST INSTRUCTION WEBPAGE FIRST.
[see test quiz page
online after 2:30pm]
WEEK 13[+1]:
- M: Maple 15 optional if you want drop another low
quiz grade;
16.1: 1, 5, 9; 21, 25;
comparison shopping:
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, no need to submit
[check here for result, with bonus problem
25 done as well].
- T: 16.2: 16.2: (f ds scalar line integrals: pp.1063-1068 midpage): 1,
3, 9, 11 [write vector eq of
line, t = 0..1];
33,
36 [ans: <4.60,0.14,-0.44>, worksheet
compares with centroid: obvious midpoint, also bonus: 33 solution].
- W:
work on Test 3 in class.
Easter Break
any car enthusiasts out there?
my brother's car.
- T:
handout on line integrals;
16.2 (F · dr = F
· (dr/dt) dt = F
·
T ds
vector
line integrals):
7 [ ∫C <x+2y, x2>
· <dx, dy>],
17, 19, 21,
27, 29a;
45, 51
(notice projection along line constant on each line segment, so can multiply
it by the length, add two separate results);
optional: 42 [ans: K(1/2-1/sqrt(30)],
48.
- W: Test 3 in?
16.3: 1, 3, 5, 11 (b: find potential
function and take difference, or do straight line segment line integral), 15
(potential function);
25, 35.
Optional
note: the final section of 16.3 on conservation of energy is really
important for physical applications but not required in this course. Enough said.
- F:
16.4: (Green's theorem) 1, 3, 17,
18
[convert double integral to polar coordinates; ans: 12
π];
[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.]
WEEK 14[-1]:
- M: handout on divergence and curl, Gauss
and Stokes versions of Green's Theorem.
Final section 16.5: 1, 5,
9, 11,
12
[easier to interpret vectorially if convert to "del, del dot, del cross"
form],
13, 15,
31 [but read 37, 38 and look at identities 23-29].
- T=F: optional handout on interpretation of circulation
and flux densities for curl and div [visualize];
Optional 16.6-7: surface integrals for fun [hemisphere
example];
Optional 16.8-9: read lightly Stokes Theorem, Gauss's law if you are
interested, when you have time;
[example1: centroid of
hemisphere,
Gauss law example, wedge of cylinder 16.7.example3;
example2: parabola of revolution (Stewart16.7.23 expanded
into Gauss/Stokes examples)].
- W=M: last day of class.
archived final exam
online;
polar coordinate example with a paraboloid inside a cylinder:
intparaboloid.mw;
CATS Teaching evaluations.
Test 3 answer key online Friday. You may
come see me to discuss your test if you wish.
Tuesday, May 6 5:30pm for last problem session in MLRC
Final Exam
Wed, May 7 [MWF1:30 slot]: 2:30 - 5:00:
Fri, May 9 [MWF12:30 slot]: 2:30 - 5:00:
email bob to switch time slots
MAPLE HW files:
maple13.mw due: Week 6
maple14.mw due: Week 10
maple15.mw due: Week 13?
*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 12:30 Fri, May 9 2:30 - 5:00; MWF 1:30 Wed, May 7
2:30 - 5:00; (switching days allowed but notify bob)
Graphing Calculator / Maple Checking ALLOWED FOR ALL QUIZZES/EXAMS
24-mar-2014 [course
homepage]
[log from last time taught with Stewart
Calculus 7e]
[Duke
website]
; it is helpful to bring a straight edge to draw lines;
eqns of lines, planes, projection, area of parallelogram, volume of
parallelopiped;
but
better!
enjoy and be safe.
any car enthusiasts out there?
my brother's car.