MAT2705 08F 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.] 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, in which case the homework
is due the following class meeting).
- M: GETTING STARTED STUFF. By Wednesday, August
27, e-mail me [robert.jantzen@villanova.edu]
from your OFFICIAL Villanova e-mail account (which identifies you with your
full name) with the subject heading "[MAT2705-XX]", where XX is your
section number 01, telling about your last math courses, your comfort level with graphing
calculators (can you do symbolic derivatives and integrals on your graphing
calculator if not in Maple?) and computers and math itself [sophomores only: how much experience you have with MAPLE (and Mathcad
if appropriate) so far], why you chose your major, etc.
Tell me what your previous math course was named (for sophomores: Mat2500 = Calc 3 or
Mat1505 = Calc 2).
[In ALL email to me, include the string "mat2705"
(no spaces) somewhere in the
subject heading if you want me to read it. I filter my email.]
In class if you brought your laptop (otherwise patiently watch bob):
1) log on to your computer and open
Internet Explorer.
(IE allows you to open MAPLE
files linked to web pages automatically if MAPLE is already open or if it
is available through the Start Menu Program listing under Math Applications, in
Netscape you must save the file locally and then open in it MAPLE using the
File Open task.)
2)
log on to 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 My Courses classroom site, and visit the link to
my course homepage from it
[
http://www.homepage.villanova.edu/robert.jantzen/courses/mat2705/ ],
3)
Open Maple 12 Standard (red not yellow icon) from the Start Menu Program
listing under Math Applications
[or click on this maple file link:
Maple example file]
[Using the File OPEN URL command in Citrix Maple, with this URL (right
click, copy shortcut) will show
you a Maple file of how to pick the viewing window for realistic exponential and sinusoidal example functions.
Clicking on the !!! icon on the tool bar with execute this worksheet
and restore the output which has been removed to make the file smaller. It
will also automatically open the two subsections, which can be closed by
clicking on the triangle at the section title. Try this at home to make sure
you can access citrix Maple.]
4) bob will quickly show you the computer environment supporting
our class. And chat up a bit the course. [cell phones on vibrate or off: bob
will give you his cell phone number in class, not on the web!] [maybe he
will try to impress you with this gee whizz!
Maple demo]
Afterclass:
5)
log on to MyWebCT and look at the Grade book: you
will find all your Quiz, Test and Maple grades here during the semester once
there is something to post.
[This is the only part of WebCT we will use this semester.]
Check out the on-line links describing aspects of the course (no need yet to look at the
MAPLE stuff). Fill out your
paper
schedule form (get a copy in class to fill out or print it out
back-to-back earlier
to fill out in advance and bring to your first class already done; consult the
dorm abbreviation
list)
to return in class Tuesday.
[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) Make sure you can access and open Maple 11 Standard in citrixweb
on your laptop. If you have any trouble, email me with an explanation of the
errors. If I receive no email, it means that you were successful. The
first time you use citrix on a computer, you must download and install the
client from the link on that web page.
You are expected to be able to use citrix Maple on your laptop or Windows
Maple in class when
needed. We will develop the experience as we go.
7) Read computer classroom
/laptop etiquette.
Then read the first paper
handout: algebra/calc background
sheet.
8)
Homework Problems:
1.1: 3, 5,
13, 33 (this links to a PDF scan of the HW problems from the book if you have
not yet purchased it);
[ "A is proportional to B" means "A = k B" where k is some constant,
independent of A and B]
(short list so you can check out our class website and read about the
course rules, advice, bob FAQ, etc, respond with your email).
It is
important that you read the section in the book from which homework problems
have been selected before attempting them.
Get acquainted with Maple11
Standard DE entry and "odetest" for problems 3,5,13 [even
33!]
[use citrixweb Maple on your
own computers]
OKAY, bob's laptop lost wireless, maybe because of low battery. We'll
examine the class web and Maple in our computer classroom tomorrow.
- T: return your schedule forms at the beginning of
class [did you use the 3 letter
dorm abbreviations?];
[feed the URL of this maple file link:
Maple example file into Maple in class following
bob's lead];
1:1: 7a [i.e., only check y1(x)], 23;
formulating DEs: 27, 29 [Hint: recall perp lines have slopes which are negative
reciprocals, the normal line is perpendicular to the tangent line and passes
through the same point on the curve; make a diagram of the given point (0,1), a "generic" function graph curve
whose normal at a point (x,y) on the curve passes through
(0,1), and draw in the connecting normal line segment between these two
points and the perpendicular tangent
line, then compute the slope of the normal line from the two points, and then from the
negative reciprocal of the derivative value, finally equate the two
to get the DE],
35, 36.
Tuesday will be our normal day for weekly quizzes: Quiz 1 Not! (try
it later without the book 20-30 minutes max: then check answer key [hand,
maple]).
- W: course information
handout;
1.2: 1, 5, 15, 19 [Hint: write piecewise linear function from graph: v1(t) for first expression,
v2(t) for second expression, solve first IVP
for x1(0) = 0, second IVP for x2(5) = x1(5)], 25 (like lunar landing problem,
see example), 39, 40 (ans: 2.4mi),
43 (pdf).
- F: on-line handout:
initial data: what's the deal?;
1.3: 3:
[hand draw in all the curves on the printout supplied by bob (rotated and
scaled to fit option on Print), maple
version],
8* : our first Maple HW problem: read the
Maple HW instruction page, then try the Maple direction field
command in this linked template to reproduce the
two given curves by using
two appropriate initial data points [Hint: notice that the y-axis
crossing points are integers!] Then include all the initial data points of
the red dots (some half-integer values); this is also described on p.29
for both Maple and Matlab, but the above template already has the
appropriate section copied into it for you; if you can find a partner to
work with in time fine, otherwise try this yourself and find a partner for
future problems, then merge your work [will not be due until we
finish chapter 1];
being careful:
11, 15 [note Dyf is just df/dy above Eq (9) in the
text]; 27
[don't worry about making a diagram];
a freebie: numerical problems caused by
vertical tangent line (figure 1.4.2).
WEEK 2[-1]:
M: Labor Day.
no class.
- T: Quiz 1 through 1.2;
1.4: 1, 5; 21, 25, 27, 29.
- W: class list contact data sheet handout (not web
published) to help form partnerships;
handout on
exponential behavior/ characteristic time [read
this worksheet explicit
plot example];
1.4: 45, 47 [recall 1.1.35, use units of
thousands of people, assume no one has heard the rumor at t=0: N(0)=0 (an
approximation in units of thousands; really N(0)=.001 would be one person
starting the rumor, or should it be two?)],
65;
Optional challenge problems:
68 (PHYS: parametric cycloid solution, obvious typo:
θ = 2t in text, not
θ = 2a)
or 69 (ME: suspension cable catenary: hyperbolic cosine)
or Torricelli's law section and 64 (ChmE: ans r = 1/35 in).
- F: check quiz 1 grades in BlackBoard gradebook accessible through
our My Course Class Home Page;
on-line handout:
recipe for first order linear DE;
1.5: 3, 7, 11, 21; 27;
21*[check both the general solution and the initial
value problem solution with the dsolve
template].
For future reference:
> deq := y ' = x y
[space implies multiplication]
> sol:=dsolve(deq, y(x))
> solinit := dsolve({deq, y(0)=1}, y(x))
[or enter DE plus IC separated by a comma, right click on output,
choose Solve DE, for y(x), or Solve DE interactively]
> y ' = x y , y(0)=1
> odetest(sol,deq)
[this returns 0 if LHS-RHS of DE is 0]
2:30 Interesting
Connelly Cinema Talk by Jesuit astronomer George Coyne, friend of dr
bob:
"The Dance of the Fertile Universe: Did God Do it?"
WEEK 3[-1]:
Weeks 3 thru 4: come by and find me in my office, tell me how things are
going.
This is a required visit. Only takes 5 minutes or less. Come say hi!
Our first test will be Tuesday of Week 4 (sep 16) unless this is a conflict
for many students in the class. Give me feedback.
- M: how to
understand the DE machine;
1.5: 17, 26 [ans: x = ½
y -2 +Cy -4],
29 [use the fact that the derivative of
the function defined by the integral is the integrand including all
multiplicative factors], 31
[show by differentiation and backsubstitution into the DE, not by rederiving it],
32 [find the constants by substituting the trial solution into the DE and
choosing the values of A and B to make the DE satisfied: put all terms on LHS
and set coefficients of the sine and cosine separately to zero],
41 [Hint:
Δt in years is the fraction of the year, so S(t)
Δt is the approximate income during this time interval, 12 percent of which
contributes to ΔA, while the interest contribution is the annual interest rate
6 percent prorated by multiplying by the fraction Δt for the interest
contribution over this shorter time period, applied to the current value of
the retirement amount A(t): 6 percent of A(t) Δt; adding these and dividing by
Δt, taking limit to get dA/dt gives the linear DE [click
here only if you have tried to get the DE and could not or its solution did not give
the correct number].
41* [use the dsolve template to solve this DE with its
initial condition; evaluate answer to question of word problem to a floating
point number and give the result in dollars and cents in a comment, then think
about significant figures and give a response with fewer significant digits].
When you are satisfied, compare with
retirement.pdf.
Maple1.mw assignment complete.
Due week between Wednesday to next Monday.
REMIND BOB to match up people who need Maple partners.
- T: Quiz 2 on linear DEs;
1.5: 37
(Use
Eq. 18 in the book; what is the final concentration of salt?);
[this mixing problem (online
handout) is an
example of developing and solving a differential equation that models a
physical situation; for another word problem example,
see this]
1.R(review): classify the odd problems 1-35 as:
separable, linear in y
(as unknown),
linear in x (as unknown), some combination of these three, or NOTA (none of the above), for
example dy/dx = y/x is all three and can be solved in three different ways;
solve 25 [it can be done in two ways by expanding out the square on the rhs
before integrating with integration constant C or by using a u-substitution
as the book did with integration constant K; express K in terms of C as given in the book supplied answers by
combining the two terms in the K solution [identity for expanding out (x+1)3 !] and then comparing with the C
solution];
solve 35 in two ways and compare the results.
Do your OFFICE VISIT yet?
- W: handout on solution of logistic DEQ
[directionfield,
integral formula];
handout on DE's that don't involve the ind
var explicitly;
2.1: 5 [go thru the solution steps
just this once with concrete numbers, remember the
integral formula, follow the steps of the book or handout derivation of the solution,
compare with the final logistic formula; use the formula for the remaining HW
problems],
9, 15 [note that by definition Bo = kMPo and
Do =
-kPo2 , and these are easily solved for M and
kM in
terms of Bo, Do, and Po to get the other two
logistic curve parameters needed to solve problem 16],
16 [ans: P = .95 M
after 27.69 months].
Optional Light Reading 1.5: Application: see how a slight generalization of
the directionfield example 1.3.3 to
include an initial time parameter and time and temperature scale parameters
has a useful physical application on pp.56-58. No need to study it, but
reading through it quickly gives an idea of how one can use a simple linear DE to
model an interesting problem.
- F: 2.1: 6
[use handout integration formula with sign reversed, or use technology
for integral and combine log terms;
answer: > dsolve({x'(t) =3 x(t) (x(t)-5), x(0)=2},x(t))
(copy and paste into the input line);
plot this
solution with technology and choose a horizontal window (negative and positive t
values) in which you see the reversed S-curve nicely, be ready to give
the approximate time interval for an appropriate viewing window
in class],
11 ["inversely propto sqrt": use β = k1/P^(1/2), δ =
k2/P^(1/2) in eq.(1)],
23 [Note dx/dt = 0.004 x (200-x) so k = 0.004,
M=200 for logistic curve
solution formula],
30, 31 [ans: find βo = 0.3 from condition dP/dt(0) = 3x10^5
using the DE at t =0, then use solution
α=0.3915 from second condition (solve this yourself or at least check numerically that it satisfies the 6
month condition) to find the limiting population as t goes to infinity (i.e.
neglect decaying exponential)].
Did you stop by bob's office yet?
WEEK 4[-1]: Maple1.mw due early this week
- M: 5:30pm MLRC voluntary Test 1 problem
session;
air resistance handout [comparison
of linear, quadratic cases; numerical solution];
2.3: 1, 2, 3, 9 [remember weight is mg, so mg = 32000 lb determines
m = 1000 in USA units]. Homework is for Friday of course!
- T: Test 1
- W: 2.4: 1 (do by hand and calculator, can check with euler.mw if you wish;
optional read about improved Euler).
maple2.mw:
instructions for this maple assignment.
[due this coming week Friday to Friday]
- F: Test 1 back graded,
answer key online, check recorded grades
on BlackBoard;
handout: why LinAlg with DE?;
[maple dsolve and DEplot for 2x2 systems
of DEQs];
Read 3.1 on linear systems (review of High School topic solving 2 or 3 linear equations in
2 or 3 variables),
do
3.1: 1, 5, 7; 9, 15, 31;
Maple2.mws due next week September 29-October 2.
> {x - y = 2, x + y = 0}
Right-click and choose "solve, solve".
WEEK 5[-1]:
- M:
handout on RREF
(Reduced Row Echelon Form, section 3.3);
3.2:
1, 3,
7; 11, 13, 15 (do a few by hand, then you may use step-by-step
row ops with MAPLE or
a calculator for the rest;
you must learn a technology method
since this is insane to do by hand after the first few simple examples);
23 [can your calculator handle this?]
- T: Quiz 3 NOT!; maybe Friday when we have
something to quiz about;
handout on
solving linear systems example;
In class open this Maple file with a partner
and use the LinearSolveTutor and enter the matrix for the system 3.2.15
given there and reduce step by step,
then solve the system (if possible), then switch positions and enter the matrix
for the system 3.2.18 given there and repeat using instead the Reduction
command template there;
3.3: 1, 3, 7, 11, 17, 23 = 3.2.13;
19* [use the tutor and record your step by step reduction, annotating each
step as in the rowredex0.mw worksheet; then check with ReducedRowEchelonForm];
To solve a linear system directly in Maple, input augmented matrix with the
palette or Matrix command and these 4 lines will do it:
> with(Student[LinearAlgebra]):
A:=Matrix([[3,8,7,20],[1,2,1,4],[2,7,9,23]]);
[you enter matrix using palette instead]
ReducedRowEchelonForm(A);
BackwardSubstitute(%);
To step by step reduce a matrix and solve the system do (pick
Gauss-Jordan):
> LinearSolveTutor(A)
Matrices for doing homework with Maple
- W:
recall
3.2.23 above never discussed;
3.2: 22
[solve using row
ops on augmented matrix to rref form, then by hand backsubstitute, check
answer with Maple];
3.3:
29 = 3.2.19 [solve
using row ops on augmented matrix to rref form, then by hand backsubstitute];
29* [now check this same problem solution by solving with
ReducedRowEchelonForm, BackwardSubstitute
of augmented matrix after inputting the Student Linear Algebra
package from the Tools menu];
MAPLE
chemical reaction problem*;
word of the day: can you say "homogeneous"?;
- F: Bring your laptop if you
cannot do row reductions on your graphing calculator.
Quiz 3 on using row reduction to solve
linear systems;
3.4: 1, 5, 7, 11, 13; 17, 21; 27, 43.
Matrix algebra is easy in Maple.
WEEK 6[-1]:
- M:
3.5: 1; 9, 11; 23 [this is a way of solving 3
linear systems with same coefficient matrix simultaneously];
30 [just substitute for the inverse and then drop parentheses and multiply
out: (ABC)-1 (ABC) = ...],
32 [multiply on the left by A-1].
Matrix multiplication, matrix inverse, determinant, transpose
> A B [note space between symbols to imply
multiplication]
> A-1
> |A| or
Determinant(A) [even absolute value sign
gives determinant]
> A%T
or Transpose(A)
- T: read on-line handout:
determinants and area etc;
3.6: (determinants abbreviated: forget
about minors, cofactors, only need row reduction evaluation to understand)
13, 17, 21
(plug in just so you've done it once);
17* [record your Gaussian elimination
steps in reducing this to triangular form and check det value against your
result];
3.5. Apps:
7* from textbook on p.197,
using the inverse matrix as described on p.196,
be sure to answer the word problem question with a text comment;
are these sandwiches cheap or not?
maple3.mw assignment
is complete; due next week (or at least the week after break if you have too
much midterm stuff going on); get it over with early, catch
up with others if not done yet
[blowing off maple means subtracting 10 points from your cum for midterm
grades: maple1 and maple2 only for midterm grades].
- W: Quiz 4;
F: 4.1: 1; 5, 7; 9; 15, 17; 19, 23; 25 (use technology in
1) evaluating
the determinant or 2) ReducedRowEchelonForm
needed
for these problems);
[HW row reductions];
hand out on the interpretation of solving linear
homogeneous systems of equations: A x = 0 [optional
read:
visualize
the vectors]
- F:
handout on nonstandard coordinates on R^2 and
R^3 [goal: understand the jargon and how
things work];
1) Using the completed first page coordinate handout
made for the new basis {[2,1],[1,3]} of the plane: graphically find the new
coordinates of the point [4,7], then confirm using the matrix multiplication
by the inverse matrix of the coordinate transformation given there. Then
find the old coordinates for the point whose new coordinates are [2,2].
2) Using the previous handout as a guide, first fill in the blanks on the
sheet handed out in class for the
basis transformation matrix and its inverse, and then for the change of
coordinates, and then use either the matrix or scalar equations of the
change of variables to calculate the other coordinates for the two given
points.
Then following the example, using a ruler and sharp
pencil, make a new -2..2,-2..2 coordinate grid associated with the new basis
vectors {[1,1],[-2,1]} and graphically represent the point [x1,x2]
= [-5,1] and find
its new coordinates [y1,y2] using the grid and confirm that this agrees with your
previous calculation. Similarly read
off the old coordinates [x1,x2] of the point whose new coordinates are
[y1,y2] = [2,-1]. Confirm that these
agree with your calculations. Put your name on it and bring it to class.
It should look like the hand version of
this maple plot. or this old jpeg
image of it.
Armenian Food Festival West Dougherty 3-6pm friday
WEEK 7[-1]:
- M:
study the handout on solving linear systems revisited;
[note only a set of vectors resulting from a linear homogeneous condition
A x = 0 can be a subspace of a vector
space; interpretation: lines, planes, ..., hyperplanes through the origin];
4.2: 1, 3, 11; 15, 19;
Look at the solution to the system 3.4.21 on p.182 (back of book p.712): what does it tell us about
the 5 vectors (namely the columns of the coefficient matrix of the linear
system) in R2? How many
independent relationships are there among the 5 vectors? How many vectors are
independent? Is this obvious if you look at the explicit components of these
vectors? [no! but if you realize that no 2 vectors are proportional, and no
more than 2 vectors can be independent in the plane, of course the answer is
obvious, this is the power of reasoning].
- T:
handout on
linear combinations, forwards and
backwards [maple to visualize];
what does the solution of a
nonhomogeneous system of equations say about the column vectors in the
augmented matrix?
4.3: 1, 3, 5, 7; 9, 13; 17, 21; 23.
- W: Quiz 5; check WebCT grades T1, Q1-4, Mpl1
are posted
4.4: 1, 3, 5, 7;
9 [this is already reduced, find solution as in class example, pull apart to find coefficient vectors of
parameters, repeat for next 4 problems reducing first],
13; 15, 17, 25 [use technology for all
HW row reductions and determinants (2x2 dets are easy by hand!)].
- F: handout summarizing
linear vocabulary; [maple
illustration of how to describe a plane in space].
[optional on-line only extra handout:
linear system vocabulary]
Fall Break. :-) Enjoy. Be safe in your travels.
Week 8[-1]:
test 2 in week 9;
- M: catch up on 4.4 hw;
Part 2 of course begins: transition back to DEs:
read p.274
on function spaces and examples 3, 6, 8, 9; equivalently read the worksheet on the
vector space of quadratic functions [quadratics.mw,
pdf];
4.7: 15 [Hint: solve c1(1 + x) + c2(1 - x) + c3(1 - x2) = 0, for unknowns c1,c2,c3;
are there nonzero solutions? if not, these are linearly independent polynomials, in terms of which any quadratic
expression in x can be written; note that any two (nonzero!)
functions of x that are not proportional are automatically linearly
independent],
16 [same approach].
Memorize: y ' = k y < -- > y = C e k x
y '' + ω2 y = 0 < -- > y =
C1 cos(ω x) + C2 sin(ω x)
- T:
handout on sinusoidal example;
5.1[up thru page 290]: 1, 3, 5, 9; 13,
17;
begin maple5.mw:
9* [use the 2nd order dsolve template to
solve the 2nd order IVP as a check].
> deq:= y'' + 3 y' + 2 y = 6 ex
> inits:= y(0) = 4, y'(0) = 5
> solgen:=dsolve(deq,y(x))
> solivp:=dsolve({deq,inits},y(x))
or right click on output of (use solve DE interactive, don't forget set
braces):
> { y'' + 3 y' + 2 y = 6 ex
, y(0) = 4, y'(0) = 5}
- W: 5.1[after page 290]:
33, 35, 39,
49 [find the IVP solution for y, set y ' = 0 and solve exactly
for x, backsubstitute into y and simplify using rules of
exponents]
5.2: 1,
11;
5.1: 49*[check your IVP solution for the highest curve in Figure 5.1.6
(highest for x > 0) using
the 2nd order dsolve template and plot it together
with the horizontal line y = 16/7 claimed by the back of book answer as the highest value of
that function; recall how to plot multiple functions in the same plot:
> plot([<expression_for_y(x)>,16/7], x=a..b, y=c..d, color=[red,blue]);
the color option is useful in distinguishing two functions when it is not
already clear which is which; is 16/7 the peak value of your
solution?].
- F: Quiz 6 similar to archived quiz on
Wednesday HW (I will ask for questions before the quiz, but please prepare
by looking at it);
read handout on complex arithmetic, exponentials
[maple commands; the complex number i is
uppercase I in Maple];
5.2:
13, 17, 21, 26 [this problem
is for fun now, but later
we learn a method to attack it];
read the optional on-line handout on
visualizing the initial value problem (IVP)
if you wish to see the interpretation of solving the IVP.
WEEK 9[-1]:
- M: MLRC 5:30; homework for wednesday:
5.3: complex roots:
8 [ans: y = exp(3x)(c1 cos(2x) + c2 sin(2x))] , 9; 17;
22, 23 (express in phase-shifted cosine form);
use technology to factor or solve for roots of polynomial equations:
> solve(r2+6 r+13 = 0) (in
general, finds all roots exactly up to fourth degree and sometimes higher
degree if lucky)
> factor(r2+6 r+13) (in
general, factors real polynomials into linear and irreducible quadratic
factors)
[Note: > fsolve(...) returns all numerical roots of a polynomial].
handouts on the
amplitude and phase shift of sinusoidal
functions and
exponentially modulated sinusoidal functions [maple
videos showing different phase shifts and frequencies]; do the short
problems at the end of each handout after studying them.
[optional handout for your amusement: power
series and DEs (chapter 11)].
- T: Test 2 on Chapters 3, 4.
- W: 5.3 higher
order DEs [ignore instructions to factor by hand or
polynomial long divide: use technology for all factoring]: 11, 13;
25, 33; 39;
49 ;
49* [check your solution with the higher
order dsolve template, and also check the IC linear system solution as in
the template, edit away all stuff not directly relevant to your new problem].
- F:
5.4:
1, 3, 13,
14 [b) the solution once put in the form x = A exp(-Kt)
cos(ω t
- δ) has the "envelope curves" x = +/- A exp(-Kt) ] ,
17;
23 [pdf].
handout on linear homogeneous 2nd order DEQ with constant positive coefficients
(damped harmonic oscillator) [on-line only: examples in
nature].
WEEK 10[-1]:
- M: Test 2 back, read answer key;
5.4: 21 (solve by hand), 21* [solve the two DEs with the same initial
conditions using Maple and plot the two solution curves together, but also plot the corresponding third solution with pure damping but no
oscillator as in the template];
handout on driven
(nonhomogeneous) constant coeff linear DEs
[complete final exercise on sheet [solution], Maple: *check with dsolve];
5.5: 1, 3, 9
[we will not cover "variation of parameters"; the book
presentation of the method of undetermined coefficients is a recipe with
little justification (educated guess), instead the handout shows exactly how and why one gets
the particular solutions up to these coefficients].
- T: handout on damped harmonic oscillator driven by sinusoidal driving function
[Maple specific example]
[maple resonance plots: general ,
specific (frequency much higher than
resonance)];
5.5: 33, 38 [y = ( exp(-x)(176 cos(x)+197 sin(x)) )/85
- (6 cos(3x)+7
sin(3x))/85]; [not many of the book driving functions are physically
interesting here]
5.6: 8. Ignore the book instructions and solve this problem by hand for the
initial conditions x(0) = 0 = x'(0). Read the
worksheet about the interpretation of
the solution.
- W: [ >
x, -x, x sin(x) Enter and right click on output
to plot together to see oscillation and its envelope. or use bob's templates
in each problem.]
handout on beating and
resonance
[Tacoma Narrows Bridge collapse:
engineering
explanation;
Wikipedia;
Google (choose video finds midway down page)];
maple5.mw due next week (5 problems now complete);
5.6: 1 [rewrite as a product of sines using the cosine difference formula:
cos(A) - cos(B) = {-2 sin((A-B)/2)} sin((A+B)/2), the expression in {} is the
sinusoidal amplitude, look at the plot of x and +/- this amplitude function
together, then repeat for the HW
problem * at the end of the example worksheet and include only that HW problem
section in your maple5 worksheet],
11 [convert steady state solution to phase shifted cosine at
end of
calculation], 17 [pdf],
23.
- F: Quiz 7; study the pdf solution of 5.6.17:
17 [pdf];
recall
coupled system of DEQs [now revisit
it (paper handout giving a preview of what we are about to embark on),
bob will discuss details on Monday];
Watch the
MIT Eigenvector
3 minute video [there are 5 frames which then repeat, so stop when you
see it beginning again---from their
LinAlg course]; see the 5 possibilities of the video in this
Maple DEPlot
phaseplot template;
if you like, play a computer game with the
Duke U applet lining up the vectors [red is x, blue is A x,
click on matrix
entries to change (try on HW matrices!), click on end of red vector and
drag around an approximate circle to see corresponding blue vector];
since I am not assigning calculation HW, please take this fun assignment
seriously.
WEEK 11[-1]:
- M:
please read quiz 7 answer key;
Maple5 is complete and due any time
through next Monday;
6.1: 1, 9 , 13, 19; 27, 25;
[go thru process: use determinant: |A-λI|, solve (solve if complex roots) to find characteristic equation and eigenvalues
of a 3x3 or higher dim matrix];
- T: Quiz 8: examine resonance for quiz 7 problem
(replace ω = 2 by parameter ω, find amplitude
as function of ω and peak value if any, as in example 5.6.17 from Friday
log);
For the matrix A = <<1|4>,<2|3>> entered by rows,
from the two independent integer component eigenvectors {b1 = <1,1>,b2 =
<-2,1>}, make the basis changing matrix
B = <b1|b2> whose columns are these vectors, use the two coordinate transformations
X = B Y and Y = B-1X
to find
the new coordinates of the point [x1, x2] = [-2,4] in the plane and to find the point
whose new coordinates are [y1, y2] = [2,-1]; finally evaluate the matrix
product by hand AB = B-1AB to see that it is
diagonal and has the corresponding eigenvalues in order along the diagonal.
[Remember this? (the geometry of
diagonalization)];
6.2: 1, 9 ("defective"); 13 (upper
triangular!, det easy), 21 ("defective"); [use Maple for det and
solve for eigenvalues, then by hand find eigenvectors]
34 [just write out the quadratic characteristic eqn and solve with the quadratic formula, think about
real distinct roots; Delta is the discriminant; what if b = c
when the matrix is "symmetric"?];
- W:
handout on the geometry of diagonalization
and 1st order linear
homogeneous DE systems (2-d example: real eigenvalues [phaseplot]);
Repeat this handout exercise for the matrix of 6.2.1 (namely
A = <<5|-4>,<2|-1>>
input by rows) and the initial condition x(0) = <0,1>;
solve the 3x3 system x' = A x for the matrix of
problem 6.1.19, with initial condition x(0) = <2,1,2>;
find the eigenvalues and matrix B of eigenvectors
for the matrix A = <<0|4>,<-4|0>>
(input by rows) and verify that AD = B-1A B is diagonal.
- F: handout on 1st order linear
homogeneous DE systems (2-d example: complex eigenvalues [phaseplot]);
handout on transition
from a complex to a real basis of a linear DE solution space;
Find the general solution for the DE system x ' =
A x
for the matrix A = <<0|4>,<-4|0>> (input by rows) and then the solution
satisfying the initial conditions x(0) = <1,0>; (solution on-line:
.pdf)
Repeat the process for problem 7.3.11: x ' =
A x
for the matrix A = <<1|-2>,<2|1>>,
initial conditions x(0) = <0,4>
[if you like you can quickly edit the DEPlot
phaseplot template to reproduce the back of the book diagram].
WEEK 12[-1]:
- M: handout on
1st order linear
homogeneous DE systems (2d example: purely imaginary eigenvalues);
using the
eigenvector technique, find the general solution for the DE system x ' = A x
for the matrix A = <<1|-5>,<1|-1>> (input by rows) and then the solution
satisfying the initial conditions x(0) = <1,2>; note the solutions are
easily obtained with dsolve (*check
your hand solution this way!);
Read 7.3 [especially compartmental analysis example 4, which has complex
eigenvectors (more examples)];
do 7.3: 37 [closed 3 tank system with
oscillations (explanation,
text example), plug into Eq(22), solve by
eigenvector method].
- T:
PDF hand solution to 7.3.37 3 tank hw;
Quiz 9 on 2x2 DE system IVP with
distinct real eigenvalues (like handout,
or summary of technique);
Read 7.1 except for examples 5-7 (opposite of reduction of order, not
needed);
read
handout on reduction of order (optional
on-line handout on phase spaces);
7.1: 1, 8 [first let = [x1,x2,x3,x4] = [x,y,x',y'], then re-express the 2
DEs
replacing x1" by x3' and x2" by x4', adding the definitions
x1' = x3, x2' =
x4, then write the 4 DEs in matrix form x ' = A
x + F],
Note: substituting x = B y , leads to B y
' = A B y +F hence y ' = B-1A B y + B-1F
so we can also solve nonhomogeneous linear DE systems in the same way,
adding the new components B-1F of the driving
vector function to the new decoupled DEs;
7.2: 5, 9.
These are very short problems, no solving required, just rewrites, to make
sure you understand matrix notation.
- W: Convert a 2nd order nonhomogeneous DE
x'' + 3 x' + 2 x = cos(t), x(0)=1, x'(0)=2
to a first order system and solve as instructed in the
handwritten reduction of order
exercise.
- F: online reduction of order
exercise solution;
note the higher order
system dsolve template
> {x1''(t) = x2(t), x2''(t) = x1(t), x1(0) = 1,
x1'(0) = 0, x2(0) = 0, x2'(0) = 0} [enter, right click and use Solve
DE Interactive]
handout on extending eigenvalue
decoupling (to nonhomogeneous case, and second order);
solve x''(t) = A x(t) + F(t),
where x(t) = <x1,x2>, F(t) =
<0,cos(t)>, A = <<0,4>,<4,0>>, x(0)=<1,0>, x'(0)=<0,0>
[solution: .mw, eigenvector
details].
WEEK 13[-1]:
- M: Hw solution: [pdf,
mw]
MLRC Test 3 problem session 5:30pm;
handout on 2 spring 2
mass system; ready handout so you understand, and then solve the stated
initial conditions for the system initially at rest at equilibrium to obtain
the result on the handout.
- T: Take home
Test 3 on chapters 5, 6, and eigenvector solution of first order
homogeneous DE systems : on-line
here 2pm
Tuesday (typos corrected Monday post T-day);
paper copy in class; read long version of
test instructions;
Due Friday Week 13;
(no need to work on it during T-break);
to plot several functions together, recall
plotting template;
recall the DEPlot
phaseplot template (or the short version:
directionfieldtemplate.mw); if you check your
results with Maple print out the worksheet and attach to your test as
indicated in the test instructions above.
T-Day!
Be safe in your travels.
WEEK 13[+1]:
- M: Continue working on Test 3. Remember the
corrections: x1(0) = 11 instead of 9 in problem 2, change
"a)" to "c)" at the end of problem 1f).
- T: back to second order systems:
In class group exercise with 2 spring, 2 mass system and gravity, easy
numbers; for HW, check out the solution:
[.pdf,
.mw];
keep working on the test.
- W: continue working on the test.
- F: previous 2 spring system but gravity
replaced by sinusoidal pull:
solve x''(t) = A x(t) + F(t),
where x(t) = <x1, x2>, F(t) =
<0, cos(2t)>, A = <<-5,2>,<2,-2>>, x(0) = <0,1>, x'(0) = <0,0>
(pdf,
.mw).
WEEK 14:
- M: Section 7.4 discusses resonance for
coupled spring systems;
Find the response (particular) solution for the previous system for a
general frequency ω in place of the value 2 in the previous problem (pdf,
.mw);
rewrite the system in first order form as a first order DE for the 4
component state vector = initial data vector = < x(t), x'(t)>.
- T[F]: 7.4: (3 spring 2 mass system) 3, 9 (undriven/driven 3
spring system)
[7.4.9 continues
7.4.3; can use direct substitution of x1 = c5 cos(3t), x2 = c6 cos(3t) into
DEs to determine undetermined coefficients c5,c6 since there is no damping term to mix cosines and sines;
also solve the IVP with inits: x(0) = 0, x'(0)
= 0: .mw].
reduction of order will not be used to treat damping on the final exam, but
if you are interested in the finishing touches to this
approach of handling coupled damped harmonic oscillator systems, read the
final section of the worksheet on the
2 mass 2 spring system or of
the 2 mass 3 spring system [note that
these systems without damping can be solved
for arbitrary values of the
system parameters just using the quadratic equation and 2x2 matrix
linear system solution]
- W: CATS evaluations;
last final exam:
Final Exam: Pages 1,2, [answer key:
Pages 1,2,
MAPLE]
Problem session? Monday afternoon 5:30pm? last minute questions.
T: Dec 16 8:00-10:30am final exam in usual MWF room with your laptop.
The final exam will consist of a first order and a second order linear
homogeneous system of DEs in the plane with the same coefficient matrix
having
real eigenvalues. Should be easy. Anyone who does not check their solutions
with Maple is shooting themselves in the foot. Problem 2 (but with no
driving force) is a good practice problem for the second order case.
I try to keep a class or so ahead with the homework for
those of you who might anticipate?
Weeks 3 thru 4: come by and find me in my office, tell me how things are
going. This is a required visit. Only takes 5 minutes or less.
*MAPLE homework log and instructions [asterisk
"*" marked homework problems]
Test 1: ; MLRC problem session
.
Test 2:
; MLRC problem session
.
Test 3: Take home out
, in ; MLRC problem session
.
Final Exam: MLRC problem session
FINAL EXAM:
2705-01: MWF/T 1:30 class: Tuesday, Dec 16 8:00 -- 10:30
MAPLE and G.Calc. CHECKING ALLOWED FOR QUIZZES, EXAMS
10-dec-2008 [course
homepage]
[log from last time taught]
extras
no class.
Be safe in your travels.