MAT2705 16F homework and daily class log
Your homework will appear here each day as it is assigned, including some PDF
and/or Maple problem solution files or PDF notes, with occasional links to
some MAPLE worksheets when helpful to illustrate some points where technology can be
useful. [There are 56 class days in the semester, 4
each normal week, numbered consecutively below and labeled
by the (first initial of the) day of the week.
Friday will usually be the quiz day. Monday quiz makeup day.]
It is your responsibility to check homework here. (Put a favorite in your
browser to the class homepage.) You are responsible for any hyperlinked
material here as well as requesting any handouts or returned tests or quizzes from classes
you missed. Homework is understood to be done by the
next class meeting (unless that class is a test, in which case the homework
is due the following class meeting).
- W (August 24): GETTING STARTED STUFF. By Friday, August 26, e-mail me [robert.jantzen@villanova.edu]
from your OFFICIAL Villanova e-mail account (which identifies you with your
full name) with the subject heading "[mat2705]",
telling about your last math courses, your comfort level with graphing
calculators and computers and math itself, [for sophomores only] how much
experience you have with Maple (and Mathcad if appropriate) so far, why you chose your major, etc,
anything you want to let me know about yourself. Tell me
what your previous math course was named (if at VU: Mat1500 = Calc 1, Mat1505 = Calc
2, Mat2500 = Calc3).
HINT: Just reply to the welcome
email I sent you before classes started.
[In ALL email to me, include the string "mat2705" somewhere in the
subject heading if you want me to read it. I filter my email.]
DURING CLASS (THIS IS THE FIRST DAY PLAY PART)
On your laptop if you brought it:
1) Open 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:
[
http://www.homepage.villanova.edu/robert.jantzen/courses/mat2705/ ],
3)
Open Maple 2016 if you already
have it
[or click on this maple file link:
firstdayplay.mw] bob will discuss
a few things to remember the rest of the course:
firstday2015.pdf
[don't worry
about page 2 or this worksheet firstday2015.mw;
we will come back to it later in the course]
4) bob will quickly show you the computer environment supporting
our class. And chat up a bit the course. [cell phones on vibrate or off: bob
will give you his cell phone number in class] [maybe he
will try to impress you with this gee whizz!
Maple demo; naahh...we'll leave this
only to the curious among us.]
AFTER CLASS (THIS IS THE HOMEWORK)
5)
log on to My Nova, choose the student tab, and go to
BlackBoard and look at the Grade book
for our course: you
will find all your Quiz, Test and Maple grades here during the semester once
there is something to post.
[This is the only part of BlackBoard we will use this semester.]
In the Student Tab, click on the double person icon to the right of our
class line to get to our photo class roster [look at the photo class roster
to identify your neighbors in class!]
and click on my home page URL
under my photo. Click on our class URL there.
Check out the on-line links describing aspects of the course (no need yet to look at the
MAPLE stuff).
[You can
drop by my office St Aug 370 (third floor, Mendel side, by
side stairwell) to talk with me about the course if you
wish and to see where you can find me in the future when you need to.]
6)
Download Maple
2016
for Windows or Mac
if you haven't already done so and install it on your laptop when you get a
chance (it takes about 15 minutes, I will help you in my office if you wish). If
you have any trouble, email me with an explanation of the errors.
You are expected to be able to use Maple on your laptop when
needed. We will develop the experience as we go.
7) Read computer classroom
/laptop etiquette and check out the
university academic integrity site. Browse our
class homepage and read the linked pages. The homework problems are few so
that you can get familiar with the website.
8)
Homework Problems:
1.1: 3, 5,
13, 33 (this links to a PDF scan of the HW problems from the book if you
have not yet purchased it);
[memorize!: "A is proportional to B" means "A = k
B" where k is some constant,
independent of A and B]
(short list so you can check out our class website and read about the
course rules, advice, bob FAQ, etc, respond with your email).
It is
important that you read the section in the book from which homework problems
have been selected before attempting them.
Optional: get acquainted with
Maple clickable calculus DE entry and "odetest" for problems 3,5,13 [even
33!]
9) Fill out the paper schedule form bob handed out in class. [see
handouts]; use the 3 letter
dorm abbreviations to return in class the next class day.
Proportionality statements must be converted to equalities with a
constant of proportionality introduced:
y is
proportional to x: y ∝ x means y
= k x . [y is a multiple of x]
y is
inversely proportional to x: y ∝ 1/x
means y = k/x .
y is
inversely proportional to the square of x: y ∝ 1/x2
means y = k/x2
- F: return your schedule forms at the beginning of
class;
check your data entered by bob on sign-up sheet---correct errors;
short Quiz 0 [not for grading];
1:1: 7a [i.e., only check y1(x)], 23 [see
Maple plot (execute worksheet by clicking on the !!! icon on the
toolbar)];
formulating DEs: 27,
29 [Hint: recall perp lines have slopes which are negative
reciprocals, the normal line is perpendicular to the tangent line and passes
through the same point on the curve; make a diagram of the given point (0,1), a "generic" function graph curve
whose normal at a point (x,y) on the curve passes through
(0,1), and draw in the connecting normal line segment between these two
points and the perpendicular tangent
line, then compute the slope of the normal line from the two points, and then from the
negative reciprocal of the derivative value, finally equate the two
to get the DE],
35, 36;
read the paper
handout: algebra/calc background
sheet [online only: more rules of
algebra NOT!];
on-line handout:
initial data: what's the deal?
WEEK 2[-2]:
- M: 1.2 (antiderivatives as DEs): 1, 5, 15,
19 [Hint: write piecewise linear function from graph: v1(t) for first expression,
v2(t) for second expression, solve first IVP
for x1(0) = 0, second IVP for x2(5) = x1(5);
soln: pdf,
mw],
25 (like lunar landing problem,
see example), 39, 40 (ans: 2.4mi),
43 (pdf)
[convert to appropriate units!].
Tomorrow is the
first meeting of Students Against Sweatshops if you are interested
(6:30pm, free pizza!).
- T: 1.3:
3:
[hand draw in all the curves on the full page paper
printout supplied by bob:
projectable image]
11,
15-16 [note Dyf is just df/dy above Eq (9) in the
text]; 27
[don't worry about making a diagram, just view this worksheet: .mw];
Maple HW not due till the week after we finish chapter 1, there is time next
week to get a partner and ask bob questions about how to use Maple;
8* : our first Maple HW problem: read the
Maple HW instruction page, then try the Maple direction field
command in this linked template to reproduce the
two given curves by using
two appropriate initial data points [Hint: notice that the y-axis
crossing points are integers!] Then include all the initial data points of
the red dots (some half-integer values); this is also described on p.29
for both Maple and Matlab, but the above template already has the
appropriate section copied into it for you; if you can find a partner to
work with in time fine, otherwise try this yourself and find a partner for
future problems, then merge your work [will not be due until we
finish chapter 1].
- W:
Read the
course info handout. Office hours are set;
use classlist contact info
(paper only) and MyNova class photos to get to know others in your class,
form Maple partnerships of 2 or 3 (4 with permission), no exceptions, you
can change partners with each assignment;
1.4 (separable DEs): 1, 5; 21, 25, 27, 29
[ignorable unless you are curious: example
2: technology hint, other
complications visualized].
-
F: Quiz 1 (see quiz archive
for an idea of what to expect);
handout on
exponential behavior/ characteristic time [read this worksheet
explicit plot example, execute
worksheet first with !!! icon on toolbar];
1.4: 45 (if your cell phone were waterproof: "Can you hear me now?"
attentuation of signal--characteristic length),
47 [recall answer to 1.1.35, use units of thousands of people, assume no
one has heard the rumor at t = 0: N(0) = 0 (an approximation
in units of thousands; really N(0) = .001 would be one person
starting the rumor, or should it be two?)],
65 [CSI problem (Newton's law of cooling): after you solve this
problem yourself, consult both [pdf, .mw];
please read Maple worksheet to see
the power of mathematics to solve all such problems at once;
please read the PDF to see how to convert the word problem line by
line to a mathematical problem].
WEEK 2[+1]:
M: Labor Day:
revisit the river crossing problem and the
oven heating problem -
T:
handout:
recipe for first order linear DE [plot];
1.5: 3, 7, 11, 21; 27;
21*[check both the general solution and the initial
value problem solution with the dsolve
template].
For future reference:
> deq := y ' = x y
[space implies multiplication]
> sol:=dsolve(deq, y(x))
> solinit := dsolve({deq, y(0)=1}, y(x))
[or enter DE plus IC separated by a comma, right click on output,
choose Solve DE, for y(x), or Solve DE interactively;
rightclick "Simplify, Simplify" or "Simplify, Symbolic" (with
radicals) may be necessary to simplify the result]
> y ' = x y , y(0)=1
Use function notation to change independent variable:
> y '(t) = t y(t) , y(0)
= 1
- W: M: online handout:
how to
understand the DE machine (example of switching variables);
online
discussion: antiderivative
functions defined by definite integrals;
1.5: 17, 26 [ans: x = ½
y -2 +Cy -4],
29 [use the fact that the derivative of
the function defined by the integral is the integrand including all
multiplicative factors (antiderivatives
as definite integral functions, more explanation of error
function)],
31
[show by differentiation and backsubstitution into the DE, not by rederiving it],
32 [find the constants by substituting the trial solution into the DE and
choosing the values of A and B to make the DE satisfied: put all terms on LHS
and set coefficients of the sine and cosine separately to zero],
41 [Hint:
Δt in years is the fraction of the year, so S(t)
Δt is the approximate income during this time interval, 12 percent of which
contributes to ΔA, while the interest contribution is the annual interest rate
6 percent prorated by multiplying by the fraction Δt for the interest
contribution over this shorter time period, applied to the current value of
the retirement amount A(t): 6 percent of A(t) Δt; adding these and dividing by
Δt, taking limit to get dA/dt gives the linear DE [click
here only if you have tried to get the DE and could not, or its solution did not give
the correct number].
41* [use the dsolve template to solve this DE with its
initial condition; evaluate answer to question of word problem to a floating
point number and give the result in dollars and cents in a comment, then think
about significant figures and give a response with fewer significant digits].
When you are satisfied, compare with the hand written solution
retirement.pdf (don't submit a Maple worksheet without consulting this
to be sure your text comments are correct).
Maple1.mw hw due any time next week
including the weekends, but preferably be familiar with using Maple to check
your work before the next quiz (read Maple HW page).
- F:
Quiz 2 [you will need to check your DE solution with Maple];
1.5 (tank mixing problems):
37
(Use
Eq. 18 in the book or the boxed equation in the online handout); extra questions: what is the final concentration of salt?
How does it
compare to the initial concentration and the incoming concentration?);
[solution pdf,
mw]
[this mixing problem (online
handout) is an
example of developing and solving a differential equation that models a
physical situation; for another word problem example,
see this];
1.R(review): classify the odd problems 1-35 as:
separable, linear in y
(as unknown),
linear in x (as unknown), some combination of these three, or NOTA (none of the above,
i.e., from skipped section 1.6), for
example dy/dx = y/x is all three and can be solved in three different ways;
don't solve them but:
solve 25 [it can be done in two ways by expanding out the square on the RHS
before integrating with integration constant C or by using a u-substitution
as the book did with integration constant K; express K in terms of C as given in the book supplied answers by
combining the two terms in the K solution [identity for expanding out (x+1)3 !] and then comparing with the C
solution];
solve 35 in two ways and compare the results.
WEEK 3[+1]: - M: handout on solution of logistic DEQ
[directionfield,
integral formula,
characteristic time];
2.1: 5 [go thru the solution steps
just this once with concrete numbers, remember the
integral formula, follow the steps of the book or handout derivation of the solution,
compare with the final logistic formula; use the formula for the remaining HW
problems],
9,
15 [READ CAREFULLY: note that by definition 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 M and
kM needed to use the solution formula to solve problem 16],
16 [ans: P = .95 M
after 27.69 -> 27.7 -> 28 months].
-
T:
handout on DE's that don't involve the ind
var explicitly;
2.1: 6
[use handout integration formula with sign reversed, or use technology
for integral and combine log terms;
answer: > dsolve({x'(t) =3
x(t) (x(t)-5),
x(0)=2},x(t))
(copy and paste into the input line, rightclick choose RHS, then rightclick
choose Plots, Plotbuilder, choose horizontal window based on characteristic
time);
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)].
REMEMBER. Maple1.mw due any
time this week through the weekend. Upgradable after submission. Email bob
when you run into trouble. Read instructions.
- W: air resistance handout
(example of a piecewise defined solution and the importance of dimensionless
variables);
[optional reading to show what is possible: comparison
of linear, quadratic cases; numerical solution
for any power];
2.3: 1, 2, 3,
9 [remember weight is mg, so mg = 32000 lb determines
m = 1000 in USA units, convert final speed to mph
for interpretation!],
22 [this is the tanh case: vterminal = 20.7 ft/s, t
= 8 min 5 s]. [soln,
.mw].
4:30pm C-Center Cinema:
Fate of the Universe talk.
- F: Quiz 3 (linear DE);
check online answer key to Quiz 2;
check Blackboard grades against 2
quiz grades so far;
2.4: 1 (do by hand and calculator, can check with euler.mw if you wish:
soln);
optional: read about improved Euler in textbook;
maple2.mw:
instructions for this second maple assignment.
[due the week after Test 1 roughly; make sure you have at least one
partner and follow the submission instructions].
WEEK 4[+1]: Test 1 on Friday;
Maple 1 HW due today.
Maple 2 due any time next week thru the weekend;
check out the quiz
3 answer key
online; come talk to bob if you
did not do well or feel unprepared for Test 1;
check Blackboard to match
your grades with those on your quiz/tests;
-
M: handout: why LinAlg with DE?;
[maple dsolve and DEplot for 2x2 systems
of DEQs];
Read 3.1 on linear systems (review of high school topic solving 2 or 3 linear equations in
2 or 3 variables),
do
3.1: 1, 5, 7; 9, 15, 31, 33, 34.
-
T: handout on
RREF (Reduced Row Echelon Form, section 3.3)
[see bob's first 2x2 example,example
3];
3.2: 1, 3, 7; 11, 13,
15
(do a few by hand, then you may use step-by-step row ops with MAPLE or a calculator for
the rest) ;
[Matrices for doing some of tonight's homework with Maple
preloaded, will be explained next time];
you must learn a technology method since this is insane to do
by hand after the first few simple examples);
23 [can your calculator handle this?]
To be explained next class:
Maple:
Right click on augmented matrix entered with the Matrix
palette, choose Solvers and Forms, then Row Echelon Form, then Reduced
to get the RREF form of the matrix, then right click on new
matrix and choose Solvers and Forms, then Linear
Solve, and you get the RHS of the solution with no LHS variables
assigned to these expressions because Maple has no clue what variable names
you wish to use.
See Example 3:
row ops with MAPLE.
-
W: Bring
laptops to class for hands on row reduction tutor experience;
handout on
solving linear systems example;
we always want to do a full reduction, not the partial Gauss
elimination reduction also described by the textbook;
in class open this Maple file with a
bob as a partner
and use the LinearSolveTutor and enter the matrix for the system 3.2.15
given there and reduce step by step,
then solve the system (if possible), then switch positions and enter the matrix
for the system 3.2.18 given there and repeat using instead the Reduction
command template there;
You can get to the Linear
Solve Tutor from the Tools Menu:
Maple
Tools Menu: Tutors, Linear Algebra, Linear System Solving: Gauss-Jordan elimination
3.3: 1, 3, 7, 11, 17, 23 = 3.2.13 (refers
back to system in 3.2 HW) [Matrices for doing tonight's homework with Maple
preloaded]
19* [use the tutor and record your final result, compare with
the right click solution].
Th: 5:30 voluntary test 1 problem
session in MLRC.
-
F: Test 1 on first order
DEs: one linear DE, one separable DE, some questions on
sinusoidal periods
and exponential characteristic times (see
Test Archive).
WEEK 5[+1]: -
M: 3.2: 22 [solve using row
ops on augmented matrix to rref form, then by hand backsubstitute, check
answer with Maple];
3.3:
29 = 3.2.19 [solve
using row ops on augmented matrix to rref form, then by hand backsubstitute];
29* [now check this same problem solution by solving with
right click RREF
of the augmented matrix, then right click Linear Solve;
MAPLE
chemical reaction problem* [pdf] (what can you
find out on the web about interpreting this chemical reaction?
chem site reaction balancer
gives balanced reactions but
no description, this one of our exercise turns out to be interesting);
word of the day (semester really): can you say "homogeneous"?
[online handout only]
Get used to this word, it will be used the rest of
the semester. It refers to any linear equation not containing any terms
which do not have either the unknowns or any of their derivatives present,
i.e., if in standard form with all the terms involving only the unknowns on
the left hand side of the linear equation, the right hand side is ZERO. [nonhomogeneous
means the RHS is nonzero]
-
T: finally matrix multiplication!
[handout];
3.4: 1, 5, 7, 11, 13; 17, 21; 27, 43.
Matrix algebra is easy in Maple
[see here for how to do Matrix stuff in Maple]. <<<<<<<<<<<
-
W: handout
example for matrix inverse algorithm;
3.5: 1; 9, 11;
23 [this is
a way of solving 3 linear systems with same coefficient matrix
simultaneously, as in the alternative
derivation of the matrix inverse];
30 [just substitute for the inverse and then drop parentheses and multiply
out: (ABC)-1 (ABC) = ...],
32 [multiply on the left by A-1].
Matrix multiplication and matrix inverse, determinant, transpose [or
right-click and use Standard Operations menu]
> A B [note space between
symbols to imply multiplication]
> A-1
> |A| [absolute value sign
gives determinant, or just right click]
Memorize:
.
Switch diagonal entries. Change sign off-diagonal entries. Divide by
determinant.
Always check inverse in Maple if you are not good at remembering this.
-
F: Test 1 back, check out answer
key;
check BlackBoard grades;
"I believe in second chances, I drop the
lowest of the first two test grades if the successive test grade is higher."
Quiz 4 row reduction
solution of linear systems;
optional on-line handout:
determinants and area etc ;
[optional: why
the transpose?
(to combine vectors we arrange them into rows of matrix,
the transpose maps columns to rows, rows back to columns)];
Read
explanation of why we need
determinants;
3.6: (determinants abbreviated:
forget
about minors, cofactors, only need row reduction evaluation to understand)
13, 17, 21
(plug in just so you've done it once);
17* [use
this example to record your Gaussian elimination
steps in reducing this to triangular form and check det value against your
result];
3.5. Application: 7* from textbook on p.201,
using the inverse matrix as described on p.200,
be sure to answer the word problem question with a text comment;
And also the question: are these sandwiches cheap or not? [test to see if
you are paying attention]
[this is the
final problem of maple3.mw. make
sure you find a partner or join a 2 partner group for this assignment]
WEEK 6[+1]: -
M: linear
independence of a set of vectors;
now we look at linear system coefficient matrices
A as collections of columns A = < C1| ... |Cn >,
matrix multiplication on the right by a column of
coefficients as taking linear combinations of those columns,
A x = x1 C1 + ... + xn Cn
and solving homogeneous linear systems of equations A x = 0 as
looking for linear relationships among those columns (nonzero solns=linear
rels) ;
4.1: 1; 5, 7;
9; 15, 17; 19, 23; 25
[use technology in
both 1) evaluating the determinant and 2)
rightclick menu to the ReducedRowEchelonForm
needed
for these problems];
[HWready row reductions for these problems];
hand out on the interpretation of solving linear
homogeneous systems of equations: A x = 0
[optional
read:
visualize
the vectors from the handout].
-
T:
vector spaces and subspaces;
study the handout on solving linear systems revisited
[remember the original one: solving linear systems example];
[note only a set of vectors x resulting from the solution of a linear homogeneous condition
A x = 0 can be a subspace of a vector
space; interpretation: lines, planes, ..., hyperplanes through the origin];
4.2: 1, 3, 11; 15, 19.
-
W: 4.3: book catches up with us: linear
independence (A x = 0),
express a vector as a linear combination (A x =
b); span of a set of vectors;handout on
linear combinations, forwards and
backwards [maple to visualize];
[optional worksheet to read: what does the solution of a
nonhomogeneous system of equations say about the column vectors in the
augmented matrix?]
4.3: 1, 3, 5, 7; 9, 13; 17, 21; 23.
-
F: Quiz 5 identical to 15S, numbers changed; Maple3.mw is due the week after break;
bases
of vector spaces, subspaces;
4.4: 1, 3, 5, 7;
9 [this is already reduced, find solution as in class example, pull apart to find coefficient vectors of
parameters, repeat for next 4 problems reducing first],
13; 15, 17, 25 [use technology for all
HW row reductions and determinants (but 2x2 dets are easy by hand!)].
handout on nonstandard coordinates on R2 and
R3 [goal: understand the jargon and how
things work]
[in class practice on blank grid [
.mw,
.pdf] with b1=
<3,-1>, b2= <2,2> ; what are new coords of <-2,6>?
what point has new coords <2,-1>?] ;
1) Using the completed first page coordinate handout
made for the new basis {[2,1],[1,3]} of the plane: graphically find the new
coordinates of the point [4,7], then confirm using the matrix multiplication
by the inverse matrix of the coordinate transformation given there. Then
find the old coordinates for the point whose new coordinates are [2,2].
2) Using the previous handout as a guide, first fill in the blanks on the
corresponding blank worksheet handed out
in class for a new
basis transformation matrix and its inverse given on that sheet, and then for the change of
coordinates, and then use either the matrix or scalar equations of the
change of variables to calculate the other coordinates for the two given
points. [Let the larger grid squares of a centimeter determine your unit
tickmarks!]
Then confirm your matrix calculations graphically by following the completed example:
use a ruler and sharp
pencil,to make a new -2..2,-2..2 coordinate grid associated with the new basis
vectors {[1,1],[-2,1]} and graphically represent the point [x1,x2]
= [-5,1] and find
its new coordinates [y1,y2] using the grid and confirm that this agrees with your
previous calculation. Similarly read
off the old coordinates [x1,x2] of the point whose new coordinates are
[y1,y2] = [-2,-1]. Confirm that these
agree with your calculations. Put your name on it and bring it to class
to hand in next class.
[If you are at all confused, here is a
completed worksheet for the in
class practice example above.]
[If you are feeling ambitious, you could also edit the Maple version
of the example to see that your hand work is correct.]
FALL BREAK :-) Enjoy. Be safe in your travels.
© dr bob
enterprises
Week 7[+1]: -
M: hand in worksheet;
handout summarizing
linear vocabulary for sets of vectors;
[optional on-line handout:
linear system vocabulary
for linear systems of equations ]
Part 2 of course begins: transition back to DEs:
read the 4.7 subsection (p.279)
on function spaces and then examples 3, 6, 7, 8;
if you are curious, example 5 illustrates the partial fraction decomposition
needed in engineering integration;
read the worksheet on the
vector space of (at most) quadratic functions [quadratics.mw,
pdf];
4.7: 15 [Hint: solve c1(1 + x) + c2(1 - x) + c3(1 - x2) = 0, for unknowns c1,c2,c3;
are there nonzero solutions? if not, these are linearly independent polynomials, in terms of which any quadratic
expression in x can be written; note that any two (nonzero!)
functions of x that are not proportional are automatically linearly
independent],
16, 17 [same approach];
[hand in late basis change worksheet if you
missed Friday class before break.]
-
T: Memorize: y ' = k y < -- > y = C e k x
y '' + ω2 y = 0 < -- > y =
C1 cos(ω x) + C2 sin(ω x)
["omega" = ω is the angular frequency "radians per time unit" as opposed to
just "frequency" f =
ω/(2π ) or "revolutions or cycles per time unit" as in physics, but in our class we
will just say "frequency" for ω, assumed to be expressed in radians per time
unit or converted to revolutions per time unit as convenient;
see also damped
harmonic oscillators and
RLC circuits, and
Hertz, not just a rental
car company but "cycles per second"]
5.1[up thru page 295, plus Example 7]:
1, 3, 5, 9; 13, 17.
8pm Idea Accelerator
Talk "Tell Me Your
Climate Story" by Devi Lockwood in library basement: a recent Harvard
graduate who has been traveling the world, collecting people's stories on how
climate change has affected them (ACS approved).
-
W: handout on sinusoidal example;
repeated root plot;
5.1[after page 295]:
33, 35, 39,
49 [find the IVP solution for y, set y ' = 0 and solve exactly
for x, backsubstitute into y and simplify using rules of
exponents]
5.2: 1,
11;
5.1: 49*[check your IVP solution for the highest curve in Figure 5.1.6
(highest for x > 0) using
the 2nd order dsolve template and plot it together
with the horizontal line y = 16/7 claimed by the back of book answer as the highest value of
that function; recall how to plot multiple functions in the same plot:
> plot([<expression_for_y(x)>,16/7], x=a..b, y=c..d, color=[red,blue]);
(let a = 0 here to plot only x > 0)
the color option is useful in distinguishing two functions when it is not
already clear which is which; is 16/7 the peak value of your
solution?].
-
F: Quiz 6 (solving IVP given
a basis of solutions);
online handout:
wronskian and higher order constant coefficient linear homogeneous DEs;
read handout on complex arithmetic, exponentials
[maple commands; the complex number i is
uppercase I in Maple];
5.2:
13, 17, 21, 26 [this problem
is for "fun" now, but later
we learn a method to attack it];
[optional: read the on-line handout on
visualizing the initial value problem (IVP)
if you wish to see the interpretation of solving the IVP]
WEEK 8[+1]: -
M:
complex roots: handout on the
amplitude and phase shift of sinusoidal
functions and DE example [.mw];
extras: [shorter presentation:
pdf], 4 quadrant amplitude-phase shift examples:
[mw,
pdf].
5.3:
8 [ans: y = exp(3x) (c1 cos(2x) + c2 sin(2x))] , 9; 17;
22,
23 (express in phase-shifted cosine form,
see handout).
Remember the first day handout on
checking DEs and angles around the unit circle.
-
T: 5.3 higher
order DEs [ignore instructions to factor by hand or
polynomial long divide: use technology for all factoring]:
11, 13;
25, 33; 39;
49 ;
49* [check your solution with the higher
order dsolve template, and also check the IC linear system solution as in
the template, edit away all stuff not directly relevant to your new problem].
handout on exponentially modulated sinusoidal functions
[maple
videos showing different phase shifts and frequencies];
do the short problem at the end of the handout (ignore complex C and W unless you
are EE or physics oriented) after
reading it. [soln: pdf,
mw].
-
W: 5.4: 1,
3 [use meter units!], 13,
14 [b) the solution once put in the form
x = A exp(-Kt) cos(ω
t
- δ) has the "envelope curves"
x = ± A exp(-Kt) ] ,
17;
23 [pdf]
(this is a useful application problem, please study the solution carefully
after trying to solve it yourself);
handout on linear homogeneous 2nd order DEQ with constant positive coefficients
(damped harmonic oscillator)
[on-line only: examples in
nature].
Th: 5:30 voluntary MLRC problem session for Test 2.
-
F: Test 2 on chapters 3,4;
VSMT presents The Addams Family! Fri (8,12)/Sat(8).
WEEK 9 [+1]: -
M: 5.4:
21 (solve by hand),
21* [solve the two DEs with the same initial
conditions using Maple and plot the two solution curves together, but also plot the corresponding third solution with pure damping but no
oscillator as in the template];
handout on driven
(nonhomogeneous) constant coeff linear DEs
[complete final exercise on this handout sheet, Maple: *check this with dsolve,
(consult after doing it by hand first:
pdf
solution; the algebra works if you are careful!)];
5.5: 1, 3, 9 [see PDF solution after you try this yourself:
version 1,
version 2]
[we will not cover "variation of parameters"; the book
presentation of the method of undetermined coefficients is a recipe with
little justification (educated guess), instead the handout shows exactly how and why one gets
the particular solutions up to these coefficients].
ignore this: EE/Physics majors if you are interested: RLC circuit application: RLC circuits [maple
plots]
-
T:
catchup day. in class: 5.4,5.5 HW discussion. After class, go back and do
those HW problems that you did not get to.
For second order
DEs, if a RHS driving function term satisfies the homogeneous DE, multiply
the corresponding trial function (the most general solution of the
homogeneous DE for which that term is a particular solution) by the
independent variable (repeat if the new trial function still satisfies the
homogeneous DE).
[not
many of the book driving functions are physically interesting here in this
exercise set]
Do a new version of 5.5.9: a slight change
to this problem to make it physically interesting describing an overdamped
spring system at rest to which an impulse force is applied, from which one
can calculate the maximum displacement which occurs: y''+3y'+2y
= x exp(-x), y(0) = 0 = y'(0). Find the
maximum of y(x) for y≥0.
[solution:
part1, part2,
mw];
5.5: 33, 38.
-
W: handout on damped harmonic oscillator driven by sinusoidal driving function
[1 sheet printable version];
[in class example.mw pdf:
resonance
calculation side by side], maple resonance plots: general, specific];
[Optional for the curious, Wiki:
driven
harmonic motion,
amplitude plots:
they introduce ζ = 1/(2Q) = k0/(2 ω0) with critical value 1.]
5.6: 8. Ignore the book instructions and solve this problem by hand for the
initial conditions x(0) = 0 = x'(0)
[read the
Maple worksheet about the interpretation
of the solution; hand solution];
15, 17. Follow the handout procedure
as in this particular case,
find the amplitude function of the frequency and plot it to compare with the
back of the book. Evaluate the natural frequency and Q value for
each [soln:
pdf,
mw; 17more-pdf].
-
F:
Quiz 7 on "driven damped harmonic oscillator system"; catch up on regular and Maple HW. Enjoy the weekend.
WEEK 10[+1]: -
M: handout on beating and
resonance [with HW explanation];
5.6: 1 [rewrite as a product of sines using the cosine difference formula:
cos(A) - cos(B) = {-2 sin((A-B)/2)} sin((A+B)/2), the expression in {} is the
sinusoidal amplitude envelope function [and its period is T = 2(2
π)/(A-B)], look at the plot of x and +/- this amplitude function
(the envelope) together in this beating
example worksheet, then do:
5.6.1*:
as explained here, repeat for the HW
problem at the end of this same beating
example worksheet (be sure to plot one
full period of the envelope function to see exactly 2 beats) and include only that HW problem
section in your maple5 worksheet],
11 [then convert steady state solution to phase shifted cosine at
end of
calculation];
23 (earthquake!):
consistent units: sec, ft, lbs; first evaluate Hookes' law constant, then
find x''+10 x = A0 ω2
sin(ω t), convert frequency to ω
= 2 π /2.25 radians/s, set A0 (=3in) = 1/4 ft. Solve. Find
amplitude of response oscillation in inches. [soln]
Maple plotting of multiple functions
>
x, -x, x sin(x) Enter and right click on output
(choose Plot Builder, 2D Plot)
to plot together to see oscillation and its envelope. or use bob's templates
in each problem. We need this for envelopes!
If you are curious watch the Google linked video of resonance NOT! (but
see the engineering explanation linked PDF for the detailed explanation):
[Tacoma Narrows Bridge collapse (resonance NOT!):
Wikipedia;
Google (choose a
You-Tube video)];
a real resonance bridge problem occurred more
recently: the Millenium Bridge
resonance.
-
T: Transition back to linear algebra:
Fun.
[In
class click on the Duke U
applet link, and then the URL title to see if it works for you.]
Watch the
MIT Eigenvector
4 minute video [there are 6 frames which then repeat, so stop when you
see it beginning again---from their
LinAlg course]; see the 6 possibilities of the video in this
Maple worksheet DEPlot directionfield
phaseplot template;
then play a computer game with the
Duke U applet lining up the vectors [red is x,
blue is A x,
click on matrix entries to change, click on tip of red vector and
drag around an approximate circle to see corresponding blue vector]; try
first with the default values, then try for a11=0, a12=1,
a21=1, a22=0, then try for the
matrices of 6.1: 1,2,9. See if you can guess the eigenvector directions
and the corresponding eigenvalues (all integer triangles locating the
vectors and integer eigenvalues) for
these matrices; write
down a simple representative eigenvector (with the smallest integer
components, say) and its eigenvalue that you can read off from the applet as
explained within the introductory webpage to compare with your matrix
calculations for these textbook HW problems . Then find them by the
eigenvector process.
6.1: 1, 2, 9 [1, 9 mw]. Ignore
polynomial division discussion, use technology for roots of polynomials!
Recall old handout
coupled system of DEQs and its
directionfield [motivation: direction fields for Maple help visualize eigendirections of a 2x2 matrix]
matrix
examples: A = <<0,1>,<1,0>> [done]; <<2,0>,<1,3>>; <<1,2>,<2,4>>.
-
W: Maple
eigenvector analysis:
3x3 example from the day
before (repeated
eigenvalues:
<<9,
-6, 6> | <4, -1, 4> | <0, 0, 3>>);
we begin HW in class:
6.1: 13, 19;
21, 25 (upper triangular so diagonal values are eigenvalues!),
27 (complex!);
[do everything by hand for 2x2 matrices; for 3x3 or higher, go thru process: use Maple determinant: |A-λI|
= 0 (right click on matrix, Standard Operations), then solve to find characteristic equation and
its solutions, the eigenvalues,
back sub them into the matrix equations (A-λI)
x = 0 and solve by rref, backsub, read off eigenvector basis of
soln space, NO POLYNOMIAL DIVISION!].
-
F: Quiz 8 (see 15S quiz 9);
diagonalization [now revisit
coupled system of DEQs in a handout giving a preview of what we are about to embark on:
why diagonalization?];
For the matrix A = <<1|4>,<2|3>> entered by rows,
from the two independent integer component eigenvectors {b1 = <1,1>,b2 =
<-2,1>}, make the basis changing matrix
B = <b1|b2> whose columns are these vectors, use the two coordinate transformations
x = B y and y = B-1
x to find the new coordinates of the point <x1, 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.[multiply b1 and b2 by A to see what multiple
you get in each case---the eigenvalue];
[solution: remember this? (now
it is the geometry of diagonalization)];
6.2: 1, 9 ("defective", see
direction field plot); 13 (upper
triangular!, det easy), 21 ("defective"); [use Maple for det and
solve for eigenvalues, then by hand find eigenvectors]
34 [just write out the quadratic characteristic eqn and solve with the quadratic formula, think about
real distinct roots; Delta is the discriminant; what if b = c
when the matrix is "symmetric" and what kind of special matrix is this if
Delta is zero?].
Mendel Award Talk 2pm at the
Jake Nevin Field House, usually terrific lectures! Don't miss it.
WEEK
11[+1]: -
M:
handout on the geometry of diagonalization
and first order linear
homogeneous DE systems (2-d example: real eigenvalues [phaseplot]);1)
Repeat this handout exercise for the matrix of 6.2.1 (namely
A = <<5|-4>,<2|-1>>
input by rows) and the initial condition x(0) = <0,1>; [make a
rough drawing of the new axes and the initial data decomposition along the
eigenvector directions, or use the phaseplot worksheet to make the diagram];
2)
solve the 3x3 system x' = A x for the matrix of
problem 6.1.19, with initial condition x(0) = <2,1,2>
(this worksheet shows how the Eigenvectors Tutor works);
3)
find the eigenvalues and matrix B of eigenvectors
for the matrix A = <<0|4>,<-4|0>>
(input by rows) and verify that AD = B-1A B is diagonal.
-
T:handout
on 1st order linear
homogeneous DE systems (complex eigenvalues) (2-d example: [phaseplot]);
handout on transition
from a complex to a real basis of a linear DE solution space [4 page double
handout print];
Find the general solution for the DE system x ' =
A x
for the matrix A = <<0|4>,<-4|0>> (input by rows) and then the solution
satisfying the initial conditions x(0) = <1,0> (solution
on-line:
.pdf);
Repeat the process for problem 7.3.11: x ' =
A x
for the matrix A = <<1|-2>,<2|1>>,
initial conditions x(0) = <0,4> (solution
on-line: .mw);
[if you like you can quickly edit the DEPlot
phaseplot template to reproduce the back of the book diagram for that
problem].
-
W: Read 7.3
optional read with many extras,
only for the curious:
[compartmental analysis: real
eigenvalues: example 2 (open
case+comparison with closed case: real/complex eigenvalues) ,
complex
eigenvalues: example 4 (closed
case);(more examples 7.3.35,36)];
do 7.3: 37 [closed 3 tank system with
oscillations (explanation,
text example4.mw), plug into Eq(22),
solve by eigenvector method] (solution
on-line: .mw,
.pdf);
choose
one problem from 7.3: 27-30 (solution
on-line: .mw).
for complex eigenvector matrices that
do not come out in "rationalized" form but involve a product or quotient of complex
numbers, right click on matrix, select "map
command onto" and insert the command "evalc" to
evaluate to a complex number, expressing each entry its real part plus i times
its imaginary part. [or see
this worksheet].
-
F: Quiz 9 [like 15S quiz 10, complex eigenvector DE system solution];
catch up on Maple hw;
look for previous
HW solution links for past few days as: (solution
on-line: .pdf or .mw).
WEEK 12[-1]: -
M: summary handout on
eigenvalue decoupling so far;
handout on extending eigenvalue
decoupling (to nonhomogeneous case, and second order);
online
examples handout [.mw]; go over this
handout to see how we can easily extend our present tool to cover the
nonhomogeneous case, and when we have a second derivative instead of a first
derivative;
solve x''(t) = A x(t) + F(t),
where x(t) = <x1(t),x2(t)>,
for A = <<-5/2|3/2>,<3/2|-5/2>>, F(t) = 0, x(0) = <1,0>, x'(0) = <0,1>,
make a hand plot of the eigenvectors and new coordinates y1,y2, the initial data position vector x(0) and the velocity vector x'(0) with tail at the tip of the
initial position vector;
express the mode variables y1(t), y2(t)
in phase shifted cosine form so you can compare the amplitudes and
phaseshifts of these two oscillations.
Check your solutions with Maple by solving the corresponding scalar
differential equations for x1(t), x2(t).
Optional.
The following two commands will show you the two relevant plots of your
results if you are curious (replace unknown variables by their expressions
that you find):
> plot([x1(t), x2(t),t
= 0..2π]) a nice figure 8 ish design!!
> plot([x1(t), x2(t)],t
= 0..2π, color = [red,blue])
.
5:30- voluntary MLRC test 3 problem session.
-
T: Take home
Test 3 is linked here; please read the short instructions on the test and the
long instructions online; this is not a collaborative effort; start in
class;
here is a stripped down example
for the direction field and solution plot versus t, all you need for
problem 3.
[takehome test 3 out, due Mon, Dec 5]
T-Day break
WEEK 13[-1]: -
M: Continue working on Test 3 in class; correct
problem 3 initial conditions as stated in the email: x1(0) = 4, x2(0)
= 5 to agree with the rest of the problem. Delete the last sentence of 1 b)
requesting plot #3, so there are only 4 required plots. [corrected Test 3 is linked here].
-
T: in class: do day
46 HW exercise (following this handout
example), then repeat together with driving function: F(t)
= < 0,50 cos(3t) > ;
[interpretation
coming soon---this represents an equivalent
2 mass 3 spring system: beautiful
Maple animations, and
the JPG movie.]
HW: continue working on
Test 3.
-
W:
2 mass spring systems: theory plus
worked examples;
[days 46, 49 exercise description:
drivenDEsystemExercise.pdf,
solution included in previous line PDF);
plots: figure8curvemodesplot.mw
(results only pdf print: figure8curvemodesplot.pdf
);
figure8curve.mw
(derivation); vertically hung
with gravity?];
7.4: 3, 9 [same system but different numbers, with/without driving term, same procedure as above,
namely:
driven 3 spring,
2 mass system, easy numbers, resonance (do parts a-f; wait for parts
g,h)].
Catholic Relief Services (student) Ambassadors present the film:
The Freedom Film- Indifference is Not an Option at 5:30 today
in Driscoll 134. Make an effort to support your fellow students in spreading
awareness of some realities of the outside world.
All grades thru
Maple2 uploaded to BlackBoard. Check for accuracy.
Cum with lowest quiz,
test 1 if less than test 2 deleted.
See Day 38 for Millenium Bridge
Resonance video.
-
F: Example of how keeping
the parameters general allows you to answer simple questions about system
behavior as they change: previous
toy system with gravity;
For the following problem, break up into
groups of at least 4. Let 1 or 2 people enter the DEs with the sinusoidal
vector driving function and identify the response vector function; the rest
go through the method of undetermined coefficients and then recombine to
find this response vector; then return to part f) and put the decoupled
variables into phase shifted cosine form to identify the amplitudes of each
free mode to make the 3 double arrow diagram with the accompanying
parallelogram/rectangle.
7.4.3,9: resonance: do part g) in class without initial conditions
considering resonance by replacing the driving frequency 3 by omega and only
find the particular (response) solution; check your solution for omega = 3
with back of book formula or Maple (part of general solution with frequency
omega). For fun, plot also the sqrt of the sum of squares of the two
amplitudes as a function of omega, this is the maximum displacement from the
origin in the phase space of the two position variables (of course with
infinite vertical asymptotes without damping).
[pdf
resonance only hand solution, maple plot]
complete problem solution: pdf (no
resonance),
maple (resonance too)
for the
example problem I have been developing instead some remarks about
visualizing the response modes:
handouts/figure8curve.mw
WEEK
14[+1]: -
M: Test 3 due; (extension
by email request possible);
handout on example
mass spring system resonance
calculation;
back to work:
handout on reduction
of order with exercise
[mixed order DE system example: pdf, mw]
[in book you
can read 7.1 First Order Systems, examples 3,4 (examples 5-7 are the
opposite of reduction of order, ignore)]; homework consists of very short problems, no solving required,
just rewrites, to make sure you understand matrix notation:
7.1: 1, 2, 8 [first let x = [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; [solutions
for fun??]
7.2: 5, 9.
-
T: in class: damped coupled
oscillators without reduction of order exercise: we derive by hand in class
the solution to the decoupled 2nd order equations [pdf exercise];
HW for the homogeneous solution only: 1) Evaluate the derivative of your vector solution and stack
together to form the initial data 4 vector and identify the vector
coefficients of the 4 arbitrary constants which appear.
2) reformulate
this problem using reduction of order, find the 4x4 coefficient matrix
eigenvectors and the solution as an arbitrary linear combination of the
eigenvalue exponentials multiplying eigenvectors and convert to real form by
taking real and imaginary parts of the complex vector solutions to compare
with the previous solution as bob did for the undamped case of his example
system.
Solution:
redorderwithoutredexercise3.pdf [redorderexercise.mw]
[Ignore previous version, yesterday only:
redorderwithoutredexercise2.pdf]
[Ignore this: This is just the 2
equal mass, 3 equal spring system, explored completely here:
2mass3spring.mw and
ep3_7-4-3spring.mw]
Solution to first order problem from previous day:
redorderexercise2.pdf.
- W: see solutions for
previous days;
Last final exam 15S.
- F: Choose
7.4.11
[template for the final exam] or the "7.4.Application" warm up problem 1 on page 445
describing lateral earthquake vibrations of a two story building. [earthquakes?] Work with multiple people,
choosing which of these problems you prefer.
Test 3 final deadline
today in class unless you have contacted bob with extenuating circumstances.
- M:
CATS?
[11:30 section 002, login to
BlackBoard here or via
My Nova, do central column "Blue Course Evaluations";
12:30 section
001 do paper forms]
Discussion of previous day problems, previous final exam.
Final Exam MLRC voluntary problem
session when?
Final Exam Saturday (see below); switching sections
possible with bob's permission.
- scroll up for current day
Weeks 2 and 3 thru 4: come by and find me in my office, tell me how things are
going. This is a required visit. Only takes 5 minutes or less.
If you are at all confused, try to do this way before Test 1 in week 4.
*MAPLE homework log and instructions [asterisk
"*" marked homework problems]
Test 1: week 4
Test 2: week 9
Test 3: Take home out-in week 12
Final Exam: MLRC problem session Friday May 1, 3pm
FINAL EXAM:
[switching between these slots with permission]
2705-01 (12:30class): Sat, Dec 17 1:30 - 4:00
2705-02 (11:30class): Sat, Dec 17 10:45 - 1:15
MAPLE and G.Calc. CHECKING ALLOWED FOR QUIZZES, EXAMS
27-aug-2016 [course
homepage]
[log from last time taught]
7.1: 1, 8 [solutions
for fun??]
© dr bob
enterprises
CATS? 