MAT2705 18F 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). It will not be collected.
- M (August 27): GETTING STARTED STUFF. By Wednesday, August 29, 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 quickly. 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
and check out our class photo roster, and visit the link to
my course homepage from it by clicking on my home page URL under my
photo and then on our class homepage, or directly:
[
http://www.homepage.villanova.edu/robert.jantzen/courses/mat2705/ ],
3)
Open Maple 2018 if you already
have it
[or click on this maple file link to download the worksheet, then open it:
firstdayplay.mw]
here is an
example of a PDF handout:
firstday2015.pdf
[here is an
example of a Maple worksheet;
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.]
Find 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
2018
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
- T: return your schedule forms at the beginning of
class;
check your data entered by bob on sign-up sheet---correct errors;
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?
- W: 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!].
- F: course info handout;
Quiz 1 [see the
archive for similar quiz 1
from the 16f semester];
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].
WEEK 2[-1]:
M: Labor Day
- T: Quiz one
answer key;
class directory list (paper only for
privacy reasons);
use classlist contact info
(paper only) and MyNova class photos to get to know others in your class or
the other one,
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].
- W:
handout on
exponential behavior/ characteristic time [cooking
roast in oven remarks]
[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].
If curious: revisit the
river crossing problem and the
oven heating problem.
- F: Quiz 2 (separable DEs);
online ahandout:
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
WEEK 3[-1]:
- 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
after This Wednesday through Thursday next week, but preferably be familiar with using Maple to check
your work before the next quiz (read Maple HW page).
- T:
1.5 (tank mixing problems ChE majors especially):
37
(Use
Eq. 18 in the book or the boxed equation in the online handout); extra questions
to answer: 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.
- W: 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].
- F: Quiz 3 (linear DEs);
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)].
4pm Villanova Room
One Book
Villanova and Peace Award Talk. Don't miss it!
[It was the best one
I have attended in many years, see a
brief 24 minute
version here.]
WEEK 4[-1]: maple1.mw due anytime through approximately Friday (so
you are ready for using it on Test 1)
- M:
air resistance handout
(example of a piecewise defined DE and 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]
- T:
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].
- W:
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.
- F: Quiz 4;
handout on
RREF (Reduced Row Echelon Form, section 3.3)
[see bob's example
3 using Maple];
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];
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?]
You can get to the Linear
Solve Tutor from the Tools Menu:
Maple
Tools Menu: Tutors, Linear Algebra, Linear System Solving: Gauss-Jordan elimination
To just get the result instead:
Click on augmented matrix entered with the Matrix
palette, then from the right pane menu choose Solvers and Forms, then
Row Echelon Form, then Reduced
to get the RREF form of the matrix, then from the right pane menu 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.
WEEK
5[-1]:
- M: Bring
laptops to class for hands on row reduction tutor experience;
handout on
solving linear systems example
[.mw];
we always want to do a full [Gauss-Jordan] reduction, not the partial Gauss
elimination reduction also described by the textbook;
the Matrix
palette inserts a matrix of a given number of rows and columns;tab between
entries;
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;
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].
5:30 voluntary test 1 problem
session in MLRC. Come ask any last minute questions.
- T: Test 1 on 1st order DEs;
come early for first class, stay later in last class if you need a bit more
time.
- W: 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 pane RREF
of the augmented matrix, then right pane 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]
- F: no quiz (test week);
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]. <<<<<<<<<<<
WEEK 6[-1]:
- M: 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."
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.
- T: 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]
- W:
linear
independence of a set of vectors; [motivating
example]
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];
[HW ready 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].
- F: Quiz 5 [like 16F Q4];
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.
Try to finish up Maple2 this weekend.
Try to submit Maple 3 by the end of next week if you can.
WEEK 7[-1]:
- M: 4.3: book catches up with us:
linear
independence (A x = 0),
express a vector as a linear combination (A x =
b)
[or does b belong to a linear relationship with
the colmns of A?];
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.
- T: bases
of vector spaces, subspaces; [counterexample]
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.]
- W:
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].
check out the Practice Quiz 6
[with answer key after you try it yourself] before Friday.
HW worksheet solution.
- F: Quiz 6 on new coordinate grids in the plane.
FALL BREAK :-) Enjoy. Be safe in your travels.
WEEK 8[-1]:
- M: 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.
- T: 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?].
- W: 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]
- F: Quiz 7 [see 16F quiz 6 but you find the solution basis functions
yourself];
complex roots: handouts on 1) the
amplitude and phase shift of sinusoidal
functions,
2) 4 quadrant amplitude-phase shift examples: [mw,
pdf].
and 3) DE example [.mw];
5.3:
8 [ans: y = exp(3x) (c1 cos(2x) + c2 sin(2x))] , 9; 17;
22,
23 (express in phase-shifted cosine form,
see first handout).
You should turn in Maple 3 by Monday (be ready to
use Maple on Test 2!),
and should have started on Maple 5.
WEEK 9[-1]:
quiz 7 makeup, ask bob; pick up quiz 6 if absent
Friday
- M: MLRC voluntary problem session at 5:30pm; no
office hour 12:30-1:45;
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 (or just right context menu to get the solution), 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
[EEs read the complex exponential stuff]
[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].
- T: Test 2 on chapters 3,4.
- W: handout on linear homogeneous 2nd order DEQ with constant positive coefficients
(damped harmonic oscillator)
[on-line only: examples in
nature];
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];
We will work on
17 in
class to reproduce the Maple results.
- F:
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],
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]
[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]
WEEK 10 [-1]: check
blackboard grades; answer key for quizzes, test 2
- M: 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];
5.6: 8. Ignore the book instructions and solve this problem by hand for the
initial conditions x(0) = 0 = x'(0) (at rest at equilibrium).
[Read the
Maple worksheet about the interpretation of
the solution. Wiki:
driven
harmonic motion,
amplitude plots:
they introduce ζ = 1/(2Q)]
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.
solns: 8: [.mw,
.pdf]; 15, 17: [.mw,
.pdf]
- T: 5.6:
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 described in the above link as well as 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 use the right
context menu on the 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! [ ex (2 cos(x) +sin(x))
you need a space before the left parenthesis!]
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 (You-Tube video)];
a real resonance bridge problem occurred more
recently: the Millenium Bridge
resonance.
Line up to VOTE!
- W: Transition back to linear algebra:
[In
class watch bob use the Maple worksheet
ExploreEigenDrag.mw to
graphically determine the eigenvectors and eigenvalues of three 2x2
matrices. 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>>];
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 that Maple worksheet 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!
- F: Quiz 8 (like 16F quiz 7);
Maple
3x3 matrix eigenvector example handout [.pdf,
.mw];
more examples;
6.1: 13, 19;
21, 25 (upper triangular so diagonal values are eigenvalues!),
[Maple input for eigenvalue derivation];
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! compare to Maple's Eigenvector result,
context menu only a].
[repeated eigenvalues:
<<9, -6, 6> | <4, -1, 4> | <0, 0, 3>>]
WEEK 11[-1]:
- M: 6.2: 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 hint: 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?].
- T:
handout on the geometry of diagonalization
and first order linear
homogeneous DE systems
(2-d example: real eigenvalues [phaseplot]),
this is the solution of yesterday's diagonalization problem)
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.
- W:
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,
.pdf);
[if you like you can quickly edit the DEPlot
phaseplot template to reproduce the back of the book diagram for that
problem].
Th-Sat VST
All My
Sons
- F: Quiz 9; 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 only); (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 context on matrix, select "map
command onto" and insert the command "evalc" to
evaluate to a complex number, expressing each entry as its real part plus i times
its imaginary part. [or see
this worksheet].
Nov 16
2pm Mendel Award
Talk [background][arrive
by 1:30 to be sure to get a seat]
Pharmacokinetics:
multicompartmental analysis of drug passage in the body is like a multitank
problem.
WEEK 12 [+1]:
- M:
summary handout on
eigenvalue decoupling so far;
in class open and execute this Maple worksheet: eigenvector DE solution
workshop:
x ' =
A x etc, bring paper and pen,
laptop;
7.3: 31 (be sure to maximize the third variable, exactly using Maple then
numerically, make sure your plot of the 3 variables in a "decay window"
agrees), 37, 39.
- T:
handout on extending eigenvalue
decoupling
(to nonhomogeneous case, and second order, ignore bottom
half for now);
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).
T-Day break
WEEK 12[+1]:
- M: in class
together do day
45 HW exercise (following this handout example [.pdf
,
.mw]), 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.]
MLRC test 3 voluntary problem session 5:30.
- T: Take home
Test 3 will be linked here after
class; we will start it in class;
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
[read it carefully]
for the direction field and solution plot versus t, all you need for
problem 4.
- W: Continue working on Test 3 in class.
- F: Last day to continue working on Test 3 in class.
WEEK 13 [+1]:
- M:
7.5:
2 mass spring systems: theory plus
worked examples [figure8curve.mw];
[solution of days 45, 46 exercises; description
parts a) thru k):
drivenDEsystemExercise.pdf]
plots: figure8curvemodesplot.mw
(results only pdf print: figure8curvemodesplot.pdf
);
HW: vertically hung
with gravity? do parts a) -d).
[insane, see this
calculation for the eigenvalues;
challenge for when you have more time: show the slower mode
eigenvector has same signed components, while the faster mode eigenvector
has oppositely signed components, using this crazy approach; removing the
third spring does make things easier!
you can more easily get manageable expressions for the eigenvectors]
[2 mass 3 spring system: beautiful
Maple animations, and
the JPG movie]
- T:
7.5: 3, 9 [same system
s previous day 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)].
- W: Takehome Test 3 back? [You may request an extension by email if you
feel more time would allow you to do a better job. Tell me your expected
submission date.] Continue working on yesterday's problem.
OFFICE HOURS
POSTPONED TODAY till at least 3pm for dept job hiring meeting. Let me know
if you want to stop by.
- F:
handout on
day 50 example used to do a
mass spring system resonance
calculation [slider
exploration];
last concept:
handout on reduction
of order with exercise
[optional: 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],
[solutions
for fun??]
7.2: 5, 9.
Days 51, 52
7.5.3,9 exercise solution: [pdf,
mw]; resonance analysis:
pdf
WEEK 14[-1]:
- M:
YouTube videos on damped oscillators
[tuned mass dampers!? short exercise];
reduction of order handout:
scalar exercise soln,
vector exercise without
reduction [soln],
[soln with reduction].
- T[F]: Earthquake!?;
reduction of order
version of 7.5.3_9;
archived exam exercise.
- W: Deadline for extended period return of Test 3 in class;
CATS teaching evaluation first 20 minutes of class;
Final Exam
discussion [archived
exams].
Th: MLRC voluntary problem session 5:30pm??
Maybe again next Wednesday 5:30pm?? for those taking Thursday exam?
Sa: 04 exam
10:45br, Th: 05 exam 2:30
- 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 4-5
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]
2705-04 (10:30class): Sat, Dec 15 10:45 - 1:15
2705-05 (11:30class): Thur, Dec 20 2:30 - 5:00
MAPLE and G.Calc. CHECKING ALLOWED FOR QUIZZES, EXAMS
14-aug-2018 [course
homepage]
[log from last time taught]
