23F homework and daily class log
Jump to current date!
[where @ is
Your homework will appear here each day as it is assigned, with occasional
links to some MAPLE worksheets when helpful to illustrate some points where
technology can be useful. [There are 56 class days in the semester, numbered consecutively below and labeled
by the (first initial of the) day of the week.]
It is your responsibility to check homework here
(not all problems are in WebAssign). You are responsible for any hyperlinked
material here as well as requesting any handouts or returned tests or quizzes from classes
you missed. Homework is understood to be done by the
next class meeting (unless that class is a test, in which case the homework
is due the following class meeting). WebAssign "deadlines" are at 11:59pm of the
day they are due, allowing you to complete problems you have trouble with after
class discussion. [If Friday is quiz day, HW due Friday has a WebAssign deadline
of midnight Monday.] The WebAssign extension tool can be used to request more time or more attempts for a given
homework assignment, so persistence can give any student 100 percent credit on
the homework cumulative grade.
Read the textbook (watch
embeded videos). Do the homework assignments on time to keep up. Use the Ask
Your Instructor tool when you are stuck. Ask for extensions in due date and
number of tries till you get 100 percent correct.
WebAssign homework management/grading is required,
giving you access to an incredible wealth of multimedia tools together with the
online e-book textbook you can access from any internet connection.
WebAssign deadlines are suggested to keep students on track, but
extensions in time and number of tries are always granted.
are labeled by chapter and section and problem number, and identify
each WebAssign problem. Optional problems which are not available in WebAssign will be square bracketed
and are helpful for your learning. Check this homework page for hints and some linked Maple worksheet
solutions. Use the Ask Your Instructor tool to get help on any
problem for which you cannot get the correct answer. There is no reason for
anyone not to get 100 percent credit for the homework assignments. This
is the most important component of learning in this class, doing problems to
digest the ideas.
Read what Villanova students say about the most important
things you can do to succeed here.
If you have any questions, drop by my office St Aug 370 (third floor
facing Mendel by side stairwell) or just come to see where
you can find me in the future when you might need to. I welcome visitors.
- W: GETTING STARTED STUFF Wednesday August 23: Introduction and Overview
We will access our
e-textbook/HW WebAssign portal through
BlackBoard with your laptop or phone.
"Ask your Teacher" "Request Extension';]
Lecture Notes: 5.0 Riemann definition of a definite
integral etc the first minilecture!
the first Maple Worksheet = MW
Make sure you read my
sent before the first class, and register with
WebAssign (immediately, if not already done) and by the weekend, reply to that
email with your
schedule attached as described there, including a bit of introduction of you
to bob, as described in that email.
Make sure you download Maple 2023 to your
local computer, available by clicking here and install it when you get a
chance (it takes about 15 minutes or less total), 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.
I will give one-on-one tours of Maple in office hours to those who are
No problem if you never used it before,
as is the case for all entering freshmen.
Explore the on-line resources. Browse the pages linked to our class home
online handout on functions and differentiation
to see what background is assumed.
Do the "Getting Started with WebAssign" assignment. Notice
the "Ask Your Instructor" button!
- Th: Lecture Notes 5.3 Fundamental Theorem of Calculus, Review
plus derivative/differential notation (pages 4-6) [summary]
Read 5.3. Do 5.3: 7, 9, 13,
37, 41 in the WebAssign
see textbook section opening
video examples in Maple when reading the section;
think about these online
handouts on rules of algebra
rules of differentiation.
- F: Quiz 1 on 5.3 (take home open resource quiz but no collaboration, due
Monday in class);
when stuck on a quiz, email me your question);
5.4 indefinite integration and net change (pages 3-4);
velocity example of net change and the
fundamental theorem of calculus:
3, 7, 17, 39, 41, 53 [hint: separate into
two integrals], 58, 60, 69,
tabular data: 75 (consistent units! hrs, secs).
- M: Lecture Notes
5.5a change variable in integral;
summary online handout: "u-substitution";
extending our short list of function derivatives:
chain rule for scaled integration
variable in a function;
5.5 1, *, 7, 13, 19, 21, 25,
separate into two integrals], 65 [express as a definite integral in u].
Tuesday Drop Add period ends [free phone app for great scans
of quizzes: Adobe
- W: Lecture Notes
5.5b symmetry and transforming definite integrals; [example: bell
curves; page two]
89, dimensionless variable discussion:
(please read worksheet for comments, use palette to enter subscripted variable "C_0" in WebAssign,
remember math is case sensitive),
92, 93, 97
- Th: Review problems [no review problems are
on WebAssign; see e-text: Review section, skip over true false and concept
4 (remember Riemann definition!), 7 (visualization is
important), 15, 21, 23, 31, 41,
53, 59, 64, 65, 67, 69,
72 [Maple gives you the value of the
limit, your task is to derive it using the fundamental theorem of calculus;
the worksheet explains why this limit is interesting for an optional read];
- F: paper class lists distributed;
Office Hours midday serve nearly
Quiz 2 (posted online
for after class, paper copy handed out in class);
Areas between Curves;
6.1: 1. 3, 11, 13, 17, 25, 57, 58;
numerical root necessary?;
Income inequality is a serious problem today, made painfully clear by COVID.
Read Applied Project "The Gini Index", the applied project following section
6.1 (inserted as a result of the lobbying of our very own Math&Stat Dept
prof. Klaus Volpert) which is important in quantifying income and wealth
[If curious checkout
this Gini index calculus application only using calculus you already
know, appearing as an Applied Project in the text after this section,
inspired by our own Klaus Volpert in the Math/Stat Dept.]
# plot 2 graphs together, or use drop second expression onto PlotBuilder
plot of either one
Monday is Labor Day!
- W: Check Quiz 1 answer
key in archive; Quiz 2 due in class;
Note Menu Maple Tools, Tutors, Calculus 1, Volumes of Revolution;
Lecture Notes 6.2 on
Volumes (of Revolution, etc); [example]
6.2: 5. 6. 11, 51,
59 [see how to set up
problem in Maple worksheet],
67 [elliptical version of example 7],
73 [circular cross-sections];
online handout on regions of the plane: relationships between
variables [areas, volumes of revolution].
- Th: Check Quiz 2 answer
key in archive;
in class group work 6.2.61
[cap of sphere set up],
Example 9, 77 [set up], if time: 6.4.48
called a trirectangular tetrahedron; use horizontal plane cross-sections,
similar triangles or write equation of line
in plane for side face projection!],
sides of isosceles right triangle are hypotenuse divided by sqrt(2),
68, 85 (wine barrel: lots of parameters).
optional fun: apply the
pyramid volume formula to the
of Egypt using the data you find online to verify its stated volume
- F: Quiz 3 (posted online
after class, paper copy handed out in class);
Bring laptops for Maple
workshop. bob will help you adjust your palettes and show you the interface.
open up this file: datafitint.mw and
read it with a partner, then do the final section exercise with
the data from Stewart
5.4.84* on a shuttle liftoff
height calculation (convert to mile units at the end). Ask bob when you are
confused about anything.
We are skipping
6.3 on cylindrical shell volumes! [example]
WEEK 4[-2]: please use
the Extension and Ask Your Teacher tools correctly to help me help you
- M: Lecture Notes 6.4 on
Work in 1-d motion;
15 [READ THIS WORKSHEET FIRST: tricky, the extra single weight is moved a
constant distance, so the associated constant product work value can be incorporated into the
integral for the cable as a constant force term in the integrand, see this
30 [this problem is all about units!
you need to convert inches to feet to get ft-lbs in the final answer],
36 [Evaluate the work needed to build the Great Pyramid of Egypt:
setup keep maximum significant figures in part a) to obtain correct
integer for part b).] [solution]
- W: Lecture Notes
Average Value of a Function; [temperature
example][lung inhale example 23]
6.5: 1, 7, 9,
7, 9, 11, 17,
flow as an example of dimensionless variables (we return to this in
- Th: we haven't done very much to review so far!
6.R: 11 (in class), 17, 20, 28, 31,
Hint: solve for the point x = a of intersection in terms of
the slope m of the line y = m x (easy!) ,
and express m as a function of a. Then use this value to set the area over x = 0..a
equal to half the area over the entire interval. This condition is easily
solved by hand if you are careful! Plot the resulting line with the curve.
Maple group exercise in class:
6.AppliedProject: Where to Sit at the Movies (after 6.5)
Assume the result of part 1 (just trigonometry), then do parts 2-4 to get a
real life example of an average viewing angle. Here is a
worksheet maple6.mw to accomplish this.
Work on this together with one or a few partners, and finish after class, if
submit worksheet by email with subject header:
lastname-lastname-...maple6.mw and same filename attached
so I can give feedback.
- F: Quiz 4 (work, avg value but no Int by Parts yet!) [remark:
words and math]
Chapter 7:1-4 "Methods
of Integration" is obsolete in the age of computer algebra systems; there is
no general approach to finding antiderivatives.
Integration by Parts;
7.1: (integration by parts is useful in physics/math variational
problems, not so much as an integration technique that we need to worry
about, but we need to be aware of the process):
3, 5, 9,
23, 45 [example
of using Tutor];
just for fun: [Explore
WEEK 5[-2]: Test 1 Thursday
- M: Lecture Notes 7.1b on
Integration by Parts etc;
74 [dimensional discussion and HW template; first transform
with a u-sub or just do definite integral with Maple],
Optional. transforming an
integral with an unknown function.
- W: Lecture Notes
7.7 (Simpson formula
3, 10 [for both these use the ApproxInTutor
and the "Compare" button which gives all the approximations simultaneously],
30 [template for Simpson], 35
37 [template for tabular data with Simpson], 42
[use this template];
[Optional: Maple approach to
the unlinked problems, with explanations.]
[compare Midpoint versus Simpson]
Is there a desire for a 5pm problem session today in Mendel G90 to
ask any questions for me to solve? The room is booked.
- Th: Test 1 thru 7.1 (see
Come early if you can to get started.
- F: No quiz during Test week;
Lecture Notes on
7.7: (numerical error?): together:
21 [saving you tedious input in
WebAssign], catch up on past HW since no WebAssign tonight!
Aside. If you use Maple's Approximate Integration Tutor, note
other methods listed. Simpson 3/8 rule uses a 3rd degree polynomial, while Newton Cotes formulas
(see the Approximate Integration Tutor window menu) generalize
the trapezoidal and these Simpson rules to even higher order polynomials. [wiki]
Amusingly enough the proof of the error estimate for the
trapezoidal rule involves two clever integration by parts and once verified, the whole proof becomes relatively straightforward. So as I mentioned, integration by parts is essential for advanced mathematics and applications.
The Simpson's rule error estimate proof is more of the same.
WEEK 6[-2]: check
key for test 1 and study the additional pages please
- M: Lecture Notes 7.8a on Improper Integrals
(infinite limits); [normal curve
7.8 [Maple examples]: use Maple for needed
antiderivatives if you have trouble with the change of variable:
1, 3, 5, 19, 29,
31, 33 [Hint: u-sub!].
- W: Test 1 back;
5.5: 52, 53,
66 [note u = exp(x) is the only way to proceed! in other
words the exponential is inside the composed function],
[transform definite integral to new variable for practice],
78 [note dx/sqrt(x) 2 d (sqrt(x)), what does that suggest? the sqrt(x) is
inside the composed function, let it be u];
[in each case, 1) re-express integrand
in terms of u AND re-express differential in terms of u
and du, and 2) change to new u-limits of integration. Then
evaluate this completely new integral expressed entirely in terms of the new
- Th: W: Lecture Notes
Improper Integrals (infinite integrands);
7.8: use Maple for needed
antiderivatives: 37, 39, 40, 41 [odd roots in Maple require care: ab-surd!],
[int by parts]: 43, 47, 48.
- F: Weekend Take home Test 1b due Monday (see
archive for copy); please
read the test rules;
to solve an equation numerically
see this worksheet;
Improper Integral (comparison and applications); [examples]
classical gas velocity distribution];
7.8: use Maple for needed antiderivatives
57, 61, 64, 67;
A Nobel prize for an improper integral?
- M: take home test due in class;
78 [in class quickie exercise together];
Maple worksheet collection of problems:
71; what must be true of n for convergence at infinity? what
must then be true of n for convergence also at the origin?
[liters!], 82 [hyperbolic
88 [games with
89 [this is just a
change of variable problem, we don't
have to actually do the integral).
- W: Lecture Notes
8.1a on arclength; [fun: all powers
lead to special functions]
Tutor): 1, 7,
9 (simple u-sub), 14 (perfect square!),
27 (use Maple evalf of inert
form of integral or wait 25 seconds to be
to plot; solve for y, perfect square is a power function!,
do one quadrant, multiply by 4).
- Th: Test 1B back: see
8.1b on arclength functions;
problems in Maple):
41, 42 [there are three equivalent
different looking forms for the arclength function due to the ln and trig
identities; only one is accepted by WebAssign: s(x)
= ln(csc(x) - cot(x))],
45, 47 (numerical solve needed);
in class exercise:
definitions of cosh and sinh; you need
to numerically solve for a
in the interval a = 40..100,
Solve, Numerically Solve From Point, choose a = 40);
We skip 8.2 on surface area since this is not
even required in the syllabus of multivariable calculus MAT2500.
We skip 8.3 since moments will be dealt
with in MAT2500, while hydrostatic pressure seems to far afield for most of
- F: Provisional Midterm grades in BlackBoard; check that uploaded grades
Quiz 5 due Wednesday after break;
Lecture Notes 8.4a on
business calc [a little business literacy never hurt anyone!];
3, 5, 9, 12, 17 [most of these are just areas between a curve and a
horizontal line, set them up, evaluate with Maple]
- M: Lecture Notes 8.4b on blood
flow, cardiac output;
dilution method example
measure flow rate out of heart, ignorable cardiac output
21 (plug in numbers, blood
23 [dye problem with functional expression for c(t)],
AE2 like 24 [cardiac dye, use the above template for tabular
- W: Lecture Notes 8.5a on probability;
8.5: 1, 2, 5, 6,
Poisson (decaying exponential) distribution
applications (pre-tech days:
10 (median value of x: probability half below x, half above x),
11 [part c) solve for the time T such that P(t>T) = 0.05 or P(0<t<T) =
12 [use maple for integral].
- Th: Lecture Notes 8.5b on probability:
normal distributions etc;
[template: adjusting the mean to achieve a
[ignorable: optional gas tank filling example and changing variable:
mw]; [circular target]
- F: Quiz 6 on probability [use short
template for normal distribution];
open and execute together: [8.5.21 electron cloud!];
[also find s(x)], 21, 22, 23.
Test 2 thru chapter 8 in class Thursday this week (improper integrals,
Sequences and series: we don't need to be experts on the preliminaries,
the emphasis is on Taylor series at the end of chapter 11.
- M: Lecture Notes
11.1 on infinite sequences; (long section but we only need an intuitive
idea, not so much detail);
Watch 11.1 first page introduction 8 minute
video; (see also motivating sequences and series worksheet):
11.1: 1, 7, 8, 21, 27, 35, 47 (l'Hopital's rule: ∞/∞), 55 (combine ln's
n = 1..10) this is how you generate
sequences in Maple; use evalf(f(n)) if you want the decimal values
- W: Test 2 prep: ask bob to work whatever problems still confuse you.
Quiz 6 answer key is online.
- Th: Test 2 in class.
- F: Lecture Notes
11.2 on infinite series; [Maple
(short problems, I avoided tedious
entry of numerical values of first n terms):
application): 1, 3, 15,
23, 25, 29, 33, 39, 49, 59,
71 [wording misleading, limiting amount
after each injestion, not the minimal level always remaining in body],
77 [assume (1+c) is
a proper fraction, sum the series, solve condition on c, pick the
solution making the ratio a proper fraction].
- M: Lecture Notes
11.3 on the integral test [mw];
11.3: 7, 8, 11, 13, 17,
18, 19, 29, 33, 39 (use Maple integral test template).
- W: see
for Test 2 answer key please;
Lecture Notes 11.4 on
comparison tests (limit comparison test best! useful for Taylor
11.4: 1, 2, 7, 9, 11, 17, 26,
33, 41 [note the missing differential in the WebAssign comparison integral!].
rn = en ln(r) is an exponential
function of n, while n
p is a power function of n, explaining why geometric series
and p-series are the two important kinds of limit comparison series
for series whose formulas are constructed from the term number n.
The factorial n! is even faster in growing than powers or exponentials [Sterling's
approximation]: exponentials on steroids!
- Th: Lecture Notes
11.5 on Alternating Series;
11.5: 1, 2 [what is the formula for the nth term?],
5, 7, 17,
19, 37, 47.
- F: Quiz 7;
11.5b on Absolute Convergence;
[irregular signed series,
divergent alternating series]
11.5: 21, 23, 25, 27, 29.
- M: Lecture Notes
11.6 on Ratio [& Root] Tests;
11.6: 1, 3, 5, 9, 14, 15,
[notice that this a_n factors so that the factorial in the
denominator cancels the factorial in the numerator, leaving behind the nth
power of that common factor, so it is a geometric series!],
[Optional FUN! if you want to be amazed, read
worksheet of problem 50 which mentions
William Gosper calculating 17 million digits of Pi (!) and shows
dramatically how using an efficient series representation pays off.]
We are skipping 11.7 since general series convergence tests are not as
important for Taylor series, where the "absolute convergence ratio test rules" and the
ratio test plus p-series comparison is all that matters, apart from
the alternating series error estimate; now we
start the important stuff of functions represented by Taylor series.
- W: Lecture Notes
11.8 on power series (almost Taylor, finally!);
2, 4, 6, 11, 13, 15, 21, 24,
35 [Hint: just consider the additional
factors in the numerator and denominator as n goes from n to n+1 to get the
ratio of successive terms, then take its limit].
- Th: Lecture Notes
11.9 on power series tricks;
(tricks with power series):
1, 3, 5, 7,
the indexing!), 27,
34 [like example
8: explained by this Maple worksheet
and this PDF but a
calculator is sufficient].
Bessel approximations compared
Lecture Notes 11.10a
on Taylor series;
11.10 (Taylor series!):
3, 4, 5, 6, 24, 27
[recall cos(n Pi) = (-1)^n , why? think],
55 [convert degrees to radians,
use alternating series estimate!].
"Maclaurin series" = Taylor series centered at origin, term not used outside
> taylor(exp(x),x = 1, 6)
; convert(%,polynom) # up to 5th
power, centered at 1
- M: see
Quiz 7 answer key;
Lecture Notes 11.10b
on binomial series;
11.10: 35, 45, 65
(alternating series estimate)
- W: Lecture Notes 11.10c: Taylor
series loose ends;
[Taylor remainder formula:
practice but not for testing, just
to be aware);
(products and quotients of series?
not for us, leads to tan(x):
video of long division,
tantaylor; ditto for other quotient trig
series (all we can handle, missing are arcsin,arccos, use binomial
series integration to get those];
11.10: useful limits with Taylor:
68, 69 [similar to 68], 70.
for fun look at an unusual function
96 which does not have a Taylor series
at x = 0 even though it and all of its derivatives exist there].
- Th:11.11: Lecture Notes 11.11
on Taylor series apps [kinetic
energy, black body radiation]
11.11: we only need the alternating series estimates for the cosine and sine
for example, so take this first section as a read only section; the second
section Applications to Physics shows some examples of how often one or two
terms in the Taylor series are enough to give a good approximation;
WebAssign for this section; we look at the following two problems in class
worked out here],
Quiz 9 online;
11.R: in 11-21 recognize how to quickly
identify the large n behavior for many of these;
19 just use
ratio test; ratio is quotient of new factors in numerator and denominator;
22 [binomial series: expand this in terms of x = 1/n after
factoring out an n from the square roots to get sqrt(1+-1/n)
to see large n behavior from leading term in this expansion, the
look at some of the following problems that seem
reasonable: 40-44 you should be able to answer all of these (for example 43
47-54 use tricks by
manipulating simple series;
56a [like 11.10.59 it
becomes alternating after the first term, use Alt Series estimate], 59;
60 is a great application we will do
together in class. [solution].
summary: the big picture.
2pm Mendel Medal Award talk in the Villanova Room, speakers usually
- M: chapter 11 problem catchup day: bring problems you want to see solved
from the HW or quizzes, past tests.
22F Test 3: PDF [answer
- M: Lecture Notes 10.1
on parametrized curves;
parametrized curves can
be fun (tease)!;
(ignorable: execute worksheet and look at many other examples):
(use Maple to
this worksheet for
how to plot parametric curves,
but can choose WebAssign plot by
finding starting and ending points]
8 [do b) first or where is t
= 0? what is the slope of the straight line?],
9 [where is t = 0? where
is start, stop?],
17 [does x increase or decrease?],
58 a only, 3 inputs [ignore rest of problem; optional see how you can
Explore the trajectories for all forward
- W: Take home Test 3 starts in class (emphasis on power series)
[Read test rules
- Th: Lecture Notes 10.2
on calculus of parametrized curves: tangents;
1, 9, 13 [pick the best tangent line by eye], 15, 24.
template for plotting parametrized curves:
- F: Lecture Notes 10.2b
on calculus of parametrized curves: areas and arclengths
positive areas, only the sign is troublesome:
[y dx/dt>=0 so integrate this t = 0 ..Pi/2],
[-y dx/dt>=0 so integrate this t = 0 ..2 Pi],
these arclength problems can be done by hand:
47 [obvious u-sub,
factor common power outside of sqrt];
48 [perfect square].
- M: Lecture Notes 10.3
polar coordinates; [fun: Earth centric venus
(trig refresher online: trig;
Maple polar curve plots;
10.3: 3, 5, 9,
11, 17 [mulitply both sides by r]
19, 22, 25 [replace x^2+y^2
by r^2, then replace y, divide through by r],
[where does it start with theta = 0?],
43 [where does it start with
theta = 0?].
- W: Test 3 due in class;
Lecture Notes 10.4
on polar curve areas; [video example;
circle and cardioid]
3 [this traces out the loop exactly once, with a
small interval of negative r between 3Pi/4 and Pi that sweeps out
the part below the x axis],
7, 21 [plot and find exact two
where r = 0, integrate between them for inner loop],
23 [draw these by hand, both circles],
35 [like in class example].
- Th: Lecture
Notes 10.4 on polar curve arclength;
10.4: 49 [this is a semicircle of arclength Pi times radius: integrand is
51 [obvious u-sub after factoring out largest power from sqrt],
evaluation of integral! what is the angular range for r>=0?].
F: Test 3
answer key online ;
Notes 10.4 on polar curve tangents [example
10.4: 63, 69 [obvious points on circle,
derive polar coords].
(not on final exam)
WEEK 14 last day:
- M: CATS on BlackBoard; review for final:
10.R.44 polar coord
arclength and area
[limited to parametrized curve
tangent lines and slopes like 10.2:7-12, 21-24, polar curve areas and arclengths].
Weeks 3 and 4 or 5 or earlier: come by and find me in my office
SAC370, spend 5
minutes, tell me how things are
going. This is a required visit. If you are having troubles, don't wait
to see me.
Tests: week 5, 9, 12 -13? [week long take home] (tentatively)
FINAL EXAM: review Wed Dec 13, 5-6pm
log from last time
MAT1505 [MWF 09:35 PM] Fri, Dec 15 08:30 am - 11:00 am
MAT1505 [MWF 10:40 AM] Mon, Dec 18 08:30 am - 11:00 am
[exchanging exam slots
is possible; ask bob]
MAPLE CHECKING ALLOWED FOR all Quizzes, EXAMS
does anyone ever scroll down to the end?