By Wednesday January 14, 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 "[MAT1505]"
[always include this string in your subject heading if don't want your email to be lost in my overflowing mailbox]
telling about your last math courses, your comfort level with graphing calculators and computers and math itself, how much experience you have with MAPLE so far, why you chose your major, etc.
Login to the Villanova home page (click on the upper right "login" icon and use your standard VU email username and password) and check out our My Classrooms classroom site linked to your list of courses at the center of the web page, and visit the link to my course homepage from it
[ http://www.homepage.villanova.edu/robert.jantzen/courses/mat1505/ ],
and read the on-line links describing aspects of the course (no need yet to look at the
MAPLE stuff).
Fill out your paper schedule form (get a copy in class or print it out earlier) in class. If you have any questions, drop by my office St Aug 370 (third floor, Mendel side, by side stairwell) or just come to see where you can find me in the future when you need to; I welcome visitors.
bob will quickly show you the computer environment.
Homework: Explore the 2 CDs that come with your textbook if you have not already done so.
Explore the on-line resources.
Think about paper handout on algebra, and remember differentiation rules.
Homework (exploration of resources):
1. Finish reading course advice and investigate class web page links.
2. Read Stewart Chapter 0: A Preview of Calculus, sections: The Area Problem, Limit of a Sequence, Sum of a Series, and read Appendix E: Sigma notation. Do E: 1, 9, 13 just to recall sigma notation.
3. Look at the (red) TEC CD and try the first (preview) module from the Main Menu Calculus, Early Trancendentals: Approximating the area of a circle, corresponding to the first part of Stewart Chapter 0.
4. Still on the TEC CD, look at the first Homework Hint module: Chapter 5, Section 5.1, Problem 11, and see how the Hints button works.
5. Select the textbook on-line quiz site from our class home page main menu and look at the on-line quiz for section 5.1 at the textbook site (but don't attempt it yet!).
[Bonus: Figure out how to access the quizzes and tests on the other (aquablue) CD: The Interactive Video Skillbuilder. I can't seem to get my office computer to do this!
Fun: think up some new "rules of algebra NOT!" to add to the handout.]
[Note the textbook appendices A,B,C,D for review of precalculus foundations if you find yourself a bit rusty.]
> with(student);
> evalf(rightsum(sin(x),x=0..Pi,10));
> evalf(leftsum(sin(x),x=0..Pi,10));
> evalf(middlesum(sin(x),x=0..Pi,10));
> rightbox(sin(x),x=0..Pi,10); etc
[for 13*, do it yourself and save the worksheet to join with a partner later; use commands:
> with(student): # a colon suppresses output
> f:=x -> sin(x^2);
> middlebox(f(x),x=a..b,n);
> evalf(middlesum(f(x),x=a..b,n)); ]
If you have not used MAPLE much last semester, you might review this weekend by browsing up to section 2.5 in Calclabs or by executing and reading the 1500 command list worksheet.
M: Martin Luther King, Jr Day
Read 4.10; do: 4.10: 1, 5, 7, 9, 11, 13;
5.4 (thru p.403): 1, 3, 5, 7, 9.
Keep in mind the "rewrite rule" of integration: sometimes the integrand must be rewritten using algebra before applying integration rules.
5.4: 15*, 27 [rewrite rule!], 39 [Hint: break this into two separate integrals for positive and negative x], 47, 53, 55, 57, 59. Why not meet with someone this weekend to try the Maple HW so far and maybe form a homework partnership?
M: SNOW DAY, one less class?
5.5: 1, 3; 13 (reciprocal function), 15, 19, 21,
41 (separate into sum of integrals, for first integral recall arctan at bottom of p.406 box of antiderivatives, then do substitution only in second integral),
43 (divide thru in new integral after substitution),
63 (multiply thru in new integral after substitution), 65, 69, 77 (this kind of problem is why we are doing integration theory!).
FYI:
> int(x/(1+x^2),x);
> Int(x/(1+x^2),x); value(%);
> with(Student[Calculus1]);
> IntTutor(x/(1+x^2));
5.4: 64* [begun in 5.1.14]; (don't forget decimal points in data entry)
maple5.mws now complete and is due Monday Feb 2 thru Friday Feb 6 (delayed due to snow day delay);
Read handout on the substitution rule;
5.R (Review Exercises on p.431): 5, 7, 17, 23, 29,
31 [after first obvious u-sub, you have to do another one: v = ln (u) ],
32 [the obvious substitution u = 1-x^4 is not helpful, but arctrig derivative formulas should ring a bell with u=x^2],
45 [chain rule], 49 [max/min estimates]; .
Browse the remaining integrals 5.R: 9-37. You should be able to do these and check your result in MAPLE for the even numbered problems, except occasionally even MAPLE can have some trouble: badint.mws;
also 5.R: 56 [answers: a) 175/6 b) 177/6], 59, 64.
6.1: 1, 3, 7, 9, 19, 27, 33*, 39, 40, 41;
Test 1 next week: preferred day?
optional for an interested student: 69 [Better form of answer: Pi*h*(R^2-2/3*Rd+1/5*d^2);
Note: if you wanted to put tickmarks on the side of the barrel to enclose equally spaced volumes, like each quarter barrel, you would need the volume of liquid from the bottom up to x as a function of x to see at what x values its graph crosses the values V/4,V/2,3V/4. Can you do find where the tickmarks should be placed? Instead of the definite integral you need the antiderivative function which is zero at the bottom of the barrel. Extra maple credit equal to one Maple assignment for a well written worksheet that accomplishes this. This technique would be needed to put equal volume tickmarks from empty to full on any tank that has symmetry about the vertical axis. CalcLabs has a project on it.]
office visit required before Test 1, might even be helpful.
6.4: 1, 3, 7, 27, 29, 30;
be sure you read examples 4 and 5 to see why one needs the Riemann limit definition of limit to derive integral formulas for new applications.
be able to check any definite or indefinite integral exactly with MAPLE for the test
> Int(exp(-x),x=0..1);
> value(%);
> evalf(%);
> int(exp(-x),x=0..1);
> int(exp(-x),x);
6.R: problems 1-25 are review problems for 6.1 and 6.2 that also include cylinders of revolution (6.3) which are not covered and not a part of the test, but you should be able to realize when you do not have a technique to evaluate these additional problems. Part of understanding what you do know is knowing when you encounter something for which you have no mathematical tool. Use some of these for review for the test. In class: 4 [ans: 32/3], 15;
be prepared to "show all work" (justifying antiderivatives which require u-substitution, for example) on the test.
Optional*: Applied Project—Where to Sit at the Movies (p 468). Assume the result of part 1 (just trigonometry), then do parts 2-4 to get a real life example of an average viewing angle. You may do this in place of 12* if you like.
You have to be able to do by hand the simplifying power algebra in an integral like problem 20b:
v_s-avg = Int(sqrt(2*g*s),s=0..1/2*g*T^2)/(1/2*g*T^2) = 2/3*g*T.
Remember: integration by parts, like the substitution rule, is more important for changing the form of an integral than as an integration technique (technology is better than a grab bag of integration techniques/recipes);
Maple6.mws due next week, why not work on it this weekend?
come discuss it with me if you got lower than 80;
Read 7.6 to appreciate the past and future of integration in practice;
Read 7.2 example 3 (AC current is a sine wave, power delivered is the square of a sine wave, so average power requires integrating sin(x)^2), then using a u-substitution use this example to do problem 7.2.64.
7.8: 1c,d, 2a,b,d, 27, 31, 33, 51, 53, 57.
Read the worksheet 7-8-62.mws.
Test 2 W->W March 10->17? best date?
Correct Maple 5,6 if you want full credit for midterm grade.
8.5: 1, 3, 5, 7, 9, 11* [make sure you answer the textbook question with a comment and edit the template to correspond to the new problem].
Maple7-8 due week after break;
SPRING BREAK. [Test 2 Wed Mar 17. MLRC Tues Mar 16].
are your two maple worksheets upgraded to full credit for your midterm grade [Wednesday]?
> a := n->sin(n)/n;
> plot([seq([n,a(n)],n=1..100)],style=point);
> limit(a(n),n=infinity);
to create a table of numerical values:
> array([seq([n,evalf(a(n))],n=1..100)]);
27 [Hint: re-express as sum of two infinite series], 29;
recall Sigma notation: appendix E.
11.2: 35, 41, 44, 49 [Hint: a_n = s_n - s_(n-1), find series which is twice example 6],
52 a,b [Hints: a): D = h(1+r)/(1-r), b): Hint: use s=1/2g t^2 -> t=sqrt(2s/g) to convert each separate distance s up and down into a time, remember that on the first drop it only goes down, T = sqrt(2h/g) (1-sqrt(r))/(1+sqrt(r))],
53 [Hint: exclude values of c for which the |ratio| >=1], optional: 54*.
handout on changing variables in integrals [maple supplement];
in class review:
6.R (p.469): 27, 30;
7.R(p.541): 1, 9, 17, pick from 41-50 (like 43,47), 62 [see back page of book integral formula 99], 63, 65, 70 [remember at endpoints function being revolved about axis has value 0, so you have n = 4 intervals and 6 endpoints, also r = C/(2Pi) converts circumference to radius; final answer V = 4051cm^3], 71, 80;
8.R(p.583): 2, 7, 19, 20.
T: 5:30pm MLRC problem session.
I will be in my office most of the day Tuesday for special help.
work (spring systems), avg value of function over interval, integration by parts (versus u-substitution), approximate integration (error estimates), improper integrals, arclength, probability functions.
11-3: 1, 2, 3, 7, 9, 11 (cubes!), 13, 17, 19; 31, 32 (need 5 terms),
32* [use template for numerical work after evaluating error conditions by hand];
> Sum(1/n^5,n=1..infinity);
> value(%);
> evalf(%);
33*(compare "exact" maple infinite sum error with your Maple calculated error).
11-5: 1, 2, 5, 7, 15, 23, 27, 29, 32;
> a := n->(-1)^(n-1)*n^2/10^n;
> evalf(seq(a(n),n=1..6));
> evalf(sum(a(n),n=1..5));
Test 2 back, take errors seriously, try to learn from them.
Quiz 7 Not!
> a:=n->(-1)^(n-1)/n/2^n;
> S:=n->sum(a(k),k=1..n);
> array([seq(evalf([n,a(n),S(n)]),n=1..10)]);
> sum(a(n),n=1..infinity);
> evalf(%); [link to this worksheet]
catch up on Maple11.mws [maple7-8 back by monday email]
9 [just like example 3], 13, 15 [Hint: what is Int(1/(5-x),x)?, use problem 7 result],
23, 27 [Hint: this is an alternating series, so you can estimate the truncation error using the next term in the series, handle first 3 terms of series by hand, then evaluate each final term with technology to check truncation error].
> taylor(sin(x),x=0,5);
> T4:=convert(%,polynom);
11.10: 27, 31, 37, 39, 43 [Hint: this leads to an alternating series, so you can estimate the error; compare your approximate result with the numerical integral result];
35* (use template); maple 11 is now complete and due before test 3.
1, 3 [do using binomial series, and then show you get the same result from
(2+x)^(-3) = 1/2*d^2/dx^2 (2+x)^(-1) by term by term differentiation of the geometric series result for (2+x)^(-1) = 1/2*1/[1 - (-x/2)];
if you are curious about relativistic physics from your exposure to physics, read "applications to physics" in 11.12 pp. 780-781.
Easter Break starts Th.
MLRC problem session for Test 3 today at 5:30.
> deq:=diff(P(t),t)=k*P(t);
> init:=P(0)=3;
> sol:=dsolve({deq,init},P(t));
> subs(sol, deq);
> simplify(%); [worksheet]
no homework today to have another block of time undistracted to work on test 3;
in class use technology/maple to check as much of the problem solutions as you can:
http://www34.homepage.villanova.edu/robert.jantzen/courses/testquiz/04s/15504s31.pdf
you do not need your own handwork to do this checking. only the test questions on this pdf.
Remember, no collaboration. Use the MAPLE help and the course log MAPLE commands or linked worksheets to help you.
read the explanation and type in all the input lines and see how they work;
9.2: 11, first do by hand x = -2..2, y = -2..2, unit square grid on integer coordinate pairs, then compare with Maple's dirgrid=[5,5] and default grid plots (see worksheet below);
> with(DEtools):
> deq:= diff(y(x),x) = x*y(x);
> inits:=[[0,0],[1,1],[0,1/3]]:
> DEplot(deq, y(x), x=-2..2, y=-1..1);
> DEplot(deq, y(x), x=-2..2, y=-1..1, inits); [worksheet]
9.1: 9, 11.
9.1: 13;
9.2: 3-6 [visual inspection (fun)]
9.3: 1, 11, 33.
catch up on previous homework.
maple11 due this week.
note final exam room and time and date below.
11.R.50 use taylor formula to get first 4 nonzero terms;
use taylor formula to evaluate the first 4 nonzero terms of the series for cos(x) centered at x = a, where a = arccos(4/5) is the smaller angle of a 3-4-5 right triangle with cos(a)=4/5, sin(a)=3/5;
use taylor formula to evaluate the series for tan(x) up to the fourth power terms centered at x = 0.
April 29, last day of class. Final exam 1 week later Thursday 4:15pm Mendel G92. You must use either your graphing calculator or Maple for some parts of the exam. Tuesday 5pm MLRC problem session;
be aware of the archived final exam.