Class Diary for M142, Fall 2009, Jochen Denzler
Wed Aug 19:
Intro to antiderivatives. Hwk p 332, 1-13 odd (not collected)
Thu Aug 20:
application: free fall, Hwk: p332, 15,20,25,30,31,33,40,41,47,49. Area
begun; specifically area under a triangle and under a parabola.
Fri Aug 21: discussion of hwk. Getting distance from velocity is like the
area problem; since we know already that this is an antiderivative problem
also, we understand that the area problem must be closely connected to an
antiderivative problem.
Hwk: p 352, numbers 1-3, 11,20. Turn in on Wed. Also read Sec 5.1., in
particular familiarize yourself with the sigma notation by Monday
Mon Aug 24: The definite integral of continuous functions defined;
evaluation by antiderivative of the integrand. Hwk p365: 4-6,31,32 due
Thu
Wed Aug 26:
Hwk discussion. Basic properties of the integral.Hwk. p.365, 39-45, 50 due
Fri
Thu Aug 27:
Sec 5.3. Role of Mean value theorem.
Indefinite integrals; evaluation of integrals by means af antiderivatives.
Examples for algebraic simplification targeted for the purpose of integration.
Hwk: p 374, 1-28 due Mon
Fri Aug 28:
The fundamental theorem of calculus. Comments on the inconvenience of sec vs
1/cos, and alert not to integrate across infinities in examples like integral dx/x
Mon Aug 31:
Hwk questions. The chain rule reviewed, and integration by
substitution. Hwk: p392, 1-54, due Thu. (Well, you should be able to do them
all; do as much as you need to be really fit in applying the method. You may
team up to do it
Wed Sep 02:
Substitution and definite integrals. A few examples and questions.
Thu Sep 03:
Hwk discussion. Symmetry arguments. Quick review for exam.
Fri Sep 04:
EXAM 1
Mon Sep 07:
LABOR DAY
Wed Sep 09:
Exam discussion; integration by parts barely begun.
Thu Sep 10:
Integration by parts; key examples: x sin x; ln x; exp times trig. Hwk: read
the section on integration by parts
Fri Sep 11:
Integration by parts finished; various notations; integral of powers of sin x;
how to combine substitution and IBP. Hwk pg 398, 3,4,10,12,14,15,17,18,21,
22,23,25,26,33,35,37,39 due Wed
Mon Sep 14:
Discussion of Hwk questions. Trig substitutions to deal with the square root of
a quadratic under the integral.
Wed Sep 16:
Trig substitutions in full detail and larger context.
Thu Sep 17:
Partial Fraction Decomposition started: Step 1: long division if needed; step
2; factor denominator; step 3: form of PFD in the absence of repeated factors.
Refer to Sec 5.7 as well as appendix G of book. Also refer to
notes that I have posted on the main course page.
Fri Sep 18:
PFD. How to calculate coefficients: either by comparing coefficients or by
plugging in (aka cover-up); also plug-in of complex numbers to calculate
coefficients in the case of quadratic terms.
Mon Sep 21:
completing squares; finishing up integration of a PFD.
Wed Sep 23:
Examples for cover-up. How to handle repeated factors. Hwk: pg 404-405, #
10,11,13,14; 15,16 (calc coefficients also, even if the texbook says don't);
23,24,27,28 due Fri
Thu Sep 24:
Ramifications: (1) factoring x^4+1; (2) how to integrate partial fractions with
repeated quadratics.
Fri Sep 25:
hwk discussion; numerical integration overview: midpoint, trapezoidal, and
Simpson rules.
Mon Sep 28:
Case studies for Simpson, midpoint and trapezoidal rules. Hwk 3, 14-16 on pg
421 due Thu
Wed Sep 30:
Q & A; improper integrals motivated and introduced.
Thu Oct 01:
improper integrals: convergence vs divergence, and comparison
Fri Oct 02:
Review for exam; areas between curves in the plane (Sec 6.1)
Mon Oct 05:
EXAM 2
Wed Oct 07:
area defined by parametrized curve; volumes by slicing. Hwk: pg 446,
2,3,4,16,23 due Fri
Thu Oct 08:
volumes by shells; examples. Hwk: pg 459 , nymbers 39 (do it by shells and
by slices) and 42; due Mon
Fri Oct 09:
`pound cake and pudding malfunction': more on volumes by shells and slicing.
Includes a calculation of integral (exp(-x^2)dx) from -infinity to infinity
as a spinoff.
Mon Oct 12:
arclength of the graph of a function; exam 2 back
Wed Oct 14:
arclength of a curve given in parametric form; examples. Among the examples:
hanging chain; the cosh and sinh functions. Hwk: pg 466, 23-25. Also study
on pg 246 `hyperbolic functions', specifically items 1-4 and 8.
3 ways of evaluating the integral that describes the arclegth of a parabola
discussed.
Thu Oct 15:
FALL BREAK
Fri Oct 16:
FALL BREAK
Mon Oct 19:
questions; area of rotation surfaces (not in book); average of function; intro
to work as an integral
Wed Oct 21:
intro to assigned hwk problems: Hwk due on: pg 460, #52; pg 465 #14; pg 465
#6+15. And these two, which are not in the
book. Work required to send a satellite out of the reach of earth
gravitation.
Thu Oct 22:
Work; pumping out a tank; center of mass.
Fri Oct 23:
Poisseuille's law (Blood flow; pg 483) Hwk pg 480, #12 and 17a. A
5-minute motivation for series: We'll now move to section 8 of the book;
section 7 and more will be the contents of Math 231
Mon Oct 26:
Power series of sin and cos discovered by repeated integration of cos <= 1
(this is not in the book). What is a sequence, and what is a series: examples,
and how sequences relate to series.
Wed Oct 28:
convergence of sequences.
Thu Oct 29:
convergense of series
Fri Oct 30:
Review for exam
Mon Nov 02:
EXAM 3
Wed Nov 04:
geometric series and decimal expansions; harmonic series; telescoping series.
Thu Nov 05:
Comparison principles begun; examples; Exam 3 back
Fri Nov 06:
Questions about exam problems answered
Mon Nov 09:
Limit comparison test; integral comparison test; explanations and examples.
Hwk: pg 585, #5, 6-10, 19-21, 24-26; 30 (compare Example 6+7 for this).
Due Thu
--- Tue Nov 10: WP/WF drop deadine ---
Wed Nov 11:
alternating series test. Absolute convergence.
Thu Nov 12:
review of power series for sin x and cos x, this time with convergence proof;
used for calculating sin 1 (easy) and sin 4 (lengthier); geometric series as a
power series; multiplication of power series; differentiation of power series
is very easy.
Fri Nov 13:
power series of sqrt(1-x), calculated by two methods
Mon Nov 16:
Taylor series of an arbitrary function that is arbitrarily often
differentiable; it `usually' represents the function from which it was
obtained.
Wed Nov 18:
Practical theory collected:
Power series centerad at any point, and their radius of convergence. They
represent a differentiable function within their interval (or disc) of
convergence. We may plug in complex numbers. Derivatives and antiderivatives of
power series are always legitimate and don't change the radius of convergence.
Thu Nov 19:
Practical theory collected:
Radius of convergence `at most that big' because otherwise you'd encounter a
bad behavior inside the interval (or disk if you allow complex x) of
convergence
Adding and multiplying power series; radius of convergence at least as large as
the smaller of the r.o.c. of the summands / factors.
Fri Nov 20:
Practical theory collected:
Long division of convergent power series. Radius of convergence not
predictable, but at least not 0.
power series used for calculating limits instead of L'Hopital.
Mon Nov 23:
(writing this ahead of time) Example: Arclength of an ellipse; Euler's formula;
SAIS
Wed Nov 25:
EXAM 4
Thu Nov 26:
THANKSGIVING
Fri Nov 27:
THANKSGIVING BREAK
Mon Nov 30:
Wed Dec 02: STUDY DAY
Wed Dec 09: FINAL EXAM 10:15-12:15
(scheduled by university policy)
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