Class Diary for M148, Spring 2015, Jochen Denzler
Wed Jan 07:
Syllabus and formalities; Antiderivatives (definition, examples, notation)
Fri Jan 09:
odometer data from speedometer; antiderivatives and area.
Decomposing areas into thin rectangles.
Hwk 1-2 due Tuesday
Mon Jan 12:
another example: area under x^3. Proof of two power sum formulas. Summation
notation. Partition of an interval defined.
Tue Jan 13:
Partitions of an interval, and their norm (aka meshsize); tagged partitions;
Riemann sums and the Riemann integral defined and existens of Riemann integral
for continuous and piecewise continuous fcts stated.
Hwk 3,4,5. 3 and 4 are due Friday, 5 will be
collected later, with the next assignment
Wed Jan 14:
easy properties of the Riemann integral; fundamental theorem (both directions);
practical calculation with the fundamental theorem
Fri Jan 16:
Proof that integrals can be calc'd by antiderivatives. A few examples,
including one where the interval needs to be split, with absolute value under
the integral.
Mon Jan 19:
MLK DAY
Tue Jan 20:
derivatives of antiderivatives, in combination with the chain rule;
overview over integration methods; substitution method started.
Hwk 6-9
assigned and due by Friday, together with last week's #5
Wed Jan 21:
A list of fcts whose derivatives one needs to know (so one recognizes them when
antiderivatives are needed).
Substitution method for definite and indefinite integrals
Fri Jan 23: Discussion of pending hwk: questions answered. Example of
substitution where f'(x)dx is not manifestly visible and one may want to
employ the inverse function: e.g. integral (e^x-1)/(e^x+1) dx.
Hwk 6-9 extended till Monday. Hwk 10-15
due Wednesday
Mon Jan 26:
A few examples of `sneaky' substitutions: integral sqrt(1+x^2) dx with x = sinh
u; integral dx/(2 + cos x) with u = tan (x/2). [For now I would tell you the
sneaky substitution, if you need one. Later, we'll learn the wisdom to see
them.]
--- Integration by parts started (that amounts to jumping ahead in Rogawski's
book; we'll return to other material later). integral x e^x dx and integral ln
x dx
Tue Jan 27:
Integration by parts
Wed Jan 28:
Combining sub and IBP: The example integral sin(ln x) dx. Similarly
integral sin (e^x) dx leads to integral sin u / u du, which is a dead end for
algebra. Honors specific material: Preprocessing integral sin u/u du by
repeated IBP to get something more suitable for quick numerical evaluation.
Hwk 16-22 due Monday and
Hwk 23 due Friday next week
Fri Jan 30:
Q&A. Integrating powers of sin and cos.
Mon Feb 02:
Q&A. Quick review of exam coverage. Organizing trig IDs.
Integrating negative integer powers of sin and cos.
Tue Feb 03:
Trigonometric and hyperbolic substitutions: purpose and method. Preprocessing
quadratic to eliminate linear term by completing the square; selecting the
type, and the matching trig or hyp substitution.
Wed Feb 04:
EXAM 1
Fri Feb 06:
Discussion of exam. The x=tan u/2 sub to convert rational expressions in sin x
and cos x into rational expressions of u. Hwk 23 extended till Monday.
Hwk
24-27 due Wed. Hwk 28-31 due Fri
Mon Feb 09:
Q& A. Substitutions to treat rational expressions in sin u and cos u:
when are there simpler alternatives to the tan(u/2) substitution?
Tue Feb 10:
Partial Fraction Decomposition (simple linear factors in the denominator),
in discovery method inspired style. Ramifications and full theory will come
yet.
Wed Feb 11:
Q&A. Mainly about trig/hyp substitutions and pending
homework. Extension for today's hwk (up to 27) granted as needed. Will
collect Friday. Try to still get the 28-31 due Friday; but I will allow
extension of that one till Monday as needed. Holding on with new assignments
for the moment...
Fri Feb 13:
Q&A on hwk. Hwk 28-30 extended till Monday; 31 till Tuesday.
partial fraction decomposition; quadratic factors.
Mon Feb 16:
Fundamental theorem of algebra; Example: factorization of x^4+1.
General pattern of PFD (bottom-heavy) with linear and irreducible quadradic
factors.
Tue Feb 17:
class cancelled for inclement weather closure of UT
Wed Feb 18:
PFD with linear factors (possibly complex coefficients): How to calculate
recursively *in principle*. How to shortcut the calculation efficiently;
cover-up method. Plugging in values.
Fri Feb 20:
combining cc pairs into irreducible
quadratics. An example of a PFD involving irreducible quadratics.
Hwk 32-36 due Tue
Mon Feb 23:
brief note how to integrate (eg) (2x+1)/(x^2+4x+5). ---
area between curves; volumes by Cavalieri's principle. Ball; torus.
Tue Feb 24:
class cancelled for inclement weather closure of UT. Will collect hwk
tomorrow.
Wed Feb 25:
Examples; Intro to the homework. Rotation bodies are a special case
Hwk 37-40 due Monday
Fri Feb 27:
Volumes of rotation bodies by spherical shells.
Mon Mar 02: Arclength of curves in cartesian coordinates.
Hwk 41-46 due Friday
Tue Mar 03:
Surface area of rotation surfaces; a few examples.
Isoperimetric inequality for shapes in the plane and for bodies in 3D space;
they are neat to know and can be used as simple
consistency checks for calculations of volumes and areas.
Wed Mar 04:
Area and arclength in polar coordinates
Fri Mar 06:
Outline of new hwk, specifically 49 b. Hwk 47-49 due Wed
Mon Mar 09:
Hwk Pblm 35 explained. Overview of pending exam material, part 1
Tue Mar 10:
Overview of pending exam material, part 2. Density. Poisseuille flow.
Wed Mar 11:
Applications to Work. Overview over material covered on exam 2.
Fri Mar 13:
Overview over material covered on exam 2 cont'd.
Error estimates for Trapezoidal and Midpoint rules given (proved for
midpoint). Simpson rule explained and error estimate given without proof.
Mon Mar 16:
SPRING BREAK
Tue Mar 17:
SPRING BREAK
Wed Mar 18:
SPRING BREAK
Fri Mar 20:
SPRING BREAK
Mon Mar 23:
Q\&A for exam.
Tue Mar 24:
EXAM 2 (material covered only before March 10; exam scheduled late to ease the
pre-spring break crunch)
Wed Mar 25:
Birds-eye overview over power series. Hwk 50-52
due Monday
Fri Mar 27:
Improper integrals: convergence vs divergence. Comparison test (also in view
of series)
Mon Mar 30:
Sequences defined. Limit & basic theorems about them.
Tue Mar 31:
Series defined. Sequence of partial sums. Convergence or divergence. Geometric
series and its limit. Harmonic series divergent.
Wed Apr 01:
p-series. Comparison tests; including integral comparison
Fri Apr 03:
GOOD FRIDAY
Mon Apr 06:
Outline of flow diagram for testing convergence
Tue Apr 07:
Ratio test and root test. Examples. Hwk 53-58 due Fri
Wed Apr 08:
Series with terms positive and negative; alternating series test explained and
proved. Flow diagram completed.
Fri Apr 10:
Power series; adding and subtracting them; term by term differentiation and
integration.
Hwk 59-64 due Wed
Mon Apr 13:
Hints for hwk 59. Radius of convergence. Series for e^x identified.
Tue Apr 14:
Plugging in points on the boundary of the convergence interval.
Multiplication of power series and its effect on the r.o.c.
Long division of power series. Composition briefly mentioned.
Wed Apr 15:
Euler's formula and conversion between trigs and exponentials; with an example
for integrating e^(3x) cos(2x). --- Taylor series for sqrt(1+x)
Fri Apr 17:
Mon Apr 20:
Tue Apr 21:
EXAM 3
Wed Apr 22:
Fri Apr 24:
Fri Apr 27: STUDY DAY
FriMay 01: FINAL EXAM 08:00 - 10:00
(scheduled by university policy with link to calendar)
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