Mon Jan 12: Intro. How to find the area under a parabola.
Tue Jan 13: Area under parabola; sums of squares; Def of Riemann sum started: Hwk pblm 1 due Fri
Wed Jan 14: Def' of Riemann integral, with illustrations. Examples how integrable and non-integrable functions may look. Hwk pblm 2 (only 1st line) due Fri. Think a bit of the 2nd line of pblm 2; we'll discuss this informally in class
Fri Jan 16: Guided discovery of set additivity, linearity and order properties of the integral. Hwk pblm 3 and 4 due for hand-in and classroom presentation on Tue. Also think about pblm 5a-d and be prepared to ask questions if you don't understand something there.
Mon Jan 19: MLK DAY
Tue Jan 20: Mainly discussed hwk. Results for Hwk 3 and 4
Wed Jan 21: Fundamental theorem of calculus; Hwk pblm 5a-d due on Monday
Fri Jan 23: indefinite integral, antiderivatives; evaluation of some simple integrals. More Hwk available now at link above: 6,7 due Mon, 8 due Tue; more to come; look at the other ones already in case you want to work ahead or have questions (5e will come only after we have all come up with the same numerics from 5d)
Mon Jan 26: chain rule and substitution (indefinite & definite integrals). Problems 9-12 due Friday
Tue Jan 27: training in integrals (substitution): sheet of integrals
Wed Jan 28: Discussion of Hwk #5cd; inverse functions; barely started repetition (or new?) inverse trigs (arcsin, arccos, arctan) and their derivatives.
Fri Jan 30: Inverse trigs (arcsin, arccos, arctan) and their derivatives. Hwk 13,14 due Monday; Hwk 15 due Friday
Mon Feb 02: Some discussion on the Simpson rule hwk; The natural logarithm.
Tue Feb 03: Discussion of Hwk pblm 13, and of 8.
Wed Feb 04: Exponential function, and hyperbolic functions, and their derivatives and antiderivatives Hwk 16,18 due Monday; Hwk 19 due Friday next week
Fri Feb 06: Equivalence of two def's of e. Exponential function as a model for growth and radioactive decay.
Mon Feb 09: Certain volumes can be eval'd as integrals. Arclengths can be eval'd as integrals. Areas of rotational surfaces can be evaluated as integrals Hwk for presentation tomorrow: calculate the volume of a torus, as sketched in class today.
Tue Feb 10: discussed shell method for volumes; volume of torus Hwk 22-24 for discussion on Wed
---> I am posting sample solutions for the integration hwk here; links may be dead yet; they'll become active when we have discussed the problem
Wed Feb 11: Hwk 19,20,21, 25 (surface area only) due Friday
Fri Feb 13: Discussion and presentation of integration, volume, surface, arclength problems.
Mon Feb 16: Discussion and presentation of integration, volume, surface, arclength problems.
Tue Feb 17: Discussion and presentation of integration, volume, surface, arclength problems.
Wed Feb 18: First Exam Thursday Feb 19, 7pm
Fri Feb 20: Class cancelled since previous day's exam was out of class time: I'll however be around for questions at this time.
Mon Feb 23: Discussion of exam; Hwk 30, 29 for presentation tomorrow --- Written hwk: 32a by Wed; 32b1,b2 by Fri; 33 by Mon
Tue Feb 24: Pblm 30: hanging chain; Newton's method revisited
Wed Feb 25: Pblm 32a; Integration by parts.
Fri Feb 27: Pblm 32b; more examples on integration by parts Use sheet of training integrals for integration by parts. Written Hwk for Monday: 34,35. By Wednesday 40,41,42
Mon Mar 01: Discussion of Pblm 39
Tue Mar 02: Discussion of Pblm 39, and for 36: how to get formulas for area and length in polar coordinates Written Hwk for Friday: 36
Wed Mar 03: Partial Fraction Decomposition; Basics
Fri Mar 05: Partial Fraction Decomposition; Cover-up method
Mon Mar 08: SPRING BREAK
Tue Mar 09: SPRING BREAK
Wed Mar 10: SPRING BREAK
Fri Mar 12: SPRING BREAK
Mon Mar 15: Partial Fraction Decomposition; using complex numbers in the cover-up method for quadratic terms; philosophy of PFD: looking exactly at the points where the rational function goes to infinity gives the crucial information most effectively. More hints for a hidden connection between exponentials and trigs.
Tue Mar 16: Training PFD
Wed Mar 17: Training PFD; integral dx/x may be ln(-x); separating fractions in quadratic terms
Fri Mar 19: The power series of sin and cos and exp, obtained through integration. (Convergence without proof so far). Euler's formula. Complex numbers come out of the electric AC outlet.
Mon Mar 22: PFD for integral from Hwk#33; some remarks on trigs and hyps on the way Hwk 43-46 due Friday
Tue Mar 23: Factoring (x^4+1), and finding the antiderivative of 1/(x^4+1)
Wed Mar 24: Overview over integration methods; in particular: which type of substitutions for which type of integrals; trig' and hyp' substitutions
---> Solutions to #36-46 posted now; (some to become readable only later)
Fri Mar 26: Integrals: Training problems
Mon Mar 29: Integrals: Training problems
Tue Mar 30: Integrals: Training problems; included: sqrt(linear/linear) is as good or bad as sqrt(quadratic)
Wed Mar 31: Integrals: Training problems; included: quick evaluation of polynomials; the complex trick for integrating exponentials times trigs
--> As requested, solutions for training problems now posted here They should be ok, but I haven't checked'em for typos yet, so don't treat'em as divinely inspired ;-)
Thu Apr 01: EXAM APRIL 1, 7PM. -- Friday class canceled in compensation for the off-day exam.
Mon Apr 05: Exam back, and discussion
Tue Apr 06: Series: convergence and basic examples; comparison test, Leibniz criterion
Wed Apr 07: Series: conditional vs absolute convergence. Termwise addition of series, and multiplication with numbers; rearrangement of absolutely convergent sereis.
Fri Apr 09: GOOD FRIDAY
Mon Apr 12: Rearrangement of conditionally convergent series can give any value, or divergence. The geometric series in detail.
Tue Apr 13: Power series; radius (and disc) of convergence; smoothness inside disc of convergence; basic idea: comparison with geometric series. non-numeric part of 47, and 48, tomorrow; all of 47, and 49, by Friday;
Wed Apr 14: Addition, multiplication, and division of power series, and what we know about the radius of convergence of the resulting series. Hwk: 47-51 by Friday
Fri Apr 16: Understanding the radius of convergence in terms of the represented function, or in terms of the coefficients.
Mon Apr 19: Taylor series of a function; it may or may not represent the function.
Tue Apr 20: Discussion of pblms 54-56; including intro to the binomial series
Wed Apr 21: Discussion of the series for 1/(1-x-x^2) from all aspects
Fri Apr 23: THIRD EXAM THURSDAY APR 22, 7PM (Instead of Friday class)
Mon Apr 26:
Tue Apr 27:
Wed Apr 28:
Fri Apr 30: STUDY PERIOD
Mon May 03: 12:30-02:30 FINAL EXAM