181 Topic List

Math 181 Topic List

Fall 2007

Notes:

Home

Textbook Order of Topics	In Class Topics	Textbook Material	Class Day	Homework Problems
1.1-1.6 and 3.11: Functions	Hurricanes, Global Warming -- Getting Ready for the Semester	Handout	1-Sep 4
2.1-2.5: Limits	Logistics and Overview of Goals -- WWWWW	Handout
	Sequences	8.1		4,10,16,85,87
2.6: Continuity	Estimating with finite sums:	5.1		a(n)=1+3n
2.7 and 3.1-3.10: Differentiation	----- Functions	1.1-1.6		1.1: 10,24,60 1.2: 6,10,18,32 1.3: 6,22,36,56 1.4: 10,18,22 1.5:12,16,26,38,44,58 1.6: --
4.1: Extreme Value Theorem	----- Units of measurement	Discussion
4.2: Mean Value Theorem	----- Sigma notation	5.2		a(n)=2+4n
2.6: Intermediate Value Theorem	Exact Values from "infinite sums"	5.2,8.1,8.2		5.2: -- 8.1: 24,34,42,48,82 8.2: 3,10,28,36
4.6: L'Hospital's Rule and Indeterminate Forms	----- Sequences	8.1
4.3-4.5: Applications of Derivatives	----- Limits of Sequences: Convergence, Divergence	8.1
4.7: Newton's Method	----- Sequence of Partial Sums	8.2
	----- Can we pre-determine convergence?	5.3, 8.2-8.6
4.8: Antiderivatives	Riemann Sums	5.3
5.1: Estimating with finite sums	Definite Integral	5.3
5.2: Sigma Notation	Exists for Continuous Functions	5.3
5.2: Limits of Finite Sums	----- Continuity	2.1-2.6
5.3: Definite Integral and Riemann Sums	Revisit Hurricanes and Global Warming	Handout 1
54: Fundamental Theorem	Why is there a "delta x" in Riemann Sums?	5.4
5.5: Indefinite Integrals	----- F(x) function for net area has f(x) for a rate of change (MVT)	5.4 FTC-1,4.2
5.5,5.6: Substitution	Fundamental Theorem of Calculus -Part 1	5.4
5.6: Area Between Curves	----- Accumulating rates across an interval reverses differentiation	5.4
5.7: Logarithm as an Integral	Fundamental Theorem of Calculus - Part 2	5.4
	----- Antiderivatives compute definite integrals	5.4
6.1: Volumes by Cross Section	What functions have antiderivatives?	7.1-7.4
6.2: Volumes by Cylindrical Shells	We need more functions (with antiderivatives): ln, gamma, power series, tables.	5.7, 8.7-8.10
6.3: Lengths of Plane Curves	We need better ways to find antiderivatives for given functions	5.5,5.6,7.1-7.4
6.4: Areas of Surfaces of Revolution
6.5: Exponential Change	Note to Bryan: Applications and then the rest of Ch7? or the reverse?
6.5: Separable Differential Equations	List of Applications: those in book (volume, length exp. change, separable ODE), finance, economics, CD Roms, Laplace Transforms, volume of n-sphere, why derivative of volume of n-sphere is volume of (n-1)- sphere [need to think of volume in a different way], arterial fluid flow, Newton's Law for temperature, indicate how to set up DE for finding total blood volume by heating and injecting blood and measuring temperature decrease, hurricane energy, class supplies other items?
6.8: Centers of Mass

7.1: Integration by Parts
7.2: Trigonometric Integrals
7.3: Trigonometric Substitutions
7.4: Partial Fractions for Integrating any Rational Function
7.5: Tables and Computer Algebra Systems (CAS)
7.6: Numerical Integration: Trapezoid Rule
7.6: Numerical Integration: Simpson's Rule
7.6: Numerical Integration: Error Bounds
7.7: Improper Integrals
7.7: Tests for Convergence of Improper Integrals

8.1: Sequences
8.2: Infinite Series Basics
8.2: Nth Term Test
8.2: Geometric Series
8.2: Telescoping Series
8.3: Integral Test
8.3: ---- Error in Approximations
8.4: Basic Comparison Test
8.4: Limit Comparison Test
8.5: Ratio Test
8.5: Root Test
8.6: Alternating Series
8.6: ----- Error in Approximations
8.6: Absolute Value Series
8.6: Alternating Series Test
8.6: ------ Conditional Convergence
8.6: ------ Absolute Convergence

8.7: Power Series
8.8: Taylor and Maclaurin Series
8.9: Convergence of Taylor Series to generating function
8.10: Binomial Series

9.1: Polar Coordinates
9.2: Graphing in Polar Coordinates
9.3: Areas and Lengths in Polar Coordinates

List of Possible Projects and Topics for Discussion

Dark Matter
- http://brahms.phy.vanderbilt.edu/a103/labs/web_darkmatter_makefig2.shtml
- http://arxiv.org/pdf/astro-ph/0403048: very high level physics but with lots of good integration use.
- http://www.vttoth.com/darkmatter.htm
- http://www.journals.uchicago.edu/ApJ/journal/issues/ApJ/v509n2/38086/sc2.html?erFrom=8694404371479934142Guest
- http://arxiv.org/pdf/astro-ph/0506009
- http://www.phys.ufl.edu/~tanner/PDFS/Hagmann00dm.pdf
- Integrated Sachs-Wolfe effect for dark energy: http://astro.uchicago.edu/~laroque/ISW.html
Might be useful for Dark Matter: http://www.sciencedirect.com/science?_ob=ArticleURL&_udi=B6TJM-4JB9GPN-D&_user=10&_coverDate=06%2F15%2F2006&_rdoc=1&_fmt=&_orig=search&_sort=d&view=c&_acct=C000050221&_version=1&_urlVersion=0&_userid=10&md5=16640ea9364bb51151670aee7b6bd88c
Analemma Details
- http://www.analemma.com/Pages/framesPage.html
- http://members.aol.com/jwholtz/analemma/analemma.htm
- Actual photos of analemma: http://www.analemma.de/english/analem.html
Mormon Tabernacle Choir and properties of ellipses
Monte Carlo Appriximations:
- http://math.fullerton.edu/mathews/n2003/MonteCarloMod.html
- http://einstein.drexel.edu/courses/PHYS405/Monte_Carlo/index.html
- http://mathworld.wolfram.com/MonteCarloIntegration.html
Parabolic mirrors and spherical abberation: Uses Taylor Series, conic sections, properties of parabolas.
Locating sound of keyboard click using properties of hyperbolas.
Accumulating Rates: Distance by bicycle
Volume of n sphere: http://mathpages.com/home/kmath163.htm
Gamma Function: Needs convergence of integrals (series?)
Integration Techniques: CDRom disks and Fourier integrals -- maybe Fourier series too.
Partial Fractions:
- Logistic Equation
- Laplace transforms: http://tutorial.math.lamar.edu/AllBrowsers/3401/IVPWithLaplace.asp
- Integration via irreducible denominators
- Use for odd powers of secant.
- Summation of simple series
- Power Series for any rational function
- linear differential systems like resonant circuits and feedback-control systems
- As an application of L'Hopital's Rule http://www.maa.org/editorial/euler/How%20Euler%20Did%20It%2044%20partial%20fractions.pdf
Series:
- Spherical aberration and parabolic mikes, mirrors and telescopes
  - Important for telescopes, cameras, glasses, laser eye surgery -- anything involving lenses.
  - http://jpsalaser.com/laser_processing52.html
  - http://www.math.gatech.edu/~xchen/teach/zp1.html
  - http://www.mellesgriot.com/products/optics/fo_3_2_1.htm
- Power Series for any rational function
- Conway's Scouts in the desert.

Fundamental Concepts for 181

Accumulation: Use sequences, series to motivate accumulation. Show the only easy sums, extend to limit structure. Also Money, Population: Linear, exponential, logistic equation, decay, temperature, money stuff from business calculus.
Error Bounds
Fundamental Theorem
Sequences
Series
Applications
Methods of Integration
- Partial Fractions: Laplace Transform Inverses, Derive Logistic function
- Point is to reduce to denominators that are IRREDUCIBLE polynomials.
- Trigonometric: Fourier Transforms
Taylor Series
Power Series

Logistical Thoughts

Assign 15 homework problems per section.
- Students do any 10: First 5 worth 1 pt each others worth 2 pts each: This means "easiest 10" gives 75%. Homework is turned in on 3rd succeeding class day. Grades 0, 50, 100% per problem.
- When grading, mark things that are wrong and make students come in to work on how to fix. (Courtesy of Sue Owen)
Have 3-6 projects. Any project completely replaces homework for one or two sections.
Have a subset of exam questions come directly from homework. This should help to motivate students to do lots of problems.
Build entire lecture sequence out of what we need to know to answer some project question.
Example: Use Fourier integrals to compute CD signal: this requires trigonometric integration, sequences, series, linearity, (can use approximations to extract coefficients with accuracy levels). Tie in to Riemann sums, orthogonality (maybe motivate using Riemann sums to show perpendicularity condition)
"Scouts in the desert" which was solved by Conway.: http://www.cut-the-knot.org/proofs/checker.shtml
Project for modeling how long it takes boiling tea water to reach right temperature. Obviously fits in ODE section. What else can I use it for?
- After boiling, how does temperature decrease? Surface area of pot, thermal conductivity of metal,
Use comment cards like Spivey: what is most important topic and what was hardest to understand?
Possible Pedagogical techniques:
- Get to know students very well very quickly and force them to realize they are known quantities
- Use Blocks of 20 minutes or so in lower-division courses to deal with each of separate goals. Revisit those goals frequently
- Use Blackboard discussion for asynchronous chat
- ** Force questions on the Project to occur on Fridays. **
- ** Introduce variations on the Moore Method **
- ** Two minute verbal presentations on directed reading **
- ** 5 minutes of directed writing on a given topic for the day **

Math 181 Topic List

Fall 2007

Textbook Order of Topics

In Class Topics

Textbook Material

Class Day

Homework Problems

List of Possible Projects and Topics for Discussion

Fundamental Concepts for 181

Logistical Thoughts