Download m210/cs271 OralTuesdays

210 Oral Tuesdays
Math Topic Lists
You will need to select two topics for speeches this semester. The first topic
will be used for your first speech. It will consist of “teaching” a topic in Calculus I or II
that you are already familiar with. The second topic
will be more in-depth and be used for your second and third speeches. You will
most likely need to conduct some research to learn about your second topic choice.
Here is a list of possible topics math and suggested resources to find more You are also
welcome to suggest a topic not on this list but must then obtain approval from the
instructor.
I The following are mini-topics appropriate for your first speech. These will be
assigned. If you have not seen your topic before (depends on your professor for Calc II)
you may select again.
• Discuss the Intermediate Value Theorem and give an application.
.
• Give a “proof” (possibly intuitive) that the derivative of sin(x) is cos(x).
• Explain implicit differentiation and use it to find derivatives of inverse trig functions.
• Explain and apply the Mean Value Theorem
• Explain and apply Newton’s Method
.
• Show how the algebraic properties of ln( x) follow from the definition of ln( x) as an
integral. (Section 5.6) Note: mention why the integral in the definition exists!
• Explain using Riemann slices how to find arc length and surface areas.
• Explain how one can use integration to find the force due to hydrostatic pressure on
some 2d shape in a fluid.
• Explain what a continuous probability distribution is and how to find its mean, give an
example (exponential distribution is a good choice), and apply it (for example, to
waiting times).
• Explain how one can define work with non-constant forces and do examples.
• Sow how the shell and washer methods may be used to find volume.
.
• Explain and apply Euler’s method.
• Discuss Logistic growth vs exponential or bounded growth. Note: use direction fields
AND show how to find analytic solutions. Focus most of the time on logistic growth.
• Show how mixing problems are modeled by linear differential equations.
• Use sequences to show how population growth of rabbits is modeled by Fibonacci
numbers. (# 65 pg 712).
• Explain and prove the convergence properties of a Geometric Series. Find an
application.
• Show various ways to calculate power series from known power series and use it to find
series representations of transcendental numbers (  , e, ln(2) , etc..). Perhaps estimate
these numbers and find the error of your estimation.
• Quickly derive the binomial series and (1) connect it to the Binomial formula
(2) apply it to special relativity (Example 3 pg 780)
• Use power series to easily calculate and approximate integrals and discuss the error of
your approximation.
• Explain in detail what eigenvalues are and how they relate to determinants. Do an
application.
• Explain when and how matrix inverses can be calculated and used to solve linear
equations
II Resources/suggested problems for 2nd/3rd talk.
• Choose a problem marked Medium or hard on the Math Fun
Facts site: http://www.math.hmc.edu/funfacts/. This requires Approval. Several
problems listed below are there.
• Choose a set of problems/ideas from the various sections of Burger and Starbird’s “The
Heart of Mathematics” Ch 2-7. Choose something challenging here. Easy topics will
be rejected. There chapters are put into their separate categories in the lust of topics
below.
Here are some specific suggestions. Many of these are in the resources above.
Linear Algebra/Abstract Algebra
• Research how linear algebra is used in fitting curves to data.
• Research how covariance matrices and eigenvalues can be used in image
processing OR statistics.
• Research how eigenvectors can be used in constrained optimization problems.
• Which numbers can be “constructed” by a straight edge and compass?
• Research Symmetry in wall patterns and group theory.
• Introduce us to coding theory.
• Introduce us to cryptology (see Dr. Feil).
Number Theory/Discrete Math
• Introduce us to Sterling numbers.
• Research Catalan numbers
• Research Fibonacci numbers
• Research game theory. Show how game theory can compute values of games with no
fixed point strategy. (see Dr. Kretchmar).
• Research Conway’s definition of surreal numbers and their applications to games. (see
Dr. Kretchmar)
• Discuss the mathematics of the card game set.
• Discuss the mathematics of the permutation game.
• Research how to play Hackenbush.
• Research how to play the dots-and-boxes game
• Research utility theory. Show how utility theory can rationalize someone “wasting”
money by buying lottery tickets.
• Find a solution for the infamous 3-way duel.
• Introduce us to the Firing Squad Synchronization Problem and give a solution.
• Introduce us to Ramsey Theory and computing van der Waerden numbers.
• Research how historically seats for the US Congress have been apportioned to different
states (what algorithms are used to assign seats based on population and which are fair).
• Find out how to compute the relative voting power in a weighted voting
system (where each voter has a set number of votes to use).
• Research the various different voting methods and which ones are fair.
• Ch 2 of Burger/Starbird
• Choose one of Martin Gardner’s puzzles (there are hundreds and hundreds). (see Dr.
Kretchmar)
• Choose any problem from the Dueling Idiots text (pending approval). (see Dr.
Kretchmar)
Calculus/Topology
• Research the properties of Cantor’s set (for example, it is perfect, closed, uncountable,
it corresponds 1-1 with the interval [0,1] yet has total length 0!).
• Research the Banach-Tarski paradox and explain sets that have no well defined size
(“measure”).
• Research the Fundamental theorem of Algebra and show how complex numbers can be
used to prove it.
• Show how contour integrals can be used to instantly compute hard integrals using
imaginary numbers.
1 x is rational
• Explain why the function f  
on [0,1] does not have a well defined
0 x is irrational
Riemann integral and explain why measure theory is needed.
• Applications of Fourier Series: Exhibit a function on [0,1] that is continuous but not
differentiable at any point.
• Applications of Fourier Series: Isoperimetric problem. Which shapes enclose the
biggest area? Answer: A circle
• Applications of Fourier Series: Cool numerical summation formulas.
• Research alternate notions of how divergent series can be said to “converge”
• Research the Stone-Weierstrauss Theorem (How polynomials approximate functions).
• Research different sizes of infinity. (Ch 3 of Burger/Starbird)
• Research Fractals, Chaos, fractional dimension (Ch 6 of Burger/Starbird)
• Research the Mandelbrot set.
• Prove that  and e are irrational. Exhibit numbers that for which their irrationality is
unknown.
Geometry/Topology
• Platonic solids
• Research non-Euclidean geometry and its uses.
• Research Klein bottles
• Ch 4 of Burger/Starbird
• Research Knot theory. (see Dr. Ludwig and Ch 5 of Burger/Starbird )
Logic
• Research Gödel’s incompleteness Theorem
• Discuss some of the truth-telling puzzles in Smullyan’s book Satan, Cantor, and
Infinity
Probability/Statistics
• Research and explain Heisenberg’s inequality and what it means in Physics
• Research the Gamma function with its applications to probability and cool
mathematical properties.
• Discuss the solution for the Monty Hall problem.
• Discuss the problem of asking an embarrassing question on a survey and obtaining
accurate results.
• Research the Central limit Theorem. Show how this if the Fundamental Theorem of
Statistics.
• Research Monte Carlo simulation.
• Research Random Walks and their strange outcomes.
• Research Bayesian decision theory.
• Ch 7 of Burger/Starbird
• Problem from the Fifty Challenging Problems in Probability text. Dr
Kretchmar has a copy. Requires approval.