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Extreme Calculus. Paul Zorn Saint Olaf College and MAA. MAA. Math. Assoc. of America. founded 1915 about 25,000 members worldwide focused on college-level math publications meetings professional development, outreach advocacy for mathematics
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Extreme Calculus Paul Zorn Saint Olaf College and MAA
MAA KC Tech Expo
Math. Assoc. of America • founded 1915 • about 25,000 members worldwide • focused on college-level math • publications • meetings • professional development, outreach • advocacy for mathematics • last 3 presidents from Minnesota … this has to stop KC Tech Expo
MAA headquarters, Washington DC • 1529 18th St NW • built 1903 • residence of Charles Evans Hughes, later Chief Justice of US Supreme Court, Secretary of State in Harding administration, unsuccessful Presidential candidate KC Tech Expo
Founded in Missouri Benjamin Franklin Finkel, Kidder Institute, 1894 KC Tech Expo
Extreme story problems Problem 4, AMM Vol 1. No.1, 1894 KC Tech Expo
Extreme story problems 2 AI = length of one board; GO = “radius” KC Tech Expo
Extreme story problems 3 Answer: area = 196,020 acres (approx) … more than 300 square miles KC Tech Expo
Extreme extremization 4 Problem: Maximize ab subject to a+b=constant Maximize abc subject to a+b+c=constant Answer: a=b or a=b=c or … . Reason: AM-GM inequality: KC Tech Expo
Extreme extremization 5 Problem: Maximize xyz subject to xy + 2xz + 2yz = constant Question: Why would we do this? Guesses? KC Tech Expo
Extreme extremization 5’ Problem: Maximize xyz subject to xy + 2xz + 2yz = constant Answer: xy*2xz*2yz = 4xxyyzz; it’s largest when xy = 2xz = 2yz , so x=y=2z KC Tech Expo
Extreme extremization 5’’ Problem: Maximize subject to Why? Guesses? KC Tech Expo
Extreme extremization 5’’’ Answer: Maximize subject to (instead) Thus KC Tech Expo
Extreme extremization 6 Problem: Find “pie-shaped” region with largest area and total fence-length 100. r * theta r theta r Length = 2r+r*theta = 100 Area= r^2 * theta / 2 KC Tech Expo
Extreme numerics … the harmonic series KC Tech Expo
Extreme blends How does f(x) = x + sin(x) behave? What about f(x) = x + a sin(x) ? f(x) = x + sin(a x) ? KC Tech Expo
Extreme Newton February 1998 June 14 KC Tech Expo
Cubic polynomials … anything new to say? All cubics through these points have same integral as the quadratic June 14 KC Tech Expo
Newton’s method on cubics … some surprises June 14 KC Tech Expo Extreme Calculus
Extreme e June 14 20 KC Tech Expo Extreme Calculus
Definite integrals 6/10/2014 KC Tech Expo Extreme Calculus
How big are these numbers? 6/10/2014 KC Tech Expo Extreme Calculus
Approximating e 6/10/2014 KC Tech Expo Extreme Calculus
Might e be rational? 6/10/2014 KC Tech Expo
On a Rolle -- Gauss/Lucas June 14 KC Tech Expo
Juan Sanchez de Cotan 1603 KC Tech Expo