Freshman Seminar Calc
Introduction to Mathematical
70-122 Software Calculus II Accounting
Led by Nam Ho
January 16, 2015
Plan for Today
Outline of what recitations will be
Summary of key ideas f
21-235: Sample problem write-up 1
August 23, 2013
Problem (Pugh p42 #12(c): If x = A|B is a cut in Q, show that
x = l.u.b.cfw_r : r A.
Solution: Let x = A|B be a cut in Q, and let S = cfw_r : r A. The proof
involves two substeps. We must show x
Sample proof that y > 0 > 0 u |u y| < |u3 y 3 | < :
Let y R be given and let > 0 be arbitrary. Set M = |y| + 1. Set
= min(1, 4M 2 ). Let u R and assume |u y| < . Then |u| < |y| + < M
|u3 y 3 | = |(u y)(u2 + uy + y 2 )| |u y| 3M 2 < 3M 2 < ,
Theorem (Stirlings formula): As n ,
Proof (modied from R. Michael, Amer. Math. Month. vol. 115 (2008) pp. 844-5):
Step 1. (Motivation for a rescaling)
ex xn dx.
n! = (n + 1) =
The function fn (x) = xn ex satises fn (x) = (nxn1 xn )ex and
Founded in 1912 as Diamond Walnut Growers
1997, appoints Michael Mendes CEO.
2005, Michael Mendes takes Diamond public
In the 2007 Super Bowl, Diamond ran this
Pop Secret, Kettle Foods, Harmony Foo
21-235: Mathematical Studies
Lectures: MWF 10:30-11:20 am in Scaife Hall room 208
Instructor: Robert Pego
Office: Wean Hall 6127
email: rpego AT cmu.edu
Office hours: TTh 3:30-5, or by arrangement
Assignment 1, d
J. Cummings and D. Handron
This midterm is due at 1159pm on Friday 25 February 2005. You may not collaborate.
Please contact us if you nd a typo or want a question claried.
1. (Spivak, 2-23) Let A = cfw_(x, y) R2 |x < 0,