2nd Midterm, Calculus 2 Sec 6
04/07/2011
You can solve as many exercises as you want from each one of the sets.
Exercises for 10 points are elementary and require short calculations, exercises for 15/20 points are intermidiate and exercises for 25/30 poin
1st Midterm, Calculus 2 Sec 6
02/17/2011
The test is made of 3+1 sets of integrals. Exercises from the rst set
are worth 10 points, from the second set 15 points, from the third set 25
points. The last two exercises are taken from an integration bee and t
Final exam, Calculus 2 Sec 6
05/10/2011
You can solve as many exercises as you want from each one of the sets.
You are allowed to use formulas from the list included with the test, but you
have to justify every other calculation.
Motivate your answers! I
Assignment 5
1) Determine for which real values of p the following integrals are convergent:
xp ln x dx
0
+
xp ln x dx
xp ln (x + 1) dx
0
The number
is positive. It might be helpful to remember that limx0
ln (x+1)
x
2) Determine for which real values of p
Homework chapter 9:
15, 19, 21, 25, 27, 29,31. 33. 35, 39. 51, 53
19 21 25 27 29
33 35 39 51 53
61, 63
Remember: Hour exam Wed Nov 9
1
2
High standard of living = high consumption of energy per person
3
4
Chapters 9 and 10: Thermodynamics
Thermodynamics:
Chapters 9 and 10: Thermodynamics
Thermodynamics: the conversion of energy amongst its various forms
th
it
(mechanical, chemical, electrical, etc.) and the relation
of these conversions to macroscopic variables (i.e.
temperature, pressure, and volume) .
S
New Science and Engineering Graduate School in New York City
Mayor Bloomberg is offering free land and $100M in infrastructure spending
Columbia building a new $450M building for neuroscience in Manhattanville Campus
Remember: Hour exam Wed Nov 9
Reading
Read:chapter6
Homework:11,13,1719,21,27,29,39,43,47,51,55
LastClass
VelocityDistribution:MaxwellBoltzmanndistribution
shape,changewithT,threeaverages
IntermolecularCollisions
collisions / sec = 4
N 2 RT
d
V
M
RealGasBehavior
ForO2 at298K:4.14x109 s1 !
con
Reading assignment: chapter 6 Equilibrium
From Last Class
Ptot = ntotRT/V
=
+
Ptot = P1 + P2 = (n1 + n2)(RT/V)
P1 = 1 * Ptot
Temperature is a measure of kinetic energy of molecular motion
PV 2
= ( KE ) avg ,mole
n
3
PV
= RT
n
KEavg , mol
3
= RT
2
KE depen
From last class
Ideal gas law from experiment
gas law from experiment
hyperbola
P = (h * Hg) * g
(h
P = (76cm * 13.6g/cm3) * 980cm/s2
P = 1*105kg/m.s2
1
Charles Law
All gases extrapolate to zero volume at the same T, if P is held constant
T (K) = 0oC + 27
CHAPTER 5: Gases
)
Reading assignment: 5.1 to 5.11 inclusive
Homework: 21, 23, 27, 31, 37, 41, 53, 57, 61, 71, 83, 93
Three distinct physical phases of matter:
Solid
Liquid
Gas
Experiments Used to Refine Theory
Used to Refine Theor
Atmospheric Chemistry
1
Molecular Orbitals of H2
Molecular orbitals are derived by a superimposition of the atomic
orbitals belonging to the valence shells of all atoms, as illustrated
by H2.
As two atoms approach each other, the orbitals overlap and
interfere either constructi
The Aufbau (Building Up) Principle
3s < 3p < 3d 4s at Z = 20, Ca
(life becomes complex!)
Covalent Bonding
A covalent bond is one in which the atoms are held together by a
shared electron pair.
e.g. H2
+
e
e
+
In essence, a pair of electrons that concent
Wave-Particle Duality
If electromagnetic radiation has dual character and can be regarded
as being particle-like, is it possible that matter could have wavelike
properties?
In 1924, Louis de Broglie made a revolutionary proposition: Small
particles of mat
Hyperbolic functions (CheatSheet)
1
Intro
For historical reasons hyperbolic functions have little or no room at all in the syllabus of a calculus
course, but as a matter of fact they have the same dignity as trigonometric functions. Unfortunately this can
Integral of a rational function
1
Intro
Most calculus textbooks present this topic as the method of partial fractions. They usually contain a
bunch of apparently unrelated examples (sometimes not exhaustive) and do not focus on the purpose
of this method.
Repurposing an HP Calculator
Lab : Hello World
Computer Science and Computer Engineering Gateway Project
Stephen A. Edwards
Fall
Abstract
In this project, you will write new rmware for an HP b calculator.
is is an example of embedded programmingcoding so
Lecture 6.
Welcome to the last lecture on Matlab. This lecture will extend the capabilities of
Matlab to solving polynomials as well other calculus applications.
This lecture will cover the following topics:
repr
Lecture 5. Imaging and making histograms
Welcome to lecture 5 of Matlab. Today we are going spend some time on how to create
multidimensional matrices and their applications in Matlab.
This lecture will cover the
Lecture 4. Creating Functions and Plotting
Welcome to lecture 4 of Matlab. Today we will invest some time on the fun part of
Matlab, creating your own functions and graphing them.
This lecture will cover the
Lecture 3. ControlFlow Statements
While Matlab can be used as a calculator, it is still fundamentally a programming
language. The building blocks of any programming language are the controlflow
statements.
This lec
Lecture 2. Matlab as a Matrix Calculator
In the previous lecture, we explored the capabilities of Matlab as a scalar calculator.
The true power of Matlab lies in its ability to manipulate matrices.
This lecture
Lecture 1. Matlab as a Calculator
Welcome to the first lecture of Matlab. This lecture will focus on introducing the
basic functionalities of Matlab to the students as well as the fundamentals of
programming. The f
Trigonometry CheatSheet
1
How to use this document
This document is not meant to be a list of formulas to be learned by heart. The rst few formulas
are very basic (they descend from the denition and/or Pythagoras theorem) and you might want
to memorize th
Taylor Series and Polynomials
1
Motivations
The purpose of Taylor series is to approximate a function with a polynomial; not only we want to
be able to approximate, but we also want to know how good the approximation is. We are already
familiar with rst o