Trapezoidal Rule and Midpoint Rule Error Bounds
1. Trapezoidal Rule
Theorem 1.1 (Error Bound for Trapezoidal Rule). Assume f (x) is continuous on
[a, b]. Let KT = max of f (x) on [a, b] and n be the number of subintervals used
KT
T
in the approximation.
Integration by parts proof
Theorem: Let F and G be dierentiable on [a, b] and assume that F and G are continuous on
[a, b]. Then
F (x)G (x) dx = F (x)G(x)
F (x)G(x) dx
(Important: G(x) can be any antiderivative of G (x)
As I did in class, we often rephra
Full Name:
MATH 122: Sample Final Exam
Section:
Thursday, December 10, 2009
Show all work and justify your answers. Your solutions should read nicely and be legible. They
should not be composed of regurgitated fragments of your mind scattered throughout t
Section:
Full Name:
I Thursday, December 10, 2009
MATH 122: Sample Final Exam
Show all work and justify your answers. Your solutions should read nicely and be legible. They
should not be composed of regurgitated fragments of your mind scattered throug
Math 21122
Integration, Dierential Equations and
Approximations
Syllabus, Fall 2012
Time and Place: WEH 7500
MWF 8:309:20, 9:3010:20
Instructor: Dr. Greggo M. Johnson
Oce: Wean Hall 8122
Phone: 4122681504
Email: [email protected]
Class webpage: ht
Carnegie Mellon University
33106 Fall12
Physics Department
Roy A. Briere
Work and Gravity
There are so many signs that it is easy to get confused when studying gravitational potentials.
I know I made at least one sign error in lecture in 2012, so I am wr
MATH 122: Extra Credit Solutions
2
2
1. f (x) = ex so f (x) = 2xex . Using this information and the equation f g = f g + g f we get
2
2
2
2
2
2xex g = 2xex g + ex g g ex (2x 1) = 2xex g
2x
g
g =
2x 1
2x
g=0
g
2x 1
Using an integrating factor (x) =
1
ex 2
Cockrell School of Engineering
The University of Texas at Austin
20162018 Transfer Guide for Austin Community College Students
Degree Programs
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PHYS2102 (Aktas) Practice Test 2 Hill, Ryan  [email protected]
Problem 1: Point charges located at 3, 8, and 11 cm along th
6/25/17
The Coulomb Balance
Equipment Needed
Calipers, Digital
Coulomb Balance, Sgt.W.
Jack, Table
Laser, HeNe
Lead, Banana Jack (3)
Level, Bullseye
or 6 in torpedo
Power Supply, Pasco SF9585A
Power Supply, Laser HeNe
Resistance Substitution App.
Ringsta
THE UNIVERSITY OF TEXAS AT AUSTIN
ME 333T Engineering Communication
Fall 2010 COURSE DESCRIPTION AND SYLLABUS
INSTRUCTORS:
Professor Moore
[email protected]
Office: ETC 5.140
Office Hours: M 4 5:30, W 3 4:30
Professor Webster
[email protected]
Algebra II
Assessment
Eligible Texas Essential
Knowledge and Skills
Texas Education Agency
Student Assessment Division
January 2014
STAAR Algebra II Assessment
Mathematical Process Standards
These student expectations will not be listed under a separate r
Cindy Raisor Fall 2004 Texas A&M University
Template for Technical Memo
To:
From:
Date:
Subject:
Name and title of recipient (Physics TA)
Your name and title (relationship to recipient)
Date submitted
Meaningful phrase that announces the topic
Introducti
33106 Fall 2012
R.A. Briere
Summary Of Some Thermodynamic Equations
Abstract
This note summarizes the equations for Q, W , and U frequently used in thermodynamics. You may nd the table on the last page useful.
1
1.1
Thermodynamic Quantities
Systems, Equa
Carnegie Mellon University
33106 Fall11
Physics Department
Roy A. Briere
PostExam IV Review Items (for Final)
The Final Exam is inclusive; it will cover all of the material in the course:
Scalars, Vectors, Kinematics (up to 3D), Circular Motion
Force
Carnegie Mellon University
Physics for Engineering Students I
Fall 2012
R.A. Briere
Center of Mass and Elastic Collisions
1
Center of Mass
We can dene the centerofmass coordinate and velocity for a twobody system as:
Xcm =
m A xA + m B xB
mA + mB
m A v
Carnegie Mellon University
Physics for Engineering Students I
Fall 2012
R.A. Briere
Chapter 2 Examples
These problems are old favorites of mine from the 10th edition. The rst one is now 264 in the
13th edition; I didnt see the second one in the new editi
Carnegie Mellon University
33106 Fall11
Physics Department
Roy A. Briere
Exam I Review Items
Chapter 1: Units, Scalars, Vectors, .
Physical quantities have a type: scalar and vector are enough for us.
If asked for a vector (velocity, momentum, .):
Give
o
C ourse2 4101F undamentals f M echanicalEngineering
#2 (75 minutes)
Qniz
Monday,N ovember19,2012
Rules
t
You h ave7 5 m inutest o c omplete he t est
i
( you c an c ontinueon t he backsof pages,f necessary).
All work is to b e d oneo n thesep ages
q
CLE
C ourse 2 4l0l F undamentals o f M echanical E ngineering
Homework # 3
Due b y 2 :30 o n W ednesday,S eptember 1 9 ( at t he s tart o f c lassl ecture)
Assigned P roblems t o b e h anded i n f or a g rade:
Chapter #4 in the textbook by Wickert:
1. P robl
o
C ourse2 4101F undamentals f M echanicalEngineering
Homework#7
O
Due b y 2 :30PM on W ednesday, ctober31
AssignedProblemsto b e h andedi n for a grade:
7
#1) Problem .28 (from y ourtext)
#2) Problem .33(from y our text)
7
Note: T heproblemsb elowa re a
/
E
o
F
2
Course 4101 undamentalsf M echanical ngineering
Ilomework #5
O
Due b y 2 :30P M on W ednesday, ctober10
Sol*h'ong
AssignedProblemsto b e h andedi n for a g rade:
#1)
3/91 The 64.41bcmto slidesdown the curvedpath in the
yeltical plane. If the cr
Carnegie Mellon University
33106 Fall11 UPDATED
Physics Department
Roy A. Briere
Exam III Review Items
We have nished so far:
Kinematics: descriptions of motion with x, v, a, t, g, T, R, etc.
i,
Dynamics: predicting future motions withF = ma, using F, f,