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
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 21-122
Integration, Dierential Equations and
Approximations
Syllabus, Fall 2012
Time and Place: WEH 7500
MWF 8:30-9:20, 9:30-10:20
Instructor: Dr. Greggo M. Johnson
Oce: Wean Hall 8122
Phone: 412-268-1504
E-mail: greggo@math.cmu.edu
Class webpage: ht
Carnegie Mellon University
33-106 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
33-106 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
33-106 Fall11
Physics Department
Roy A. Briere
Post-Exam 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 3-D), Circular Motion
Force
Physics I for Engineers (33-106) Exam IV
Carnegie Mellon University
December 05, 2011
Name 50 l, 1ng EON/C Section
Signature g Q Q :
Directions: This is a closed-book exam; no notes or other aids are permitted.
Do all problems; make sure your (30p)!
Carnegie Mellon University
33-106 Fall11
Physics Department
Roy A. Briere
General Review Items
There is already very good problem-solving advice in the textbook. I am just adding
some emphasis and remarks based on issues Ive seen while teaching this cours
Physics I for Engineers (33-106) Exam 1
Carnegie Mellon University
September 24, 2012
Name S O Lg T T g 2V5 Section 2:
Signaturem '
Directions: This is a closedbook exam; no notes or other aids (and no calcu
lator!) are permitted. Do all problems; make
Physics I for Engineers (33-106) Exam III
Carnegie Mellon University
November 12, 2012
Name SOL UL l CLUVQ Section %
Signaturem
Directions: This is a closed~book exam; no notes or other aids are permitted.
Do all problems; make sure your copy of the e
Physics I for Engineers (33-106) Exam III
Carnegie Mellon University
November 14, 2011
Name
Section
Signature
Directions: This is a closed-book exam; no notes or other aids are permitted.
Do all problems; make sure your copy of the exam has all problems i
Carnegie Mellon University
33-106 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,
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 center-of-mass coordinate and velocity for a two-body 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 2-64 in the
13th edition; I didnt see the second one in the new editi
Carnegie Mellon University
33-106 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 4-101F 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 4-l0l 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 4-101F 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 4-101 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