December 15, 2011 ~ F I Total Points: 200
Follow directions carefully:
0 Write your name section number and instructor's name on this test and on each of the 8 answer sheets.
a Number the answer sheets 1 through 8
0 Do each problem on ONE answer sheet as
Cholesky Decomposition
Cholesky decomposition is a special version of LU decomposition tailored to handle symmetric matrices more eciently.
For a symmetric matrix A, by denition, aij = aji . LU decomposition is not ecient enough
for symmetric matrices. Th
1
Functions, Graphs and Limits
1.1
The Cartesian Plane
In this course we will be dealing a lot with the Cartesian plane (also called the xy-plane),
so this section should serve as a review of it and its properties.
y-axis
6
Quadrant 2
Quadrant 1
- x-axis
dy
dy dv
=
dx
dv dx
or
[f (g(x)]0 = f 0 (g(x)g(x)
A crucial case we will come across a lot is when y = [g(x)]n . In this case,
y 0 = n[g(x)]n
1
g 0 (x)
This is called the general power rule.
Examples:
1. h(x) =
p
2x + 3 = (2x + 3)1/2 . Then
1
h0 (x) = (
2. Not every function is dierentiable everywhere. In fact, you cant take the derivative
at points
of discontinuity,
with corners,
or with vertical tangents.
Note that saying that a function isnt dierentiable at points of discontinuity is equivalent to
3. Find the derivative of h(x) =
p
x
.
x+1
solution: We use the quotient rule:
p
p
( x)0 (x + 1) (x + 1)0 ( x)
0
h (x) =
(x + 1)2
1
x 1/2 (x + 1) (x1/2 )(1)
= 2
(x + 1)2
1 1/2
x + 1 x 1/2 x1/2
2
= 2
(x + 1)2
1 1/2
x + 1 x 1/2
2
2
=
(x + 1)2
4. Let f (x) =
October29,2007> ' g ] ' MATH m. TESTZ'.
A
. VNAME:
(2.54.2) I . v y . SECTION"
Do'all problems in the spaces provided. Mark Your answers clearly and write them in simplified form. You must
show all appropriate work in order. to receivécredit for an
Gauss Elimination
Gauss elimination is a way to formalize the eliminate and substitute process we looked at in the
last lecture. As before, it is a two-step process:
Forward elimination eliminate one of the unknowns at a time and continue until the rst so
Cholesky Decomposition Example
Example: Consider the circuit in Figure 1, where R1 = R2 = R3 = R4 = 5 and R5 = R6 =
R7 = R8 = 2. In addition, V1 = V2 = 5. Solve for the loop currents i1 , i2 , i3 and i4 .
Figure 1: Circuit for Example 1
Using Kirchos law,
March 31, 2006 MATH [Al TEST 2A NAME:
SECTION:
(2.3-4.2)
You may NOT use calculators of guy type on this exam.
Do each problem in your exam booklet. All answers should be in simplest form and marked clearly.
You must show all appropriate work in order to
\
October 36, 2006 V MATH [IZfTEST 2 _ NAME:
A
(2.4-4.2) ' ' j SECTION:
!
Instructions: _
Do all problems in'the spaces provided. Mark your answers clearly and write them in simplied form. You must
show all appropriatework in order to receive credit for a
mm MATH 11215373
SECFEGN:
Imitataas:
Do all problems in the spams provided. Mark m answers clearly and write them in simplied mn. You must show all
appropriatc work in order to receive credit for an answer. You may um graphing calculators, but show work
A
September 26, 2005 MATH 1 laTEST 1 Name:
P2 - 2.3 Section:
Instructions:
Do all problems in the spaces provided. Mark your answers clearly and write them in simplied form. You must show all
appropriate work in order to receive credit for an answer. You
May 12 MATH 112 Total Points: 200
2012 FINAL EXAM
Follow directions carefully:
0 Write your name section number and instructors name on this test and on each of the 6 answer
sheets.
0 Number the answer sheets 1 through 6
0 Do the problems on the answer sh
_.wwwwwm
MATH 112
FINAL EXAM
December 1 4, 2006
Total Points: 200
Follow directions carefully:
0 Write your name section number and instructors name on this test and on the booklet.
0 Answer problems in order. Mark your answers clearly. Show all appropria
May 11 MATH 1 12/1 13 Total Points: 200
2013 FINAL EXAM
Follow directions carefully:
0 Write your name section number and instructor's name on this test and on each of the 7 answer sheets.
c Number the answer sheets 1 through 7
0 Do each problem on ONE an
B
December 2006 , 1 1 2 " 3 NAME:
(4.3-4.5, 8.1, 5.1-5.6, 5.8, 6.1, 6.5) SECTION:
Instructions:
Do all problems in the spaces provided. Mark your answers clearly and write them in simplied form. You must show
all appropriate work in order to receive credi