Math 115 Section 03 (Barsamian) Quiz 4
Ohio University, Tuesday, 28 September, 2004
Q4:
20
Attendance:
16
Quizzes:
80
Course:
Name (print):_
96
Course Percentage:
%
Course Grade:
f(x)
1 The graph of a function f ( x ) is shown at
right.
(a) On the graph o
M1A1 Mechanics: Test 1
26 January 2015
1. Consider the motion of a point on the surface of a rolling wheel of radius R
r(t) = rc (t) + R cos[(t)]
+ R sin[(t)]
|,
where the position of the wheel centre is given by rc (t) = xc (t)
+ R
|. Except for
part (
M2AA3, Introduction to Numerical Analysis
Dr Robert N
urnberg
Solutions to Problem Sheet 1
1. Note that ka bk2 = ha b, a bi = kak2 + kbk2 2 ha, bi. Adding yields the desired result.
2.
q1 =
a1
= e1 .
ka1 k
v 2 = a2 ha2 , q 1 i q 1 = (0, 3, 0, 0)T ,
v
q 2
.
Question 1. The functions f (x) and g(x) are defined for all x by
f (x) =
x2
x+2
,
+x+1
g(x) = f (x2 ).
Find the even part of f (x) and the even part of g(x), which we write as fe (x) and ge (x).
Express the dierence fe (x) ge (x) in simplest form.
Answ
M2AA3, Introduction to Numerical Analysis
Dr Robert N
urnberg
Solutions to Problem Sheet 4
1. (i) Normal equations
T
A A=
1 + k2
1
1
1 + k2
" 1+k #
2+k2
1+k
2+k2
AT A x = AT b x =
T
A b=
and
1+k
1+k
.
.
(ii) Q R via Givens rotations
1 k
0
1
k 1
0
G12 =
.4
Question 4. Given events E, F, G :
(i) Prove that (E C )C = E.
(ii) If F C and G are disjoint, prove that (E \ F )C \ G = E C \ G.
(iii) If E and F are elements of a -field, F, prove that E \ F 2 F.
Answer.
(Must take care with direction of implication
M1P2 Algebra test I solution
Let U and W be the following subspaces of
4
:
U = Spancfw_(1, 1, 1, 2), (2, 1, 0, 5),
W = Spancfw_(0, 1, 1, 1), (3, 2, 4, 1)
1. Find a basis B for U \ W .
2. Find a basis B2 for U + W such that B B2 .
Answer. Let v 2 U \ W . T
M3S11 / M4S11 GAMES, RISKS AND DECISIONS
EXERCISES (non-assessed)
1. Show that for any real numbers a, b, c, d, e, the two-person zero-sum game
with pay-off matrix
a
c
c
a b
d c
e c
b
d
e
always has a pure -strategy saddle-point.
If a finite two-person ze
M1M2 - Mathematical Methods II
Test 2 Monday, 2 February 2015
1. Using the method of variation of parameters (or reduction of
order), derive the general solution of the following dierential
equation:
d2 y
dy
4
+ 4y = 0.
dx2
dx
Show that the functions in t
M2AA3, Introduction to Numerical Analysis
Dr Robert N
urnberg
Solutions to Problem Sheet 3
1. Assume that C is an eigenvalue of A Rnn . Then there exists a z Cn \ cfw_0 such
that A z = z. Denoting the complex conjugates of and z by ? and z ? , we obtain t
.(-"
Question.
(a) Vectors x, y, z 2 R2 are such that |x + y + z| = |x| + |y| + |z|. Prove that
x, y and z are multiples of the same vector. (You need to give complete details of
your argument, but you can use any result from lectures without proof.)
(b)
Math 115 Section 03 (Barsamian) Quiz 5
Ohio University, Friday 1 October, 2004
Q5:
20
Attendance:
19
Quizzes:
100
Name (print):_
Course:
119
Course Percentage:
1 (This is exercise 3.2#2) Match the functions with their corresponding graphs.
y
y
y
x
graph 1
Page 1 of 4
Math 115 Section 03 (Barsamian) Midterm 1 Solutions
( x + 2 ) ( x 0 ) 0 , makes it clear that the important x-values
1 Rewriting one of the factors and rearranging,
( x 3)
are -2, 0, and 3. The sign chart becomes
zero
zero
undefined
neg
neg
po
Math 115 Section 03 (Barsamian) Quiz 3
Ohio University, Wednesday, 22 September, 2004
Q3:
20
Attendance:
12
Quizzes:
60
Course:
Name (print):_
72
Course Percentage:
%
Course Grade:
1 Use the technique of completing the square to rewrite the function f ( x
M2AA3, Introduction to Numerical Analysis
Dr Robert N
urnberg
Solutions to Problem Sheet 0
1. Let A = (a1 an ) Rmn .
(i) (AT )ij = Aji = (ai )j . ith row of AT is aTi .
m
m
X
X
(ii) (B A)ij =
Bik Akj =
Bik (aj )k = (B aj )i .
(AT
k=1
m
X
C)ij =
k=1
T
(A )
M2AA3, Introduction to Numerical Analysis
Dr Robert N
urnberg
Solutions to Problem Sheet 7
1. The following MATLAB function can be used to plot all the required graphs. Note that
for [a, b] = [5, 5] the error kf pn k diverges as n .
function sheet7q1(a,b,
M2AA3, Introduction to Numerical Analysis
Dr Robert N
urnberg
Solutions to Problem Sheet 2
1. If g(x) = a + bT x + 21 xT A x, then
[ g(x)]i =
g
(x) = bi + 12 [A x]i + 12 [AT x]i = bi + [A x]i
xi
as AT = A ;
that is, g(x) = b + A x. In addition, we have th
M2AA3, Introduction to Numerical Analysis
Dr Robert N
urnberg
Solutions to Problem Sheet 5
1. f (x) = x2 is an even function, and d = . Therefore bm = 0, m = 1 ,
Z
Z
2
2
2 2
2 sin(m x)
x cos (m x) dx =
x
x sin(m x) dx
am =
0
m
m 0
0
Z
Z
4
4
cos(m x)
1
M1P1 Analysis test question
Week 17
(a) Define what it means for a sequence (an )n1 of real numbers to converge to a R.
(b) Prove carefully from first principles that the sequence 1, 2, 3, . . . (i.e. the
sequence (an )n1 with an = n) does not converge to
Galois Theory
Lectured in Autumn 2016 at Imperial College by Prof. A. Corti.
Most humbly typed by Karim Bacchus.
Caveat Lector: unofficial notes. Comments and corrections should be sent to
[email protected] Other notes available at wwwf.imperial.ac.uk/kb514
Protactinium234
Uranium-238
Question 11 (2 points)
Which statement about freshwater use is FALSE?
Question 11 options:
As the population increases, water scarcity and sanitation issues
will also increase.
The World Health Organization estimates that 1 in
Question 5 options:
A smooth, clear surface such as a parking lot or other paved
area
A forested area with lots of undergrowth and a variety of trees,
shrubs, and forest floor plants
A pretty flower bed full of colorful roses
An area with several rows of
Electromagnetic induction is a recently discovered technology
that is used to clean sulfur dioxide emissions from coal-fired
power plants.
Argentina has very good outdoor air quality due to the fact
that 100% of their vehicles have catalytic converters th
electricity
neutrons
Question 10 (2 points)
Which of the following isotopes would be the safest to handle in the
shortest period of time?
Question 10 options:
Thorium-226
Thorium-234
Uranium-234
Which of the following is NOT a component of coal reserves (areas
where coal is formed)?
Question 38 options:
low oxygen
cold temperatures
high pressure
all of the answeres are components of coal
reserves
buried plant matter
Question 39 (2 points)
What co
Carbon sequestration technology, which limits the emission of the
greenhouse gas carbon dioxide (CO2) into the atmosphere, is
currently used by _% of all coal-fired power plants in the USA.
Question 23 options:
9
5
1
0
0
5
0
2
5
Question 24 (2 points)
In
O2 (Atmospheric
Oxygen)
Question 34 (2 points)
Coal is composed primarily of:
Question 34 options:
Fluorine
Nitroge
n
Chlorine
Oxygen
Carbon
Question 35 (2 points)
Coal forms over:
Question 35 options:
A few months
A few years
millions of years
A few week
Earth's atmospheric circulation and prvailing winds are caused by
_ and _.
Question 25 options:
infiltration; hypoxia
industrial emissions; coal-fired power plants
the Coriolis Effect; Solar energy
electromagnetic induction; fluidized-bed
combustion
the l
The process of capturing CO2 from a large point source and storing it
in a way so that it does not enter the atmosphere is called:
Question 36 options:
burning fossil fuels
air pollution
the greenhouse effect
secondary mining
carbon capture and
sequestrat
q
The image shows a device that generates a form of renewable
energy that we have discussed in class. (Hint: this device converts
mechanical energy to electrical energy). What is this device called?
Question 42 options:
electric generator
carbon fuel cell
Question 1 (2 points)
If too much water is removed from an aquifer in coastal areas,
_.
Question 1 options:
desalinization will decrease.
the water table will rise.
saltwater intrusion will occur.
the water table will drop.
infiltration from runoff will i
True
False
Question 32 (2 points)
Rank the following sectors by the total amount of greenhouse gases,
which each emits into the atmosphere. Rank from greatest amount
of greenhouse gases emitted to least.
Question 32 options:
Industrial processes; Electric