The document proves that the integral of a function f(x,y) over a rectangle is zero if and only if at least one pair of the rectangle's sides has integer length. It shows this by evaluating the integral directly and seeing that it equals zero only when one of the factors in parentheses is zero, which occurs when one of the side lengths has integer value. It then extends this result to higher dimensional spaces, showing that if a region is divided into subregions each with at least one integer edge length, then the original region must also have this property.
We define the definite integral as a limit of Riemann sums, compute some approximations, then investigate the basic additive and comparative properties
The second Fundamental Theorem of Calculus makes calculating definite integrals a problem of antidifferentiation!
(the slideshow has extra examples based on what happened in class)
We define the definite integral as a limit of Riemann sums, compute some approximations, then investigate the basic additive and comparative properties
The second Fundamental Theorem of Calculus makes calculating definite integrals a problem of antidifferentiation!
(the slideshow has extra examples based on what happened in class)
A Quest for Subexponential Time Parameterized Algorithms for Planar-k-Path: F...cseiitgn
The field of designing subexponential time parameterized algorithms has gained a lot of momentum lately. While the subexponential time algorithm for Planar-k-Path (finding a path of length at least k on planar graphs) has been known for last 15 years. There was no such algorithms known on directed planar graphs. In this talk, I will survey this journey of designing subexponential time parameterized algorithms for finding a path of length at least k in planar undirected graphs to planar directed graphs; highlighting the new tools and techniques that got developed on the way.
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On Extendable Sets in the Reals (R) With Application to the Lyapunov Stabilit...BRNSS Publication Hub
This work produces the authors’ own concept for the definition of extension on R alongside a basic result he tagged the basic extension fact for R. This was continued with the review of existing definitions and theorems on extension prominent among which are the Urysohn’s lemma and the Tietze extension theorem which we exhaustively discussed, and in conclusion, this was applied extensively in resolving proofs of some important results bordering on the comparison principle of Lyapunov stability theory in ordinary differential equation. To start this work, an introduction to the concept of real numbers was reviewed as a definition on which this work was founded.
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A Quest for Subexponential Time Parameterized Algorithms for Planar-k-Path: F...cseiitgn
The field of designing subexponential time parameterized algorithms has gained a lot of momentum lately. While the subexponential time algorithm for Planar-k-Path (finding a path of length at least k on planar graphs) has been known for last 15 years. There was no such algorithms known on directed planar graphs. In this talk, I will survey this journey of designing subexponential time parameterized algorithms for finding a path of length at least k in planar undirected graphs to planar directed graphs; highlighting the new tools and techniques that got developed on the way.
I am Manuela B. I am a Calculus Assignment Expert at mathsassignmenthelp.com. I hold a Master's in Mathematics from, the University of Warwick Profession. I have been helping students with their assignments for the past 8 years. I solve assignments related to Calculus.
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You can also call on +1 678 648 4277 for any assistance with Calculus Assignment.
On Extendable Sets in the Reals (R) With Application to the Lyapunov Stabilit...BRNSS Publication Hub
This work produces the authors’ own concept for the definition of extension on R alongside a basic result he tagged the basic extension fact for R. This was continued with the review of existing definitions and theorems on extension prominent among which are the Urysohn’s lemma and the Tietze extension theorem which we exhaustively discussed, and in conclusion, this was applied extensively in resolving proofs of some important results bordering on the comparison principle of Lyapunov stability theory in ordinary differential equation. To start this work, an introduction to the concept of real numbers was reviewed as a definition on which this work was founded.
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success in IIT-JEE, AIEEE, PMT, CBSE & ICSE board classes. The organisation is run by a competitive staff comprising of Ex-IITians. Our goal at ATC is to create an environment that inspires students to recognise and explore their own potentials and build up confidence in themselves.ATC was founded by Mr. ANURAG TYAGI on 19 march, 2001.
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I am Manuela B. I am a Differential Equations Assignment Expert at mathsassignmenthelp.com. I hold a Master's in Mathematics from, the University of Warwick. I have been helping students with their assignments for the past 13 years. I solve assignments related to Differential Equations.
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Integral Calculus. - Differential Calculus - Integration as an Inverse Process of Differentiation - Methods of Integration - Integration using trigonometric identities - Integrals of Some Particular Functions - rational function - partial fraction - Integration by partial fractions - standard integrals - First and second fundamental theorem of integral calculus
Discuss and apply comprehensively the concepts, properties and theorems of functions, limits, continuity and the derivatives in determining the derivatives of algebraic functions
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MAT-121 COLLEGE ALGEBRAWritten Assignment 32 points eac.docxjessiehampson
MAT-121: COLLEGE ALGEBRA
Written Assignment 3
2 points each except for 5, 6, 9, 15, 16, which are 4 points each as indicated.
SECTION 3.1
Algebraic
For the following exercise, determine whether the relationship represents y as a function of x. If the relationship represents a function then write the relationship as a function of
x
using
f
as the function.
x+y2=5
Consider the relationship 7n-5m=4.
Write the relationship as a function
n
=
k
(
m
).
Evaluate
k
(
5
).
Solve for
k
(
m
) = 7.
Graphical
Given the following graph
Evaluate
f
(4)
Solve for
f
(x) = 4
Numeric
For the following exercise, determine whether the relationship represents a function.
{(0, 5), (-5, 8), (0, -8)}
For the following exercise, use the function
f
represented in table below. (4 points)
x
-18
-12
-6
0
6
12
18
f(x)
24
17
10
3
-4
-11
-18
Answer the following:
Evaluate
f
(-6).
Solve
f
(
x
) = -11
Evaluate
f
(12)
Solve
f
(
x
) = -18
For the following exercise, evaluate the expressions, given functions
f
,
g
, and
h
:
f(x)=4x+2
; g(x)=7-6x; h(x)=7x2-3x+6
f(-1)g(1)h(0) (4 points)
Real-world applications
The number of cubic yards of compost,
C
, needed to cover a garden with an area of
A
square feet is given by
C
=
h
(
A
).
A garden with an area of 5,000 ft2 requires 25 yd3 of compost. Express this information in terms of the function
h
.
Explain the meaning of the statement
h
(2500) = 12.5.
SECTION 3.2
Algebraic
For the following exercise, find the domain and range of each function and state it using interval notation.
f(x)=9-2x5x+13
Numeric
For the following exercise, given each function
f
, evaluate
f
(3),
f
(-2),
f
(1), and f (0). (4 points)
Real-World Applications
The height,
h,
of a projectile is a function of the time,
t,
it is in the air. The height in meters for
t
seconds is given by the function h(t)= -9.8t2+19.6t. What is the domain of the function? What does the domain mean in the context of the problem?
SECTION 3.3
Algebraic
For the following exercise, find the average rate of change of each function on the interval specified in simplest form.
k(x)=23x+1
on [2, 2+h]
Graphical
For the following exercise, use the graph of each function to
estimate
the intervals on which the function is increasing or decreasing.
For the following exercise, find the average rate of change of each function on the interval specified.
g(x)=3x2-23x3 on [1, 3]
Real-World Applications
Near the surface of the moon, the distance that an object falls is a function of time. It is given by d(t)=1.6t2, where
t
is in seconds and d(t) is in meters. If an object is dropped from a certain height, find the average velocity of the object from t = 2 to t = 5.
SECTION 3.4
Algebraic
For the following exercise, determine the domain for each function in interval notation. (4 points)
f(x)=2x+5 and g(x)=4x+9, find f-g, f+g, fg, and fg
For.
1. Solution
Week 8 (11/4/02)
Sub-rectangles
Put the rectangle in the x-y plane, with its sides parallel to the x and y axes,
and consider the function,
f(x, y) = e2πix
e2πiy
. (1)
Claim: The integral of f(x, y) over a rectangle, whose sides are parallel to the
axes, is zero if and only if at least one pair of sides has integer length.
Proof:
b
a
d
c
e2πix
e2πiy
dx dy =
b
a
e2πix
dx
d
c
e2πiy
dy (2)
= −
1
4π2
e2πib
− e2πia
e2πid
− e2πic
= −
e2πiae2πic
4π2
e2πi(b−a)
− 1 e2πi(d−c)
− 1 .
This equals zero if and only if at least one of the factors in parentheses is zero. In
other words, it equals zero if and only if at least one of (b − a) and (d − c) is an
integer.
Since we are told that each of the smaller rectangles has at least one integer pair
of sides, the integral of f over each of these smaller rectangles is zero. Therefore,
the integral of f over the whole rectangle is also zero. Hence, from the Claim, we
see that the whole rectangle has at least one integer pair of sides.
Remarks:
1. This same procedure works for the analogous problem in higher dimensions. Given
an N-dimensional rectangular parallelepiped which is divided into smaller ones, each
of which has the property that at least one edge has integer length, then the original
parallelepiped must also have this property. In three dimensions, for example, this
follows from considering integrals of the function f(x, y, z) = e2πix
e2πiy
e2πiz
, in the
same manner as above.
2. We can also be a bit more general and make the following statement. Given an
N-dimensional rectangular parallelepiped which is divided into smaller ones, each of
which has the property that at least n “non-equivalent” edges (that is, ones that
aren’t parallel) have integer lengths, then the original parallelepiped must also have
this property.
This follows from considering the N
m functions (where m ≡ N + 1 − n) of the form,
f(xj1 , xj2 , . . . , xjm ) = e2πixj1 e2πixj2 · · · e2πixjm , (3)
where the indices j1, j2, . . . jm are a subset of the indices 1, 2, . . . , N. (For example, the
previous remark deals with n = 1, m = N, and only N
N = 1 function.) We’ll leave it
to you to show that the integrals of these N
m functions, over an N-dimensional rect-
angular parallelepiped, are all equal to zero if and only if at least n “non-equivalent”
edges of the parallelepiped have integer length.