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Khashishi
But why is it true for radians only?
Radians are the natural units for angles.
If you take a circle with radius 1, then the circumference is ##2\pi##. And there are also ##2\pi## radians in a revolution. So, the arc length of a circle radius 1 is equal to the angle in radians. The tangent function is defined... | {
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The ratio is a strong statement, my reply was commenting on the remark afterwards only.
fresh_42
If ##\tan x = x + \alpha (x)##, then ##\frac{\tan x}{x} = 1 + o(x) ##. Conversely, if ##\lim_{x\to 0} \frac{\tan x}{x} = 1 ##, then ##\tan x = x(1+\alpha (x)) =: x + \hat{\alpha} (x) ##. We just label things differently, i... | {
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nuuskur
Radians are the natural units for angles.
If you take a circle with radius 1, then the circumference is ##2\pi##. And there are also ##2\pi## radians in a revolution. So, the arc length of a circle radius 1 is equal to the angle in radians. The tangent function is defined as the ratio of lengths of a right tri... | {
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# Alternating series; first term is 0. Do I have a problem?
I have an alternate series which I want to test for convergence or divergence. The series is as follows:
$$\sum_{n=1}^\infty (-1)^n \frac{n^2-1}{n^3+1}$$
I know how to test this for convergence, but the first term is $0$ and so "$n+1$" terms are not allways... | {
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• I have no idea how I didn't think of this, it's the exact same series. – AstlyDichrar Dec 31 '16 at 0:49
• @AstlyDichrar Yes, it is the exact same series ;) – Olivier Oloa Dec 31 '16 at 0:51
• @AstlyDichrar The series is convergent by the alternating test of convergence:en.wikipedia.org/wiki/Alternating_series_test –... | {
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# Limit of average of decimal digits: $\lim\frac{1}{n} (x_1 + \dots +x_n) = constant$
I have a problem to solve in Ergodic Theory, but I am stuck and have no idea how to procedure. The problem is the following.
Prove that there exists a constant α such that for Lebesgue a.e. x∈[0,1] $\lim_{n\to\infty} \frac{1}{n} (x_... | {
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• Okey thank you very much!! I will try :) – Andreas Wicher Dec 18 '17 at 16:46
• So, pretty sure $x \mapsto 10x$ is also ergodic and lebesque measure presurving. And I guess I should use the interval [0.1, 0.2) but i don't understand right now, why the orbit should correspond to the number of digits? And do i need to ... | {
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# Two points are randomly selected on a line of length $1$
Two points are randomly selected on a line of length $1$. What is the probability that one of the segments is greater than $\frac{1}{2}$? Points can be placed anywhere between [0, 1], for example. Thanks!
• Notice that a line segment will only be greater than... | {
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If the points are on opposite sides of the midpoint(a probability $\frac 12$ event, with $P<\frac 12< Q$ say, then again with probability $\frac 12$ we have $Q$ is nearer $1$ than $P$ is near $\frac 12$,so the segment between them has length greater than $\frac 12$.
Thus the total probability Is $$\frac 12+\frac 12\ti... | {
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# Floor and ceiling functions
Floor and ceiling functions
Floor function
Ceiling function
In mathematics and computer science, the floor function is the function that takes as input a real number x, and gives as output the greatest integer less than or equal to x, denoted floor(x) or x. Similarly, the ceiling functio... | {
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The fractional part is the sawtooth function, denoted by ${\displaystyle \{x\}}$ for real x and defined by the formula[10]
${\displaystyle \{x\}=x-\lfloor x\rfloor .}$
For all x,
${\displaystyle 0\leq \{x\}<1.}$
### Examples
x Floor ${\displaystyle \lfloor x\rfloor }$ Ceiling ${\displaystyle \lceil x\rceil }$ Frac... | {
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These formulas can be used to simplify expressions involving floors and ceilings.[11]
{\displaystyle {\begin{aligned}\lfloor x\rfloor =m&\;\;{\mbox{ if and only if }}&m&\leq x
In the language of order theory, the floor function is a residuated mapping, that is, part of a Galois connection: it is the upper adjoint of ... | {
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Negating the argument complements the fractional part:
${\displaystyle \{x\}+\{-x\}={\begin{cases}0&{\text{if }}x\in \mathbb {Z} \\1&{\text{if }}x\not \in \mathbb {Z} .\end{cases}}}$
The floor, ceiling, and fractional part functions are idempotent:
{\displaystyle {\begin{aligned}{\Big \lfloor }\lfloor x\rfloor {\Big... | {
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More generally,[14] for positive m (See Hermite's identity)
${\displaystyle \lceil mx\rceil =\left\lceil x\right\rceil +\left\lceil x-{\frac {1}{m}}\right\rceil +\dots +\left\lceil x-{\frac {m-1}{m}}\right\rceil ,}$
${\displaystyle \lfloor mx\rfloor =\left\lfloor x\right\rfloor +\left\lfloor x+{\frac {1}{m}}\right\rfl... | {
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This is sometimes called a reciprocity law.[17]
### Nested divisions
For positive integer n, and arbitrary real numbers m,x:[18]
${\displaystyle \left\lfloor {\frac {\lfloor x/m\rfloor }{n}}\right\rfloor =\left\lfloor {\frac {x}{mn}}\right\rfloor }$
${\displaystyle \left\lceil {\frac {\lceil x/m\rceil }{n}}\right\rc... | {
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for x not an integer.
## Applications
### Mod operator
For an integer x and a positive integer y, the modulo operation, denoted by x mod y, gives the value of the remainder when x is divided by y. This definition can be extended to real x and y, y ≠ 0, by the formula
${\displaystyle x{\bmod {y}}=x-y\left\lfloor {\f... | {
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${\displaystyle \left({\frac {2}{p}}\right)=(-1)^{\left\lfloor {\frac {p+1}{4}}\right\rfloor },}$
${\displaystyle \left({\frac {3}{p}}\right)=(-1)^{\left\lfloor {\frac {p+1}{6}}\right\rfloor }.}$
### Rounding
For an arbitrary real number ${\displaystyle x}$, rounding ${\displaystyle x}$ to the nearest integer with ti... | {
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### Beatty sequence
The Beatty sequence shows how every positive irrational number gives rise to a partition of the natural numbers into two sequences via the floor function.[24]
### Euler's constant (γ)
There are formulas for Euler's constant γ = 0.57721 56649 ... that involve the floor and ceiling, e.g.[25]
${\di... | {
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In 1947 van der Pol used this representation to construct an analogue computer for finding roots of the zeta function.[28]
### Formulas for prime numbers
The floor function appears in several formulas characterizing prime numbers. For example, since ${\displaystyle \left\lfloor {\frac {n}{m}}\right\rfloor -\left\lflo... | {
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None of the formulas in this section are of any practical use.[34][35]
### Solved problems
Ramanujan submitted these problems to the Journal of the Indian Mathematical Society.[36]
If n is a positive integer, prove that
1. ${\displaystyle \left\lfloor {\tfrac {n}{3}}\right\rfloor +\left\lfloor {\tfrac {n+2}{6}}\rig... | {
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A bit-wise right-shift of a signed integer ${\displaystyle x}$ by ${\displaystyle n}$ is the same as ${\displaystyle \left\lfloor {\frac {x}{2^{n}}}\right\rfloor }$. Division by a power of 2 is often written as a right-shift, not for optimization as might be assumed, but because the floor of negative results is require... | {
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1. ^ Graham, Knuth, & Patashnik, Ch. 3.1
2. ^ 1) Luke Heaton, A Brief History of Mathematical Thought, 2015, ISBN 1472117158 (n.p.)
2) Albert A. Blank et al., Calculus: Differential Calculus, 1968, p. 259
3) John W. Warris, Horst Stocker, Handbook of mathematics and computational science, 1998, ISBN 0387947469, p. 151
... | {
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31. ^ a b Ribenboim, p. 186
32. ^ Ribenboim, p. 181
33. ^ Crandall & Pomerance, Ex. 1.4, p. 46
34. ^ Ribenboim, p.180 says that "Despite the nil practical value of the formulas ... [they] may have some relevance to logicians who wish to understand clearly how various parts of arithmetic may be deduced from different ax... | {
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# Probability of multiples
1. Jun 27, 2014
1. The problem statement, all variables and given/known data
An integer is chosen random from the first 100 positive integers. What is the probability that the integer is divisible by 6 or 8?
2. Relevant equations
NaN
3. The attempt at a solution
The answer is 24/100 if I ... | {
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4. Jun 27, 2014
### Pranav-Arora
If a number is a multiple of both 3 and 5, it gets added to the result twice. So you should subtract the numbers which are a multiple of $\text{lcm}(3,5)=15$ because those numbers will be added twice in the sum and you get an erroneous result.
$199$ multiples of $5$.
Replace $195$ wi... | {
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12. Jun 28, 2014
### Orodruin
Staff Emeritus
Well, in logic terms what you are doing is first summing over P=1 and then adding the sum over Q=1. This is not the same as summing over (P or Q). The sum over (P or Q) is exactly what is in the code in #3.
13. Jun 28, 2014 | {
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What is the state of the equilibrium for a second derivative equal to zero?
Considering a potential energy of $U$, and a displacement of $x$, the force is given by
$F=-\frac{\partial U}{\partial x}$.
Since equilibrium is defined as the point at which $F=0$, we can express this as $\frac{\partial U}{\partial x}=0$. T... | {
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• possibly related to physics.stackexchange.com/q/362641 – ZeroTheHero Nov 8 '17 at 15:39
• I believe you'd just go to the third derivative since to find out behavior around equilibrium in the first place we take a taylor series about that point (and normally throw away the third and higher derivatives). – Señor O Nov ... | {
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In other words, according to the general theory of stability (e.g., see Arnold's or Fasano-Marmi's textbooks) the equilibrium $x_0$ is stable (in the future) if
fixing a neighborhood $U$ of $(x_0,0)$, there exists a second neighborhood $V\subset U$ of $(x_0,0)$ such that, every pair of initial conditions $x(0)=y_0$ an... | {
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(Statement (b) is nowadays an elementary subcase of a famous theorem due to Lyapunov but a proof was already known by Lagrange and Dirichlet. As a matter of fact, the total energy $E(x, \dot{x})$ is a Lyapunov function for the system for the critical point $(x_0,0)$ when $U$ has a strict minimum at $x_0$.)
• Didn't yo... | {
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For example, if $U'''(x_0) \ne 0$, then the sign of the force does not change as $\Delta x$ goes through zero; the force is opposite the displacement in one direction and with the displacement in the other direction (which will drive the particle away from $x=x_0$).
For the force to be restoring in the case that $U''(... | {
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# How to find the limit of this recurrence relation?
$a_n$ is a sequence where $a_1=0$ and $a_2=100$, and for $n \geq 2$: $$a_{n+1}=a_n+\frac{a_n-1}{(n)^2-1}$$
I have a basic understanding of sequences. I wasn't sure how to deal with this recurrence relation since there is $n$ in the equation.
By using an excel shee... | {
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You have: $$(n^2-1)\,a_{n+1} = n^2 a_n - 1,$$ that by putting $b_n = n a_n$ becomes: $$(n-1) b_{n+1} = n b_n - 1,$$ or: $$\frac{b_{n+1}}{n}-\frac{b_n}{n-1}=-\frac{1}{n(n-1)}=\frac{1}{n}-\frac{1}{n-1},$$ so if we set $c_n=\frac{b_n}{n-1}=\frac{n}{n-1}a_n$, we end with: $$c_{n+1}-c_{n} = \frac{1}{n}-\frac{1}{n-1}.\tag{1}... | {
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-
I didn't understand what you meant with "centering a recursion around its fixed point" and why did you a product of the sequence? – FiBO Feb 21 at 16:50
"Centering around $1$" is, as explained one line below, to consider $b_n=a_n-1$. And if $b_{n+1}=c_{n+1}b_n$ for every $n\geqslant2$, then $b_n=c_nc_{n-1}\cdots c_3... | {
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EDIT: So, how could you guess the form $a_n = b + c/n$? Well, if you look for solutions to $f(z+1) = f(z) + \dfrac{f(z) - 1}{n^2 - 1}$ where $a_n = f(n)$ is a rational function of $n$, if $f(z)$ has a pole of order $k$ at $z=p$ then $f(z+1)$ has a pole of the same order at $z=p-1$. This rapidly leads to the conclusion ... | {
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# Does there exist some $C$ independent of $n$ and $f$ such that $\|f''\|_p \geq Cn^2 \| f \|_p$, where $1 \leq p\leq \infty$?
Let $f$ be a trigonometric polynomial on the circle $\mathbb{T}$ with $\hat{f}(j) = 0$ for all $j \in \mathbb{Z}$ with $\lvert j \rvert < n$. Does there exist some $C$ independent of $n$ and $... | {
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Define the sequences $(a_{n,j})_{j=0}^\infty$ by $$a_{n,j} = \begin{cases} \frac{1}{n^2} + \frac{2(n-j)}{n^3},& \text{if } j < n\\ \frac{1}{j^2}, & \text{if } j \geq n \end{cases}$$ for each $n \in \mathbb{N}$. Then (extending to $j \in \mathbb{Z}$ by $a_{n,(-j)} = a_{n,j}$) we can use the lemma to find $g_n \in L^1(\m... | {
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One immediate choice is Fejer kernel
$$k_{n}(s) = \frac{1}{n}\left(\frac{\sin(\frac{ns}{2})}{\sin(\frac{s}{2})}\right)^{2}.$$ Now $$k_{n}(s) \asymp n$$ on $$[0,\frac{1}{n}]$$, and $$k_{n}(s) on $$[\frac{1}{n}, \pi]$$, therefore $$\int_{0}^{\pi}sk_{n}(s)ds \leq C' \frac{\log(n)}{n}$$. Well, not too bad but not exactly ... | {
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# Is there notation for “some two of the three statements are true”?
There are three propositions A, B, C and another condition "some two of these propositions are true and the third one is false", or, in other words, "exactly 2 of 3 propositions are true". Using truth tables and a Karnaugh map (as discussed at How to... | {
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Using the Iverson bracket, $$[A]+[B]+[C]=2$$
-
The most symmetric definition of 'exactly one of three' I know is $$\text{exactly one of } P, Q, R \text{ is true} \;\equiv\; (P \equiv Q \equiv R) \land \lnot (P \land Q \land R)$$ This uses the fact that equivalence ($\;\equiv\;$) is associative.
Using this, we can wr... | {
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The double use of $\equiv$ as a binary and ternary operatior is hideous and confusing. – Lord_Farin Nov 18 '13 at 18:14
@Lord_Farin Obviously we have different tastes at this point. :-) If $\;(P \equiv Q) \equiv R\;$ is equivalent to $\;P \equiv (Q \equiv R)\;$, as it is in classical logic, why write the parentheses... | {
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# Is $n^\frac{1}{n}$ ever rational?
Sorry if this is a duplicate, as usual I'm struggling with how to search for this.
I was wondering to myself how to prove that you can't get a square number that is twice another square number, I.e. $$m^2=2n^2$$ and I quickly came up with a neat proof using the fact: $$\frac{m}{n}=... | {
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• What exactly does this prove? – user370967 Nov 25 '17 at 13:47
• Why so many downvotes? Peter used almost the same argument in his answer. – Ennar Nov 25 '17 at 13:49
• @Math_QED It was a hint. – Jaideep Khare Nov 25 '17 at 13:52
• It is not obvious that $n^{\frac{1}{n}}$ must be an integer or irrational. (To clarify... | {
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Suppose $n=(a/b)^n$ with $n,a,b\in\mathbb{N}$. We may assume that $a$ and $b$ have no prime factors in common. Suppose $p\mid a$, where $p$ is prime. Then $p^n\mid a^n=nb^n$. Since $p\not\mid b$, we must have $p^n\mid n$. But $p^n\ge2^n\gt n$, which is a contradiction. Hence $a$ has no prime factors, i.e., $a=1$. But $... | {
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# Linear and Angular Momentum on a wooden gate
## Homework Statement
A uniform, 4.5-kg, square, solid wooden gate 1.5 m on each side hangs vertically from a frictionless pivot at the center of its upper edge. A 1.1-kg raven flying horizontally at 5.0m/s flies into this door at its center and bounces back at 2.0m/s in... | {
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Doc Al
Mentor
The "system" you should be considering is the gate + raven. The pivot is external to the system.
Ok, now i should choose gate and raven as a system to consider whether the momentum is conserved. I classified the weight as one of the external force exerted by the earth. And if the pivot is external to the... | {
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For angular momentum, it is always important to specify the reference axis. Since there is a horizontal impulse from the pivot, it could also result in an angular impulse, changing angular momentum. To avoid this, you must choose the pivot itself as the reference axis. A force through the reference axis has no moment a... | {
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(1.1)(5)(1.5/2)=(1.1)(-2)(1.5/2)+(4.5)(1.5^2)(angular speed)/3 I don't remember that L=moment of inertia * angular velocity instead of square the angular velocity. Thank you very much.
haruspex
Science Advisor
Homework Helper
Gold Member
2020 Award
(1.1)(5)(1.5/2)=(1.1)(-2)(1.5/2)+(4.5)(1.5^2)(angular speed)/3 I don't... | {
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# Need Help With Derivative Problem
#### G537
##### New member
Question
I want to investigate the family of curves given by f(x) = x^4 + x^3 + cx^2. I understand how to solve this when c > 0 and c = 0, but I don’t know how to solve this for c < 0 using the first derivative test. If I assume that c< 0 and thus f(x) =... | {
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x1,2 = $$\displaystyle \frac{-3 \pm \sqrt{9 - 4*(8)*(-c_1)}}{8}$$ ...........................edited
You have three roots $$\displaystyle \ \ \to \ \$$ x = 0, x1 and x2
Last edited:
#### Dr.Peterson
##### Elite Member
Question
I want to investigate the family of curves given by f(x) = x^4 + x^3 + cx^2. I understand... | {
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Consider c+5 = 0, so c =-5 which means -c =-5?? -c=-5 means that c=5. So 5+5 =0??
#### G537
##### New member
Thank you all for your helpful responses. I think that your responses get to the crux of my confusion: first, is it necessary to consider cases in this question and, if not, why? It seems to me that negating t... | {
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#### Dr.Peterson
##### Elite Member
Thank you all for your helpful responses. I think that your responses get to the crux of my confusion: first, is it necessary to consider cases in this question and, if not, why? It seems to me that negating the constant c would produce different curves with different characteristic... | {
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Regarding the given solution, it simply states that “for c < 0, there is a maximum at x = 0 and minima at x = [-6 +- sqrt(36 - 96c)]/24.” So it appears they took the approach Dr. Peterson recommended above. Dr. Peterson (or others), I’m curious how you determined that it was not necessary to use cases in this problem. ... | {
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##### Elite Member
f'(x) = 4x^3 + 3x^2 - 2c1x = x (4x^2 + 3x - 2c1)
roots of f'(x) are:
x = 0
x1,2 = $$\displaystyle \frac{c_1 \pm \sqrt{9 - 4*(8)*(-c_1)}}{8}$$
Instead of $$\displaystyle \ "c_1" \$$ for the first term of your quadratic formula, I have -3, because the quadratic expression is $$\displaystyle \ 4x^2 +... | {
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This does lead us initially to three cases, but not based on the sign of c.
Case 1
$$\displaystyle \text {no real solutions} \iff (3)^2 - 4(4)(2c) < 0 \iff c > \dfrac{9}{32} \implies f(x)\\ \text {has no local maximum and a single minimum at } x = 0.$$
Case 2
$$\displaystyle c = \dfrac{9}{32} \implies f'\left ( - \d... | {
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Case 3c
$$\displaystyle 0 < c < \dfrac{9}{32}.$$
Can you do it?
#### G537
##### New member
Thank you once again to everyone for taking your time to help me! This has been extremely helpful. Jeff, your summary was immensely helpful, as it provided very clear reasoning. I also learned a bit about proper typesetting a... | {
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$$\displaystyle 0 < c < \dfrac{9}{32}.$$
Can you do it?
I approached your final 2 questions in the same way. For the case where $$\displaystyle c<\frac{9}{32}$$, I see that $$\displaystyle c=0$$ is a "special case" of this inequality in that it is the only value of $$\displaystyle c<\frac{9}{32}$$ for which f(x) has o... | {
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If my reasoning is incorrect, please let me know. Thanks again!
#### JeffM
##### Elite Member
For a polynomial of degree 4, $$\displaystyle f(x)\rightarrow\infty$$ as $$\displaystyle x\rightarrow\pm\infty$$ and so there must be an odd number of extrema. If there were an even number, then either $$\displaystyle f(x)\r... | {
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I approached your final 2 questions in the same way. For the case where $$\displaystyle c<\frac{9}{32}$$, I see that $$\displaystyle c=0$$ is a "special case" of this inequality in that it is the only value of $$\displaystyle c<\frac{9}{32}$$ for which f(x) has only one extremum, and therefore it breaks the inequality ... | {
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Except in the special case where c = 0, c < 9/32 entails that we have extrema at three different values of x. We know the middle one will be a maximum and the two end ones will be minima. We also know that there will be extrema at
$$\displaystyle x = 0, \ x = \dfrac{-3 - \sqrt{9 - 32c}}{8}, \text { and } x = \dfrac{-3... | {
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You might want to work out the general rules for polynomials of odd degree. You have the basic ideas.
For a polynomial of odd degree, there must be either no extrema, or an equal number of maxima and minima, so the number of local extrema is an even number. Consequently, there are a minimum of zero extrema and a maximu... | {
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A polynomial of even degree has at least one extremum.
A polynomial of odd degree has as many minima as maxima.
A polynomial of even degree and a positive leading coefficient has one more minimum than it has maxima.
A polynomial of odd degree and a negative leading coefficient has one more maximum than it has minima... | {
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# Finding how many terms of the harmonic series must be summed to exceed x?
The harmonic series is the sum
1 + 1/2 + 1/3 + 1/4 + 1/5 + 1/6 + ... + 1/n + ...
It is known that this sum diverges, meaning (informally) that the sum is infinite and (more formally) that for any real number x, there there is some number n s... | {
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This question came from our site for professional and enthusiast programmers.
How exact does the answer need to be? Is it acceptable to limit x to an integer? What the maximum x you need to be able to handle? – David Schwartz Jan 21 '12 at 22:06
@DavidSchwartz- I'm mostly interested in the case where x is an integer,... | {
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Example : the sum will cross the value $20$ for $n$ evaluated at $\rm floor(\rm exp(20-gamma)+0.5)= \rm round(\rm exp(20-gamma))= 272400600$ and indeed (this is not a proof!) :
$H_{272400599}=19.9999999977123$
$H_{272400600}=20.0000000013833$
-
-
Interesting thanks! It seems that Benoit Cloitre proposed in 2002 a mor... | {
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## WeBWorK Problems
### Question with two possible answers
by Daniele Arcara -
Number of replies: 5
How can I allow several answers each to be correct for one problem?
Say that I want to ask the students to integrate 2*sin(x)*cos(x), and I want them to be allowed to write any of the following answers (forgetting the... | {
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So, if you were going to try to write your own answer checker for this problem, you would want to check that the difference between a correct answer and the student answer is a constant for all x. Also, you would want to require students to enter +C to indicate that they know that the indefinite integral yields a fami... | {
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### Re: Question with two possible answers
by Alex Jordan -
Hi Daniele,
I wouldn't view this as allowing for multiple answers, in the sense of trying to enumerate lot of common antiderivatives the student might com up with. Off the top of my head, I see two options for you.
• Use http://webwork.maa.org/wiki/FormulasT... | {
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### Re: Question with two possible answers
by Danny Glin -
In the context of integrals, using the upToConstant flag is probably the right way to go since it catches different expressions even beyond the ones that you are anticipating.
If you truly do have a question with a small number of distinct correct responses, ... | {
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# Hypotheis Testing (paired T test): 1) Is my work correct? 2) How to graph in SPSS,?
I have the following Hypotheis testing problem.
The statement of the exercise is:
Experiment: Eleven different varieties of barley were considered. Of each variety, half was kiln-dried and the other half was left untreated. Then th... | {
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Part (iii) Conduct a statistical hypothesis test for the question in (i) at significance level $\alpha = 0.05.$ Include relevant SPSS output. Formulate a conclusion for the test and cite the appropriate p-value.
This is my output:
My answer: The P-value is .602. We have not enough evidence to reject $H_0$ in favor of... | {
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First, you should find the eleven differences. They are:
11.5, -7.5, 2.5, -5.0, 2.0, -2.5, -13.0, -0.5, -7.5, 7.0, 0.5
[Note: A paired t test is equivalent to a one-sample t test on the differences (testing the null hypothesis of null difference against the two-sided alternative). Because of the paired nature of ... | {
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# Calculate the value of the integral $\int_{0}^{\infty} \frac{\cos 3x}{(x^{2}+a^{2})^{2}} dx$ where $a>0$ is an arbitrary positive number.
Question: Calculate the value of the integral $\displaystyle \int_{0}^{\infty} \frac{\cos 3x}{(x^{2}+a^{2})^{2}} dx$ where $a>0$ is an arbitrary positive number.
Thoughts: I don'... | {
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Given $n\in \Bbb N$ such that $n> a$, define $\gamma (n):=\gamma _1(n)\lor \gamma _2(n)$ with $\gamma _1(n)\colon [-n,n]\to \Bbb C, t\mapsto t$ and $\gamma _2(n)\colon [0,\pi]\to \Bbb C, \theta \mapsto ne^{i\theta}$, ($\gamma (n)$ is an upper semicircle).
Observe that $S$ is the set of singularities of $\varphi$ and b... | {
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Furthermore, $$\int \limits _{\gamma _2(n)}\varphi (z)dz=\int \limits _0^\pi \varphi(ne^{i\theta})\cdot ine^{i\theta}d\theta=\int \limits _0^\pi\dfrac{e^{i\cdot 3ne^{i\theta}}ine^{i\theta}}{(n^2e^{2ni\theta }+a^2)^2}d\theta=n\int \limits _0^\pi i\dfrac{e^{i\cdot 3n(\cos (\theta)+i\sin (\theta))}e^{i\theta}}{(n^2e^{2ni\... | {
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I regret having started this.
• Does anyone know how can I make the $\vert$ in $\vert _{z=ia}$look bigger? – Git Gud Aug 29 '13 at 23:28
• \big\vert, \Big\vert, \bigg\vert, \Bigg\vert produce $\big\vert\;\Big\vert\;\bigg\vert\;\Bigg\vert$, choose your favourite size. – Daniel Fischer Aug 29 '13 at 23:41
• @DanielFisch... | {
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Since $ia$ is the only pole in the semicircular region (where $\gamma_R$ is the semicircular portion of the contour [as indicated in the picture]). Jordan's Lemma shows that the second integral vanishes as $R\to \infty$, so it suffices to compute the residue (don't forget that $ia$ is a double pole). After taking the r... | {
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$$\int_{C_R}\frac{e^{iz}}{z^2+a^2}dz$$
the integrand becomes
$$\Bigg|\frac{e^{iz}}{z^2+a^2}\Bigg| \leq \frac{\Big|e^{i R e^{R\theta}}\Big|}{R^2-a^2} = \frac{e^{-R \sin(\theta)}}{R^2-a^2}.$$
Now, you can see that $R\sin(\theta ) \geq 0$ for $0\leq \theta\leq \pi$ which insures that
$$\lim_{R\to \infty} \frac{e^{-R \... | {
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# Remainder in polynomial division
The remainder when $x^{50}$ is divided by $(x-3)(x+2)$ is of the form $ax + b$. Find the units digit of $a$.
I tried to tackle the problem using the polynomial remainder theorem but got stuck as the divisor is a quadratic expression.
Siong Thye Goh has a good idea but we need to wo... | {
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$$(-2)^{50}= -2a+b$$
$$(3)^{50}= 3a+b$$
$$3^{50}-(-2)^{50}=5a$$
$$(3-(-2))\left( \sum_{i=0}^{49} 3^i(-2)^{49-i} \right)=5a$$
• Why the $mod 10$ – nootnoot Oct 31 '16 at 16:25
• We are interested in the unit digit right? $a \mod 10$ gives us that – Siong Thye Goh Oct 31 '16 at 16:26
• this gives $5a\equiv5\pmod{10}$... | {
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Lemma $\ \ c \equiv d \pmod n\,\Rightarrow\, c^{nk} \equiv d^{nk} \pmod{n^2}$.
Proof $\$ By hypothesis $\ c = d+nj\,$ for some integer $\,j\,$ so by the Binomial Theorem $$c^{nk} = (d+nj)^{nk} = d^{nk} + (\color{#c00}nk)(\color{#c00}nj) d^{nk-1} + (\color{#c00}nj)^{\color{#c00} 2}(\cdots) \equiv d^{nk}\!\! \pmod{\!\co... | {
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Divisibility Relation On the Set $S = \{ 2, 6, 7, 14, 15, 30, 70, 105, 210 \}$: Hasse Diagram, Maximal, Minimal Elements, Greatest, Least elements
Consider the divisibility relation on the set
$$S = \{ 2, 6, 7, 14, 15, 30, 70, 105, 210 \}$$
It is given that this relation is a partial order on $$S$$.
(i) Draw the Ha... | {
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I would greatly appreciate it if people could please take the time to review this.
• your answers are correct. In the (iii) part, you mistyped that $S$ has no minimal elements, It has minimal elements as you have already found them but no least element. Aug 26 '18 at 19:29
• @AnuragA thank your for the confirmation. I... | {
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# Square Roots
1. Aug 29, 2009
### S_David
Hello,
My calculus book says that readers who are writting $$\sqrt{9}$$ as $$\pm3$$ must stop doing that, because it is incorrect. The question is: why is it incorrect?
Regards
2. Aug 29, 2009
### arildno
Because the square root of a number A is DEFINED to be the uniqu... | {
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(alternatively, "minus square root of a", in complete agreement of calling -2 for "minus two")
9. Aug 29, 2009
### slider142
The symbol $\sqrt{9}$ is shorthand for "the principal square root of 9" (not simply "a square root of 9") where the principal square root is a function. A function has only a single output for... | {
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Why is $p \land (p \lor q)$ equivalent to $p$?
How can you prove with equivalence laws that $p \land (p\lor q)$ is equivalent to $p$? I know you have to get rid of $q$, but I'm not sure how.
-
To prove using equivalence laws, we need your list of equivalence laws. There are many possible "standard" lists. – user7530... | {
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Finding eigenvectors to eigenvalues, and diagonalization
I just finished solving a problem on finding eigenvectors corresponding to eigenvalues, however, I'm not sure if it is correct. I was wondering if someone could check my work:
For the matrix $W = \begin{bmatrix} 1 & 2 \\ 3 & 2\\ \end{bmatrix}$, I must find the ... | {
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we can use Row operations to obtain a diagonal matrix similar to W
W = \begin{bmatrix} 1 & 2 \\ 3 & 2\\ \end{bmatrix} $r_1-r_2=R_1$ gives $$W = \begin{bmatrix} -2 & 0 \\ 3 & 2\\ \end{bmatrix}$$ then $R_2=2r_2$ gives W = \begin{bmatrix} -2 & 0 \\ 6 & 4\\ \end{bmatrix} now $R_2=r_2+3r_1$ gives $W = \begin{bmatrix} -2 & ... | {
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# What is the autocorrelation for a random walk?
Seems like it is really high, but this is counterintuitive to me. Can somebody please explain? I am very confused by this issue and would appreciate a detailed, insightful explanation. Thanks a lot in advance!
(I wrote this as an answer to another post, which was marke... | {
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In the context of your previous question, a "random walk" is one realization $(x_0, x_1, x_2, \ldots, x_n)$ of a binomial random walk. Autocorrelation is the correlation between the vector $(x_0, x_1, \ldots, x_{n-1})$ and the vector of the next elements $(x_1,x_2, \ldots, x_n)$.
The very construction of a binomial ra... | {
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# Prove the map has a fixed point
Assume $K$ is a compact metric space with metric $\rho$ and $A$ is a map from $K$ to $K$ such that $\rho (Ax,Ay) < \rho(x,y)$ for $x\neq y$. Prove A have a unique fixed point in $K$.
The uniqueness is easy. My problem is to show that there a exist fixed point. $K$ is compact, so ever... | {
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• Nice proof!Thank you! :)) Mar 10, 2012 at 13:53
• How do we show that f(x):=ρ(x,A(x)) is indeed continuous? Apr 2, 2012 at 9:22
• @Jacques: $\delta: x \mapsto (x,x)$ is continuous, $A$ is continuous, so $g:(x,y) \mapsto (x,A(y))$ is continuous, and $d:(x,y) \mapsto d(x,y)$ is continuous, so $f(x) = (d\circ g \circ \d... | {
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you don't need to prove completeness or define any sequence. Define a nonnegative real function $$h(x) = \rho(x,f(x) )$$ This is continuous, so its minimum is achieved at some point $$x_0.$$ If $$h(x_0) >0,$$ we see that $$h(f(x_0) ) = \rho( f(x_0), f(f(x_0 )) < \rho( x_0, f(x_0)) = h(x_0)$$ Put together, $$h(f(x_0) ) ... | {
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# Alternating sum of binomial coefficients: given $n \in \mathbb N$, prove $\sum^n_{k=0}(-1)^k {n \choose k} = 0$
Let $$n$$ be a positive integer. Prove that \begin{align} \sum_{k=0}^n \left(-1\right)^k \binom{n}{k} = 0 . \end{align}
I tried to solve it using induction, but that got me nowhere. I think the easiest wa... | {
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$$\sum_{k=0}^n (-1)^k\binom{n}{k} = 0$$
for all $$n \geqslant 1$$.
• but what happen if n its an odd number? I think it doesn't apply – FranckN Dec 18 '13 at 14:55
• Take a subset $S$ that doesn't contain $1$. If $S$ has an odd number of elements, then $S\cup \{1\}$ has an even number of elements, and vice versa. – D... | {
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Even though this question is pretty old, and the OP probably will not see the answer, I think it's worthwhile to provide a proof by induction, which the OP (and maybe others) had problems with and surprisingly no one has posted yet.
Since the statement is true for $n=1$, suppose it holds for $n=m$. Then the statement ... | {
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# *a function f:a→b is invertible if f is:* | {
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... |
De nition 5. The function, g, is called the inverse of f, and is denoted by f -1 . 1. not do anything to the number you put in). Then f 1(f… a if b ∈ Im(f) and f(a) = b a0 otherwise Note this defines a function only because there is at most one awith f(a) = b. Using this notation, we can rephrase some of our previous re... | {
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... |
is invertible if and only if f is bijective. First of, let’s consider two functions $f\colon A\to B$ and $g\colon B\to C$. f:A → B and g : B → A satisfy gof = I A Clearly function 'g' is universe of 'f'. According to Definition12.4,we must prove the statement $$\forall b \in B, \exists a \in A, f(a)=b$$. Let f : A ----... | {
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... |
Bijection f is denoted by f -1 in this case =... F−1: B → A is unique, the inverse of f to be one - and... Order of mapping we get the input as the new output pages.. Theorem 3 that functions! Called invertible if and only if ' f ' is invertible or not input d into function... Mar 21, 2018 in Class XII Maths by rahul15... | {
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"lm_q2_score": 0.8824278788223264,
"openwebmath_perplexity": 805.1243433361718,
"openwebmath_score": 0.8342097401618958,
... |
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