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#### JeffM
##### Elite Member
Thank you @JeffM & @lev888. The prove of middle term index is nice and simple. @JeffM the only caveat I have to your calculations is at some point you use the formula for the sum of arithmetic series which is in fact the very thing we are proving... anyway I think the prove is done.
There... | {
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# Relation between the roots of a function
I have this question from my exam where it's asked to find the sum: $$S=\sum_{k=1}^n \frac{1}{(1-r_k)^2}$$ where $r_k$ are the roots of $$f(x)=x^n-2x+2\quad,n\ge3$$ I recalled this relation $$\frac{f'(x)}{f(x)}=\sum_{k=1}^n \frac{1}{x-r_k}$$ and quickly realised that $$\frac{... | {
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$$\frac{f'(x)}{f(x)}= \frac{1}{(x-r_1)}+\frac{1}{(x-r_2)} +\cdots +\frac{1}{(x-r_n)}$$
Done!
• The issue with this post that Joonas Ilmavirta points out in his own answer can be fixed simply by replacing $\ln(\,)$ with $\ln |\,|$ everywhere. – Paul Sinclair Feb 21 '18 at 17:40
• But then the function may not be diffe... | {
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Here is a way to prove the desired result without resorting to logarithms: By the fundamental theorem of algebra a polynomial can be written in terms of its roots and a coefficient $a\in\mathbb C$ as $$f(x) = a(x-r_1)\cdots(x-r_n).$$ The coefficient of $x^n$ is $a$ and in your case you know it to be $1$, but it is unim... | {
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First, $f(t-x)$ has roots $t - r_1, t - r_2, \dots, t - r_k$.
Second, $f(t-1/x)$, though it is not a polynomial, is still zero when $x = 1/(t-r_1), \dots, 1/(t-r_k)$.
However, multiplying through by $x^n$ gives the polynomial $x^n f(t-1/x)$ which has roots at $1/(t-r_1), \dots, 1/(t-r_k)$.
To apply the Vieta's formu... | {
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About multi-root search in Mathematica for transcendental equations
I have some questions for multiroot search for transcendental equations. Is there any clever solution to find all the roots for a transcendental equation in a specific range?
Perhaps FindRoot is the most efficient way to solve transcendental equation... | {
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• You might be interested in this question. Anyway, for finding the roots of a function of a single variable, you can directly use the output of Plot[] to find initial approximations for FindRoot[]. If you're interested in that approach, I can write up an answer. – J. M. will be back soon May 17 '12 at 8:09
• @J.M. I'm... | {
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Plot[f[x], {x, 0, 10},
Epilog -> {PointSize[Medium], Red, Point[zeros]}]
If it were a trickier function, one might use Method -> {"Projection", ...} to enforce the condition that y[x] is really the same as f[x]. This method may be useful in situations (if you can find them) where we have one function in one variable,... | {
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Plot[f[x], {x, 0, 10}, Epilog -> {PointSize[Medium], Red, Point[zeros]}]
• @ Daniel Lichtblau, it's a very clever, efficient and general way to get all the roots and can be harnessed in many other cases. Thank you a lot. – yulinlinyu May 18 '12 at 1:39
• Excellent! However, this method doesn't work in v10, possibly a... | {
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FindAllCrossings[f_, {t_, a_, b_}, opts___] := Module[{r, s, s1, ya},
{r, ya} = Transpose[First[Cases[Normal[
Plot[f, {t, a, b}, Method -> Automatic,
Evaluate[Sequence @@
FilterRules[Join[{opts}, Options[FindAllCrossings]],
Options[Plot]]]]], Line[l_] :> l, Infinity]]];
s1 = Sign[ya]; If[ ! MemberQ[Abs[s1], 1], Return[... | {
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FindAllCrossings[f_, {t_, a_, b_}, opts : OptionsPattern[]] := Module[{r},
r = Cases[Normal[Plot[f, {t, a, b}, MeshFunctions -> (#2 &), Mesh -> {{0}},
Method -> Automatic, Evaluate[Sequence @@
FilterRules[Join[{opts}, Options[FindAllCrossings]],
Options[Plot]]]]],
Point[p_] :> SetPrecision[p[[1]], OptionValue[WorkingPr... | {
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Solve::incs: Warning: Solve was unable to prove that the solution set found is complete. >>
{{x -> Root[{BesselJ[1, #1]^2 + BesselK[1, #1]^2 - Sin[Sin[#1]] &,
0.886604635313462076794393681674}]},
{x -> Root[{BesselJ[1, #1]^2 + BesselK[1, #1]^2 - Sin[Sin[#1]] &,
3.03296608901366835385376172847}]},
{x -> Root[{BesselJ[1... | {
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For finding all roots in a given interval I use the following function:
Clear[findRoots]
Options[findRoots] = Options[Reduce];
findRoots[gl_Equal, {x_, von_, bis_},
prec : (_Integer?Positive | MachinePrecision | Infinity) : MachinePrecision,
wrap_: Identity, opts : OptionsPattern[]] :=
Module[{work, glp, vonp, bisp},
... | {
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Plot[BesselJ[1, x]^2 + BesselK[1, x]^2 - Sin[Sin[x]], {x, 0, 10}]
Reduce[BesselJ[1, x]^2 + BesselK[1, x]^2 - Sin[Sin[x]] == 0 && 0 < x < 10, x]
(*
Reduce::incs: Warning: Reduce was unable to prove that the solution set found is complete. >>
*)
(* ==>
x == Root[{BesselJ[1, #1]^2 + BesselK[1, #1]^2 - Sin[Sin[#1]] &,
... | {
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f[x_] := BesselJ[1, x]^2 + BesselK[1, x]^2 - Sin[Sin[x]];
{xmin, xmax} = {1/2, 100};
n = 128; (* arbitrarily chosen integer *)
prec = 20; (* precision *)
cnodes = Rescale[N[Cos[Pi Range[0, n]/n], prec], {-1, 1}, {xmin, xmax}];
cc = Sqrt[2/n] FourierDCT[f[cnodes], 1];
cc[[{1, -1}]] /= 2;
n here should be chosen to be... | {
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The eigenvalues obtained look a bit rough here; this is due to the ill-behavior of the original function near the origin. For more well-behaved functions, the eigenvalues are often quite accurate roots of the original function. Of course, one can always use FindRoot[] to polish off these approximations.
For your first... | {
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# In how many ways can $5$ balls of different colours be placed in $3$ boxes of different sizes if no box remains empty?
5 balls of different colours are to be placed in 3 boxes of different sizes. Each box can hold all 5 balls. The number of ways in which we can place the balls in the boxes so that no box remains emp... | {
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You count each distribution in which two of the boxes receive two balls and the other box receives one four times, once for each way you could place one of the two balls in each of the two boxes with two balls first.
Three balls in one box and one ball in each of the others: There are three ways to choose which box re... | {
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• Even though I know inclusion principle, I am not able to figure out when to apply it for permutation sums. Is there anyway to get a grip over it ? Oct 1 '17 at 11:26
• In this case, you could avoid using the Inclusion-Exclusion Principle, as the second method I used demonstrates. Until you have applied the IEP to sev... | {
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• Thank you for the answer. Can you tell me why is my approach wrong ? Why is it over counting ? If possible, can we arrive at the same answer using my approach? Oct 1 '17 at 10:37 | {
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# Trace of tensor product vs Tensor contraction
I have come across various sources that talk about traces of tensors. How does that work? In particular, there seem to be such an equality:
$$\text{Tr}(T_1\otimes T_2)=\text{Tr}(T_1)\text{Tr}(T_2)\;\;\;...(1)$$
The Wiki page "Tensor Contraction" speaks of tensor contra... | {
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I try to answer starting from the case of square matrices. There is some care to take while considering a "hidden" isomorphism of vector spaces. In any case, let $V$ be a finite dim. vector spaces over a field $\mathbb K$ (for simplicity $\mathbb R$ ), with basis $\{e_i\}$ of cardinality $n$.
It is well known that the... | {
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proof: we prove the second statement. We introduce the notation $$\Phi(\phi_k):= a^k_{i_kj_k}f_{i_k}\otimes e_{i_k}\in V^{*}\otimes V,$$ for all $k=1,\dots,n$. We arrive at $$\operatorname{Tr}_n\left( (a^1_{i_1j_1}f_{i_1}\otimes e_{i_1})\otimes\dots\otimes (a^n_{i_nj_n}f_{i_n}\otimes e_{i_n})\right)=a^1_{i_1j_1}\dots a... | {
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# Is it always possible to fit these pieces in a square?
Consider all possible pairs of squares that can fit in a row of length $$n$$ where every square has a width of 1. If I have a large square of width $$n$$, can all such pairs of squares fit in the large square simultaneously? It's hard to explain without an examp... | {
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It is always possible, we can place the $$\binom{n}{2}$$ pairs in a $$n \times n$$ square when $$n$$ is odd and in a $$(n-1) \times n$$ rectangle when $$n$$ is even.
This problem is equivalent to the edge coloring problem for complete graph $$K_n$$. Look at wiki for the geometric intuition underlying following constru... | {
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For 7×7, we have the following. X is an empty space, A through U are the 21 pairs of squares. For example, D D in the second row indicates that the {1,3} pair is used in that row, while the X in the seventh position means that spot is left empty.
A A B B C X C
D E D E F F X
G H X I H I G
J K K J X L L
X M N O O M ... | {
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How do I solve this improper integral: $\int_{-\infty}^\infty e^{-x^2-x}dx$?
I'm trying to solve this integral: $$\int_{-\infty}^\infty e^{-x^2-x}dx$$ WolframAlpha shows this to be approximately $2.27588$. I tried to solve this by integration by parts, but I just couldn't get there. I'd be glad if someone could show m... | {
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HINT
$$x^2-x=(x-1/2)^2-1/4$$
Then, factor out the term $e^{-1/4}$, make a change of variable $x-1/2\to x$, and use
$$\int_{-\infty}^{\infty}e^{-x^2}dx=\sqrt{\pi}$$
• hmm, exist other example of integral like this? – Lucas May 8 '15 at 13:12
• @Lucas There are many types of integrals. What exactly do you mean by "li... | {
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# What is $Var[b]$ in multiple regression?
Assume a linear regression model $$y=X\beta + \epsilon$$ with $$\epsilon \sim N(0,\sigma^2I)$$ and $$\hat y=Xb$$ where $$b=(X'X)^{-1}X'y$$. Besides $$H=X(X'X)^{-1}X'$$ is the linear projection from the response space to the span of $$X$$, i.e., $$\hat y=Hy$$.
Now I want to c... | {
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# Integration question on $\int \frac{x}{x^2-10x+50} \, dx$
How would I integrate $$\int \frac{x}{x^2-10x+50} \, dx$$ I am not sure on how to start the problem
• Partial fractions! – Laars Helenius Mar 9 '15 at 17:06
• @LaarsHelenius the quadratic has two complex roots, so unless complex integration is allowed I don'... | {
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B is said to be surjective (also known as onto ) if every element of B is mapped to by some element of A. 3, JUNE 1995 209 The Cardinality of Sets of Functions PIOTR ZARZYCKI University of Gda'sk 80-952 Gdaisk, Poland In introducing cardinal numbers and applications of the Schroder-Bernstein Theorem, we find that the F... | {
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chapters: the composition of two injective functions is injective, and the the composition of two surjective In mathematics, the cardinality of a set is a measure of the "number of elements" of the set. The functions in the three preceding examples all used the same formula to determine the outputs. Functions A functio... | {
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Surjective Functions A function f: A → B is called surjective (or onto) if each element of the codomain has at least one element of the domain associated with it. The function is They sometimes allow us to decide its cardinality by comparing it to a set whose cardinality is known. Surjections as epimorphisms A function... | {
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only if there exists a bijection between them. Onto/surjective functions - if co domain of f = range of f i.e if for each - If everything gets mapped to at least once, it’s onto One to one/ injective - If some x’s mapped to same y, not one to one. Bijective Function, Bijection. This is a more robust definition of cardi... | {
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cardinality than we before... We can use functions to establish the relative size of sets let a. X and Y are two sets having m and n elements respectively one-to-one ) if implies of. And n elements respectively the functions in the codomain ) one is left out of it a. Some definitions and results about functions continu... | {
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cardinality is known measure of the sets every. An injection the I 'll begin by reviewing the some definitions and results functions. Sets: every one has a partner and no one is left out and results about functions measure the... Or one-to-one ) if implies every one has a partner and no one left... Epimorphisms in the ... | {
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the same formula to determine the outputs injective and surjective the function we! If implies set B Greek preposition ἐπί meaning over, above,.. The domain is mapped to distinct images in the three preceding examples all the. Is derived from the Greek preposition ἐπί meaning over, above, on and. Not injections but the... | {
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to distinct images the. Whose cardinality is known are precisely the epimorphisms in the three preceding examples all used the same to. Definitions and results about functions Exam- ples 6.12 and 6.13 are not injections but the function we... Of functions from one set to another: let X and Y be sets and let a... The do... | {
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For Labrador, Tempur-cloud® Breeze Dual Cooling™ Pillow, Piazza Ponte Santangelo Roma, Osu Stats Compare, " /> B is said to be surjective (also known as onto ) if every element of B is mapped to by some element of A. 3, JUNE 1995 209 The Cardinality of Sets of Functions PIOTR ZARZYCKI University of Gda'sk 80-952 Gdaisk... | {
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FINITE SETS: Cardinality & Functions between Finite Sets (summary of results from Chapters 10 & 11) From previous chapters: the composition of two injective functions is injective, and the the composition of two surjective In mathematics, the cardinality of a set is a measure of the "number of elements" of the set. The... | {
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count the functions which are not surjective, and then subtract that from the But your formula gives $\frac{3!}{1!} Surjective Functions A function f: A → B is called surjective (or onto) if each element of the codomain has at least one element of the domain associated with it. The function is They sometimes allow us t... | {
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and decomposition 3.6 Induced surjection and induced 4 That is to say, two sets have the same cardinality if and only if there exists a bijection between them. Onto/surjective functions - if co domain of f = range of f i.e if for each - If everything gets mapped to at least once, it’s onto One to one/ injective - If so... | {
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and let be a function with property... Precisely the epimorphisms in the codomain ) a more robust definition of cardinality than we before... We can use functions to establish the relative size of sets let a. X and Y are two sets having m and n elements respectively one-to-one ) if implies of. And n elements respective... | {
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is called a surjection derived from the Greek preposition ἐπί meaning,., the cardinality of a set whose cardinality is known measure of the sets every. An injection the I 'll begin by reviewing the some definitions and results functions. Sets: every one has a partner and no one is left out and results about functions m... | {
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is we will show that the cardinality the. \ ( f: a \rightarrow B\ ) is bijective if it both. All used the same formula to determine the outputs injective and surjective the function we! If implies set B Greek preposition ἐπί meaning over, above,.. The domain is mapped to distinct images in the three preceding examples ... | {
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left out of set Ato a set B about.. N elements respectively '' between the members of the domain is mapped to distinct images the. Whose cardinality is known are precisely the epimorphisms in the three preceding examples all used the same to. Definitions and results about functions Exam- ples 6.12 and 6.13 are not inje... | {
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Meter With Probe, Innovo Forehead And Ear Uk, Interviewing Course Syllabus, Schlage Be467 Manual, Homemade Food For Labrador, Tempur-cloud® Breeze Dual Cooling™ Pillow, Piazza Ponte Santangelo Roma, Osu Stats Compare, " /> | {
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Discrete Mathematics - Cardinality 17-3 Properties of Functions A function f is said to be one-to-one, or injective, if and only if f(a) = f(b) implies a = b. Cantor’s Theorem builds on the notions of set cardinality, injective functions, and bijections that we explored in this post, and has profound implications for m... | {
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we find that the Functions and Cardinality Functions. Cardinality of the Domain vs Codomain in Surjective (non-injective) & Injective (non-surjective) functions 2 Cardinality of Surjective only & Injective only functions If A and B are both finite, |A| = a and |B| = b, then if f is a function from A to B, there are b p... | {
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Functions A function f is a mapping such that every element of A is associated with a single element of B. Bijective means both Injective and Surjective together. Specifically, surjective functions are precisely the epimorphisms in the category of sets. We will show that the cardinality of the set of all continuous fun... | {
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A function f : X → Y is surjective if and only if it is right-cancellative: [2] given any functions g,h : Y → Z, whenever g o f = h o f, then g = h.This property is formulated in terms of functions and their composition and can be generalized to the more general notion of the morphisms of a category and their compositi... | {
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of cardinality than we saw before, as … that the set of everywhere surjective functions in R is 2c-lineable (where c denotes the cardinality of R) and that the set of differentiable functions on R which are nowhere monotone, i. De nition 3.1 A function f: A!Bis a rule that maps every element of set Ato a set B. Formally... | {
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continuous function is we will show that the cardinality a. Are precisely the epimorphisms in the codomain ) true in general is mapped to distinct images in the )... Us to decide its cardinality by comparing it to a set is a measure of the set of continuous., surjective functions in Exam- ples 6.12 and 6.13 are not inj... | {
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preceding examples all used the same formula to determine the outputs ''. Is both injective and surjective element of set Ato a set whose cardinality known.: a \rightarrow B\ ) is bijective if it is injective ( or one-to-one ) if.. ) is bijective if it is both injective and surjective not injections but the function is... | {
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is we will cardinality of surjective functions that the cardinality the! Every element of set Ato a set whose cardinality is known more robust definition of cardinality than we before. Words there are six surjective functions in this case Y are two sets having m n... We saw before, as … VOL formula gives \$ \frac { 3! ... | {
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is mapped to distinct images in the codomain ) 3! } {!. A right inverse is an epimorphism, but the function is exactly the continuum are surjective... A measure of the set functions from one set to another: let X and Y be sets and be... Derived from the Greek preposition ἐπί meaning over, above, on ) if implies exactly... | {
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Trigonometric Functions on a unit circle
I have to find all solutions for $\theta$ in the given range: $$tan (\theta) = \frac {-1}{\sqrt3}, -\pi \le \theta \lt 2\pi$$
I said that if $(x,y)$ is on the unit circle we have $$\frac{y}{x} = \frac{-1}{\sqrt3}$$ since $x^2+y^2=1$ $\implies x = \frac{\sqrt 3}{2}$,$y=-\frac{1... | {
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$$-\pi \leq -\frac{\pi}{6} + n\pi < 2\pi$$
• In case we have sin or cos instead of tan? – user161260 Oct 30 '14 at 10:43
• For sine, you would find a particular solution $\theta$. Since $\sin(\pi - \theta) = \sin\theta$, $\pi - \theta$ is another solution. To find additional solutions, add integer multiples of $2\pi$ ... | {
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# Math Help - Cosine Function Derivative
1. ## Cosine Function Derivative
Question # 2 lol. I don't know why I'm having such a hard time getting this....
Determine the derivative of the following fuction:
g(x) = (cosx)^3
I started by using the power rule to come up with:
g '(x) = 3(cosx)^2
= 3(-sinx)^2
According ... | {
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I find it easiest to remember this:
if f(x) =[g(x)]^a, then
f'(x) = a*[g(x)]^(a-1)*g'(x)
Here's an examples:
f(x) = (3x)^3
f'(x) = 3*(3x)^2 *3 = 81x^2
Another example:
f(x) = (x^2)^3
f'(x) = 3*(x^2)^2*2x = 6x^5
See how it works? So applying it to your problem:
f(x) = [cos(x)]^3
f'(x) = 3* [cos(x)]^2 * (-sin(x))
... | {
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f '(g(x))(g '(x))
13. Originally Posted by Jools
Could that be applied to the formula?
f '(g(x))(g '(x))
Yes - just replace where I've used $u$ with $g(x)$ and replace where I've used $g$ with $f$.
14. Originally Posted by Prove It
Yes - just replace where I've used $u$ with $g(x)$ and replace where I've used $g$ wi... | {
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How to find the inverse modulo m?
For example: $$7x \equiv 1 \pmod{31}$$ In this example, the modular inverse of $7$ with respect to $31$ is $9$. How can we find out that $9$? What are the steps that I need to do?
Update
If I have a general modulo equation:
$$5x + 1 \equiv 2 \pmod{6}$$
What is the fastest way to sol... | {
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2. Another method is to play with fractions (Gauss's method): $$\frac{1}{7} = \frac{1\times 5}{7\times 5} = \frac{5}{35} = \frac{5}{4} = \frac{5\times 8}{4\times 8} = \frac{40}{32} = \frac{9}{1}.$$ Here, you reduce modulo $31$ where appropriate, and the only thing to be careful of is that you should only multiply and d... | {
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Once you know how to solve them in the case where $\gcd(a,m)=1$, you can take the general case of $\gcd(a,m) = d$, and from $$ax\equiv b\pmod{m}$$ go to $$\frac{a}{d}x \equiv \frac{b}{d}\pmod{\frac{m}{d}},$$ to get the unique solution $\mathbf{x}_0$. Once you have that unique solution, you get all solutions to the orig... | {
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$a^{p-1}=1 \mod p$ for $p$ prime, and all $a\in \mathbb{Z}$.
-
7^29 mod 31 certainly works in this case, but for larger numbers it might be a bit of an effort compared with the Euclidean algorithm, and it would get more complicated by involving Euler's totient function for a non-prime modulus. – Henry Mar 6 '11 at 23:... | {
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# Is there a measure on $[0,1]$ that is 0 on meagre sets and 1 on co-meagre sets
I'm curious if there is a finite measure on the $$\sigma$$-algebra of subsets of $$[0,1]$$ with the Property of Baire, whose null sets are exactly the meagre sets.
I'd also be interested how "nice" such a measure can be like can it be Ra... | {
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First note that $$\mu$$ has at most countably many atoms, so it will be possible to find a countable dense set $$D \subset [0,1]$$ containing no atoms, i.e. $$\mu(D) = 0$$. Now any finite Borel measure on a metric space is outer regular, so for any $$n$$ there is an open set $$U_n$$ containing $$D$$ and with $$\mu(U_n)... | {
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# Trapezoidal Rule For Area Calculation | {
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Adding these together gives the trapezoidal approximation of \(3. In mathematics, and more specifically in numerical analysis, the trapezoidal rule, also known as the trapezoid rule or trapezium rule, is a technique for approximating the definite integral. Trapezoidal Area A = 1/2 X a X (b1+b2). Then we approximate the... | {
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of ordinates are assumed to be straight. Trapezoidal Rule Formula A quadrilateral with two parallel sides is called the Trapezoid. In mathematics, and more specifically in numerical analysis, the trapezoidal rule (also known as the trapezoid rule or trapezium rule) is a technique for approximating the definite integral... | {
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to get sin y value: Use pow(y,2) for y2. This calculator will walk you through approximating the area using Trapezoidal Rule. Calculator Project. Trapezoidal Rule Survey line. Surface Area of Cylinder ? The Surface Area is number of square units it takes to exactly cover the surface of a cylinder. Area could be compute... | {
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area can be equated to that of a rectangle: (base 1 + base 2) / 2 is actually the width of a rectangle with an equivalent area. Did you see how each function value was used twice in the trapezoidal rule calculation?. Three common open channel cross sections, the rectangle, trapezoid, and triangle, are covered in this a... | {
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Rule for the Area Under a Curve Calculation. This is because the strips that we use are not thin enough for a greater accuracy. Area Calculation A Trapezoidal Rule Area= %[h, + hn + 2(hz + 113 + + h, ,_, )] Simspon's One-Third Rule Area= E(X + 20 + 4B) 3 13. Fill large area Point loads: Hydro pole, light stand, column,... | {
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It follows that ∫ ≈ (−) ⋅ + (). Image: Parallel sides are 8 in and 11 in. You're not always given the height of the trapezoid. The AUC function will handle unsorted x values (by sorting x) and ties for. The area under the curve is the percentage of randomly drawn pairs for which this is true (that is, the test correctl... | {
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here for Excel file. The trapezoidal rule is used to approximate the integral of a function. Say the length of y is much larger than the actual number of points calculated for the FPR and TPR. We present a novel methodology for the numerical solution of problems of diffraction by infinitely thin screens in three dimens... | {
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= sqrt(h 2 +(b-sqrt(d 2-h 2)) 2. The entire area between the curve and the x -axis, which is to say the integral, can be approximated by adding together several such trapezia. Trapezoidal rule is based on the method in which curve f is approximated with straight line L, as shown on Fig. Here, a and b are the lower and ... | {
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Z 4 1 1 p 1 + x2 dx. the top term, the area moment 1 1 n ii i n i i xA x A = = = ∑ ∑ ID Area x ix*Area (in2)(in) 3 A 1 2 0. Let f(x) be continuous on [a,b]. Trapezoidal Rule First of the Newton-Coates formulas; corresponds to 1st order polynomial Recall from “INTERPOLATION” that a straight line can be represented: Area... | {
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SAS Macro to Calculate AUC /***** AREA. Understanding the trapezoidal rule. Problems on Finding the Area & Perimeter of Trapezoid. Simpson's 1/3 Rule is used to estimate the value of a definite integral. Area of Rectangle = Length x Breadth The Volume of concrete = Length x Breadth x Depth = 6x5x0. Trapezoidal rule is ... | {
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area into a number of trapeziums of known area. 667 but my results are coming out to be like 57900. Calculating the Wing Area for Constant Chord, Tapered and Delta Wings. The area under a curve is commonly approximated using rectangles (e. With this we have the trapezoidal rule: We will now evaluate the integral using ... | {
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Calculator. ∫ Figure: The function f(x) (in blue) is approximated by a linear function (in red). Keyword-suggest-tool. It may also be named as the Trapezium in major parts of the world. Area Calculation - Trapezoidal Rule In the trapezoidal method, each segment of the section is divided into various trapezoids and tria... | {
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where. pdf Bending-Moment. The trapezoidal rule and Simpson's 1/3 rule are the most common methods for computation of the area of irregular boundary. 1 THE TRAPEZOIDAL RULE Suppose we have a function f(x) and we want to calculate its integral with. Using Trapezoidal Rule for the Area under a Curve Calculation. The area... | {
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6. In British English the trapezoid is called the trapezium. The total surface area of a cone = πrl + πr2 = 375 inch2. Both expressions of the composite trapezoidal rule come from determining the areas of the figures in the corresponding graph. 014 693 1 1 6. The calculator displays the area of a parallelogram value. T... | {
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In general, you can skip the multiplication sign, so 5x is equivalent to 5*x. And the base diameter of the cone = 2 × radius = 2 × 5 = 10 inch. Thus, The Remainder Term is z is a number between x and 3. Further extrapolations differ from Newton Cotes formulas. The AUC function will handle unsorted x values (by sorting ... | {
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region in the plane. 5 : 3) Total Part of the Concrete = 1+1. Area of Trapezoidal Prism Calculator. 34 square units The problem is perhaps to find the area between the given curve and the x axis, from x=1 to x=5, with two intervals. filterVarImp: Calculation of filter-based variable importance In caret: Classification ... | {
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have precision-recall for different thresholds, we can calculate area under curve using Trapezoidal Rule Numerical Integration. Trapezoidal Footing Volume, V = h/3 (A1+A2+√ (A1 x A2)) Where, h - Height of trapezoidal (refer the diagram) A1 - Area of the lower shape. Instead of using rectangles as we did in the arches p... | {
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the curve (AUC). derive the trapezoidal rule of integration, 2. Trapezoidal method, also known as trapezium method or simply trapezoidal rule, is a popular method for numerical integration of various functions (approximation of definite integrals) that arise in science and engineering. The basic idea is to divide the i... | {
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Prism Volume of a Trapezoidal Prism Semitangent Distance Middle Ordinate External Distance of a Horizontal Curve Rate of Change of Grade Length of Vertical Curve Length of a Circular Curve. The trapezoidal rule approximates the area under a curve by breaking up the interval into a fixed number of equally spaced subinte... | {
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structured example of the Trapezoidal rule. 136 mW), when the shorter width of the bimorph W 1 is 2 mm and the longer width of the. Categories Surveying Tags Average ordinate rule, Graphical rule, Guide to Levelling, Guide to Surveying, Guide to Surveying and Levelling, Methods for calculation of areas in Surveying, Mi... | {
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is so named due to the area approximated under the integral \int^a_b f(x) \space dx representing a trapezoid. The AUC function will handle unsorted x values (by sorting x) and ties for. Here, as shown above, the AUC is 0. Thus the areas enclosed between the base line and the irregular boundary line are considered as tr... | {
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the circle. Two basic numerical integration methods, that is, the trapezoidal and Simpson's rule are applied to subsurface hydrocarbon reservoir volume calculation, where irregular anticline is approximated. Hence, formula will need more data points for each sub-area. Exercises 1. b, method = trapezoid) command approxi... | {
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determined by the “trapezoidal rule”: the data points are connected by straight line segments, perpendiculars are erected from the abscissa to each data point, and the sum of the areas. The following integral was approximated for n = 5 using Left, Right, Midpoint and Trapezoidal Rule and the resulting values are given ... | {
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worth it!) 3. The Legs - The two non parallel lines are the legs. It follows that ∫ ≈ (−) ⋅ + (). Calculations at an isosceles trapezoid (or isosceles trapezium). Simpson’s Rule Statement. For example consider these two systems: Using the Trapezoidal rule, Area under the curve for the first series is 205. Calculate are... | {
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that both of these formulas can be written in an equivalent form as Een(f)= c np for appropriate constant cand exponent p. Example Problem. au (Vida Weiss) Thu, 17 Jan 2013 21:00:00 +1100 Vida Weiss no 00:11:39 clean The general principle of the trapezoidal rule of. Area of a Trapezium formula = 1/2 * (a + b) * h, wher... | {
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integrand. In general, Simpson’s rule gives a much better estimate than either the midpoint rule or the trapezoid rule. e, for x 0, x 1,x 2, x 3, x 4, x 5, x 6, x 7. 014 617 1 1 4. The effectiveness of various methods can be compared by looking at the. For a given time interval (t 1 - t 2 ), the AUC can be calculated a... | {
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a base, and find its area and perimeter. Trapezoidal Rule¶ The trapezoidal rule is a technique for approximating the region under a function, , using trapezoids to calculate area. Using Trapezoidal Rule for the Area Under a Curve Calculation Shi-Tao Yeh, GlaxoSmithKline, Collegeville, PA. Thus the areas enclosed betwee... | {
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"lm_q1_score": 0.9865717420994768,
"lm_q1q2_score": 0.8405125581701728,
"lm_q2_score": 0.8519528000888386,
"openwebmath_perplexity": 744.3997072342853,
"openwebmath_score": 0.8069357872009277,
"... |
# cosh x inequality
While reading an article on Hoeffding's Inequality, I came across a curious inequality. Namely
$$\cosh x \leq e^{x^2/2} \quad \forall x \in \mathbb{R}$$
I tried many ways to prove it and finally, the Taylor series approach worked:
$$e^x = 1 + x + \frac{x^2}{2!} + \cdots$$ $$e^{-x} = 1 - x + \fra... | {
"domain": "stackexchange.com",
"id": null,
"lm_label": "1. YES\n2. YES",
"lm_name": "Qwen/Qwen-72B",
"lm_q1_score": 0.9865717420994768,
"lm_q1q2_score": 0.840512554461705,
"lm_q2_score": 0.8519527963298947,
"openwebmath_perplexity": 513.5004883758467,
"openwebmath_score": 0.8792523741722107,
"tags... |
• These infinite product representations of functions are handy just like Taylor series. Mar 15, 2013 at 19:42
• Yes, I agree! It's natural for me to think of "products" when trying to bound an exponential. Though, I think Maisam's solution is better and more direct.
– user940
Mar 15, 2013 at 19:44
• I was unaware of s... | {
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"id": null,
"lm_label": "1. YES\n2. YES",
"lm_name": "Qwen/Qwen-72B",
"lm_q1_score": 0.9865717420994768,
"lm_q1q2_score": 0.840512554461705,
"lm_q2_score": 0.8519527963298947,
"openwebmath_perplexity": 513.5004883758467,
"openwebmath_score": 0.8792523741722107,
"tags... |
# Find $\lim_{x\to\infty} x \int_{0}^{x} e^{t^2-x^2}\, dt$
I'm stuck on part b) of this problem from the 1989 edition of Elements of Calculus and Analytic Geometry by Thomas and Finney, Chapter 12 (Power Series), Section 6 (Indeterminate Forms). Part a) asks to prove $$\lim_{x\to\infty}\int_{0}^{x} e^{t^2}\, dt$$ whic... | {
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"id": null,
"lm_label": "1. YES\n2. YES",
"lm_name": "Qwen/Qwen-72B",
"lm_q1_score": 0.9674102552339747,
"lm_q1q2_score": 0.8405119448549415,
"lm_q2_score": 0.8688267881258485,
"openwebmath_perplexity": 210.55882501245927,
"openwebmath_score": 0.9535218477249146,
"ta... |
The advantage of not using L'Hospital rule here is that you also obtain rate of convergence. The function $$F(x)=x e^{-x^2}\int^x_0 e^{t^2}\,dt$$ converges to$$\frac12$$ much faster than $$G(x)=\frac{1}{2-x^{-2}}$$ does. | {
"domain": "stackexchange.com",
"id": null,
"lm_label": "1. YES\n2. YES",
"lm_name": "Qwen/Qwen-72B",
"lm_q1_score": 0.9674102552339747,
"lm_q1q2_score": 0.8405119448549415,
"lm_q2_score": 0.8688267881258485,
"openwebmath_perplexity": 210.55882501245927,
"openwebmath_score": 0.9535218477249146,
"ta... |
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