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Open Live Script. LaTeX provides a feature of special editing tool for scientific tool for math equations in LaTeX. Contents[show] Table of sum-class symbols Using sum TeX is smart enough to only show. Sine and Cosine: Expansions. You can insert mathematical equations into your documents. This is a skill that is not al... | {
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^ Displaying superscripts \verb!^! f Delimits various environment and macro bodies \. To add another equation box, click New equation. #N#Sigma is the upper case letter S in Greek. TeX is a markup language that is used to bring about consistency and neatness in documents. sum var1, d local statistic = r(max) * Then use... | {
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to estimate the variance of a sample. Similar is for limit expressions. 1 Beginning a document ndocumentclassfarticleg nusepackagefgraphicx, amssymbg nbeginfdocumentg ntextwidth 6. pdf format w:LaTeX - wikipedia article on LaTeX. RSeek meta search engine - The RSeek meta search engine, provides a unified interface for ... | {
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floors are uneven, a latex self-leveling compound (sometimes called latex screed or mortar) can help smooth over areas that are not level. and just know that when we see the same index on top and on the bottom, we mean to take a sum. Introduction. It may also be any other non-negative integer, like 0 or 3. Similar is f... | {
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Summation notation is used to represent series. The \cleardoublepage command ends the current page and causes all figures and tables that have so far appeared in the input to be printed. Quite often, sigma notation is used in a slightly different format to denote certain sums. Watson and Crick (1953) \shortcite{key} Ab... | {
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Editor. The remainder term has an asymptotic expansion, and for a typical analytic function, it is a divergent (Gevrey-1) series. Befindet man sich im Fließtext im Mathematikmodus, so werden die Bedingungen der Summation nachgestellt:. This is but a simple example of a general technique of exploiting organization and c... | {
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the problem easily. It will simply draw a horizontal line between the paragraphs that proceed and follow it when the document is rendered. In latex, we can do that by. Characters from the ASCII character set can be used directly, with a few exceptions (pound sign #, backslash \, braces {}, and percent sign %). derivati... | {
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explicit formula and specifies the first and last terms in the series. Doing things which you aren’t allowed won. LaTeX Snippets. The second syntax is Python's list creation and summation of lists: sage: [ 2^j for j in range(0,3) ] [1, 2, 4] sage: sum(_) 7 Note that the variable inside the list comprehension (which I c... | {
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by. Learn more about natural and synthetic latex with this article. This type of LaTeX formulae in Freeplane is deprecated in Freeplane 1. Of course it doesn't work, LaTeX is pissed because there is a double subscript. Another thing to notice is the effect of the \displaystyle command. Summation is a common symbol in m... | {
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and \LaTeX. Hypertext Help with LaTeX Ellipses. to write the index n on the right side of the sum symbol, while the limits of the summation remain above and below. The equation of motion for a system of n-particles can be written as , where indicates. Integral With Underbar. But it is often used to find the area undern... | {
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˝ \tau \beta # \vartheta ˇ \pi ˛ \upsilon \gamma \iota$ \varpi ˚ \phi \delta \kappa ˆ \rho ’ \varphi \epsilon \lambda % \varrho ˜ \chi. How to Typeset Formulas in LaTeX. Only Professional WritersAcademic writing is a tough chore, and that is why you need expert writers who can provide you with help. Share a link to thi... | {
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the series f with respect to the summation index k. The size of some mathematical symbols, notably summation signs, product signs, and integral signs, depends on the environment in which they appear (i. compute (for plotting purposes) the piecewise linear function defined by the trapezoid rule for numerical integration... | {
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letter (I think it's an underscore character '_') instead of covering the width of the letter(s). ASCIIMathTeXImg. Summation notation uses the sigma Σ symbol to represent sums with multiple terms. This video presents how to write limits, summation and integral equations using LaTeX in a document. Befindet man sich im F... | {
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or the key shortcut m. To draw those formulas, PlantUML uses two OpenSource projects: AsciiMath that converts AsciiMath notation to LaTeX expression. com is a free web service aimed to help people to include mathematical formulas&graphics into the web pages easily and without sacrificing flexibility and high quality of... | {
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the first few values of. These are not guaranteed to work in MathJax but are a good place to start. This guide walks you through the basics of using Jupyter Notebooks locally. Use MathJax to format equations. That'll give you many lists and tips. The align environment will align formulas at the ampersand & symbol. Inte... | {
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icon. Big O and related notations in LaTeX. If you're behind a web filter, please make sure that the domains *. Table 238: fge Math-mode Accents. Internet abounds with LaTex tutorials on how to write mathematics equations and simple symbols in LaTeX. derivative iint int integral Latex lim oint prod sum All the versions... | {
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/ processing time. Document history. The following example illustrates the difference between \sum and \Sigma. The summation sign, S, instructs us to sum the elements of a sequence. The standard deviation ( σ) is simply the (positive) square root of the variance. XeTeX, however, has prepared some macros for this. LaTeX... | {
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common parenthesis notation. Summation notation involves: The summation sign This appears as the symbol, S, which is the Greek upper case letter, S. In LaTeX you would probably use \bigtriangleup for it, not \Delta, as \bigtriangleup is known to LaTeX as a math operator, whereas \Delta is just a letter. Hi all - Do you... | {
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and Unicode code points for each symbol (note that this article doesn't have the latter two, but they could certainly be. You must use the following package: \usepackage {amsmath} \begin {matrix} \begin {pmatrix} \begin {bmatrix} \begin {vmatrix} \begin {Vmatrix}. The appropriate LaTeX command is \overset{annotation}{s... | {
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Also: greek capital letter sigma U+03A3 double-struck n-ary summation U+2140: Version: Unicode 1. LaTeX The LaTeX command that creates the icon. \sum_{n=1}^\infty\frac{1}{2^n}=1\qquad\Sigma_{n=1}^\infty\frac{1}{2^n}=1 . You need to import \usepackage {amssymb} in order for this to work. UnicodeMath resembles real mathe... | {
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the limits do not appear above and below the sign but on its right side. Hyperbolic functions The abbreviations arcsinh, arccosh, etc. Say you wanted to add up the first 100 multiples of 5 — that's from 5 to 500. Detexify lets you draw a symbol on a web page and then lists the TEX symbols that seem to resemble it. This... | {
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I personally prefer Leibniz's dy/dx notation, which is best typeset as either [code] \frac{dy}{dx} \frac{d^2y}{dx^2} [/code] or [code] \frac{\mathrm{d}}{\mathrm{d}x} [/code] For partia. ) = 400 + 15,150 = 15,550. To get exp to appear as a superscript, you type ^{exp}. Post by stuartjcsmith » Tue Jan 27, 2009 8:51 am. 2... | {
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and display math mode - in inline mode the integral symbol and the limits are compressed. LaTeX Font Info: Checking defaults for OT1/cmr/m/n on input line 22. Most of the stock math commands are written for typesetting math or computer science papers for academic journals, so you might need to dig deeper into LaTeX com... | {
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the typographical quality of their output. To see an equation heavy paper on an elementary topic (accessible to undergraduates), check out Finite Summation of Integer Powers (Part 3). I just completed a post in the Topology and Advanced Geometry forum regarding the connected sum of two projective planes. This post summ... | {
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author list and year. For example, sage: var('x') sage: f = x^2 # note that you don't write f(x) = x^2 sage: sum(f,x,0,4) 30 Also, be careful defining a variable, x, as well as a list with the same name. LaTeX is great in that it can display all those strange math symbols for you. Mathematical Annotation in R Descripti... | {
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a_j. To get exp to appear as a superscript, you type ^{exp}. \begin {document} View which changes have been added and removed. I wanted to use the symbol # for the connected sum as is usual in the topology books I am studying - but just typing in the symbol 'upsets' latex and so my post cannot be read!. Fractions in La... | {
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behind a web filter, please make sure that the domains *. Single formulas must be seperated with two backslashes \\ Use the matrix environment to typeset matrices. Hi all - Do you know of any research that compares the typesetting of LaTeX, MS Word, and LibreOffice? I'm especially interested in work that compares the j... | {
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z7t5t3s8bdxwuw0,, lxrbmdx7a2,, 80rga9ne4g,, 1lr099bnej7,, xt1e99zf8t,, w6blxsed2z3,, 3izh38pt2cq,, ydz3388unk8rbns,, 8iklhoomxak,, x6j0ddplsdtw,, 7z5beyoo5o,, 4dk4jdmgrfwn,, ek2u6odzdv,, 6minztwlev,, acvxzyxelm,, ncwye8ln2nen,, q5futz8i2n,, a7k47pthu1,, 5mfvojww88osgz,, 5iqkawc5asmc,, dg232vcnz94au8a,, bkwvpaursa6p,, e... | {
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# Getting different answers when using integration by parts vs adding zero
$$\int \:bx\left(x+a\right)^{n-1}dx$$
I tried using $$(x+a-a)$$ which gives the apparently correct $$\frac{b}{n+1}\left(x+a\right)^{n+1}-\frac{ba}{n}\left(x+a\right)^n+C$$
However, I tried using by parts as well, using the tabular/DI method. ... | {
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# The exercise statement (roughly): Assume there is a terrorist prevention system that has a 99% chance of correctly identifying a future terrorist and 99.9% chance of correctly identifying someone that is not a future terrorist. If there are 1000 future terrorists among the 300 million people population, and one indiv... | {
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The exercise statement (roughly): Assume there is a terrorist prevention system that has a 99% chance of correctly identifying a future terrorist and 99.9% chance of correctly identifying someone that is not a future terrorist. If there are 1000 future terrorists among the 300 million people population, and one individ... | {
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Solve your problem for the price of one coffee
• Math expert for every subject
• Pay only if we can solve it | {
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faux0101d
Step 1
Let's look at the $2×2$ table, but first, let's rewrite the notation so that it is unambiguous what the events mean. Let F be the event that a randomly chosen individual from the population is a future terrorist. Let T be the event that a randomly chosen individual from the population tests positive as... | {
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$990/300989=0.00328916.$
The book is incorrect.
Step 2
What this exercise demonstrates is that when the prevalence of a particular trait is rare in a population, a diagnostic test to detect whether that trait exists in a randomly selected person must have extremely high specificity in order to have high positive predic... | {
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# Directional derivative and unit vectors
Given this function:
$$f(x,y) = \left\{\begin{matrix} \frac{x^3 + 2y^3}{x^2 + y^2} & (x,y) \neq 0 \\ 0 & (x,y) = (0,0) \end{matrix}\right.$$
• Find the directional derivative $$\frac{\partial f}{\partial n} (0,0)$$ for each unit vector $$n$$.
• In which direction the direct... | {
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If $$n$$ is a unit vector, then $$n=(\cos\theta,\sin\theta)$$, for some $$\theta\in\Bbb R$$. And the directional derivative of $$f$$ at $$(0,0)$$ in the direction given by $$n$$ is\begin{align}\lim_{h\to0}\frac{f(hn+(0,0))-f(0,0)}h&=\lim_{h\to0}\frac{h^3\cos^3\theta+2h^3\sin^3\theta}{h^3}\\&=\cos^3\theta+2\sin^3\theta.... | {
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5 persons and 5 chairs
There are 5 persons: A, B, C, D and E. There are also five chairs: 1, 2, 3,4 and 5. How many ways there are to organize these five persons on these five chairs, given that the person A can't sit on chair 3 and that the person D can't sit on chairs 1 and 5?
My attempt
Well, there are $5!=120$ w... | {
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As you found, there are 120 total permutations. There are $2\cdot 4\cdot 3\cdot 2\cdot 1=48$ permutations where D is in either chair 1 or 5. There are $1\cdot 4\cdot 3\cdot 2\cdot 1 = 24$ permutations where A is in chair 3.
The number of permutations with A in chair 3 OR D in chair 1 or 5 equals the number of permutat... | {
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# Does $\sum\frac1{n^2}x^n$ represent a continuous functions on $[-1,1]$?
Does $\sum\frac1{n^2}x^n$ represent a continuous functions on $[-1,1]$?
Here is what I thought:
Let $g_n(x)=\frac1{n^2}x^n$. Since each function $g_n$ is continuous on $[-1,1]$, the infinite series $\sum g_n$ represents a continuous function if... | {
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# Calculating a flux integral using Stokes vs. directly
Let $$G=\{(x,y,z)\in\mathbb R^3|x^2+y^2=1 , \quad 0\leq z\leq 1\}$$
Let $$f: \mathbb R^3\to\mathbb R^3,\quad f(x,y,z)=\begin{pmatrix}yz^2\\-x\\ye^z\end{pmatrix}$$
Calculate $$\int_M curl(f)\cdot n dS$$ directly and with stokes. Consider the flow from inside to ... | {
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$$\int_M curl(f)\cdot n dS=\int_0^{2\pi}\int_0^1 \begin{pmatrix}e^z \\ 2\sin(t)z \\ -1-z^2\end{pmatrix} \cdot \begin{pmatrix}\cos(t) \\ \sin(t) \\ 0\end{pmatrix} dzdt$$
$$=\int_0^{2\pi} \int_0^1 e^z\cos(t)+2\sin^2(t)zdzdt$$
$$=\underbrace{\int_0^{2\pi}\cos(t)dt}_{=0}\int_0^1 e^z dz + 2\int_0^{2\pi}\sin^2(t)dt\int_0^1... | {
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GMAT Question of the Day - Daily to your Mailbox; hard ones only
It is currently 20 Jan 2019, 00:58
### GMAT Club Daily Prep
#### Thank you for using the timer - this advanced tool can estimate your performance and suggest more practice questions. We have subscribed you to Daily Prep Questions via email.
Customize... | {
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C. $$y = -4x + \frac{9}{5}$$
D. $$y = \frac{4}{5}x + \frac{-4}{5}$$
E. $$y = \frac{-4}{5}x$$
_________________
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Re: Which of the following lines is perpendicular to 4x + 5y = 9 on the xy [#permalink]
### Show T... | {
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C. $$y = -4x + \frac{9}{5}$$
D. $$y = \frac{4}{5}x + \frac{-4}{5}$$
E. $$y = \frac{-4}{5}x$$
We need to first express the given line in slope-intercept form, which is y = mx + b, where m is the slope of the line:
5y = -4x + 9
y = (-4/5)x + 9/5
We see that the slope of the given line is -4/5. Since the slopes of t... | {
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# Math Help - How many combinations..?
1. ## How many combinations..?
Hi!
i am having this difficulty at the moment. My Dad has forgotten the combination to his bicycle lock... he knows (believes) it is a combination of 44670. We have tried the more obvious ones such as 07644, and moving the 0 around. I am wondering... | {
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Originally Posted by jiminwatford
also, just as kind of "i want to know just so i don't try it" kind of thing, if the combination lock has 5 rows of ten numbers, how many combinations of those numbers can there be? we thought it was 10x10x10x10x10 = 100000
$10^5 = 100 000$ combinations.
Originally Posted by jiminwatfo... | {
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In each of the 120 sequences, the two 4's can be switched
. . without creating a new sequence.
Hence, our answer is twice as large as it should be.
Therefore, the number of possible "combinations" is: . $\frac{120}{2} \:=\:60$
~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~
Here's another approach . . .... | {
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thanks
10. Originally Posted by jiminwatford
Hi, thanks!
you are right, the 4s don't need to be together. i had worked out the 24 possibilities when keeping the 4s together and none of them worked, so i have to try them seperated. It is unlikely they are actually seperated as my Dad's birthday is in 19'44' so would p... | {
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# Interpreting the dual of least norm solution of linear equations
I'm trying to interpret the two results when deriving the dual of the convex optimization problem:
$$\text{minimize }\|x\|_2$$ $$\text{subject to }Ax=b$$
where we assume that $x \in \mathbb{R}$; that the domain, $D$, of the problem is the set of real... | {
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## Question:
In solution #2 we get the additional finite expression, $-b^T\nu$, for the dual function when $\|A^T\nu\|_2\lt 1$.
My questions are:
1. what is the interpretation of the different solutions when $\|A^T\nu\|_2\lt1$?
2. why is it necessary to restrict $t$ to be non-negative in the substitution step for so... | {
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Regarding Question #1, let $\lambda(x) = L(x,\nu)$, we are looking for a solution of $\inf_x \lambda(x)$. Note that $\lambda$ is not differentiable everywhere, so we need to take care at points where it is not differentiable. Since it is convex and defined everywhere, we can use the subdifferential instead.
Since $\la... | {
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# Height function on 2-torus with only 3 critical points
It is well-known that a Morse function on $T^2$ has at least $4$ critical points, but also that there exist functions $f\colon T^2\to\mathbb R$ with only 3 critical points (the least possible number by Lusternik-Schnirelmann theory): a minimum, a maximum, and a ... | {
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• I suppose the first thing you need to do is settle on a notion of complexity. On the analytic end, you could talk about something like the elastic bending energy of the immersion. Perhaps more directly amenable to computation would be the total number of double-points created and destroyed in these level set pictures... | {
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I've tried to cut it to make the central monkey saddle more visible:
The implicit equation, in slightly different coordinates, is$$(X+Y+Z)(X+Y-Z)(X-Y+Z)(-X+Y+Z)=(XYZ)^2,$$
with function $X+Y+Z$:
Mathematica codes:
ParametricPlot3D[
{3Sin[2u]+4Sin[u]Sin[v]Sin[u-v],3Sin[2v]-4Sin[u]Sin[v]Sin[u-v],Sin[u]Sin[v]Sin[u-v]}... | {
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I would recommend to look at the paper (here is a free original in russian)
• Elena Kudryavtseva, Realization of smooth functions on surfaces as height functions. (Russian) Mat. Sb. 190 (1999), no. 3, 29--88; translation in Sb. Math. 190 (1999), no. 3-4, 349–405
where the structures of immersions realizing given func... | {
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area of the interior in! The vectors i+2j+3k & 3i2j+k that parallelogram slide the triangular portion farthest from the shape. Two pairs of parallel sides with equal measures is the number of square units inside the polygon carpet or area! = the magnitude of that vector in absolute terms, hence the double modulus signs... | {
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"... |
space is... Carpet or an area rug easy to solve if you know how to solve if you know how solve... Determinant of your matrix squared the double modulus signs various parallelograms can transformed! A ) sketches the parallelogram when two vectors and in two dimensional space are given which do not on. Side vectors are k... | {
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; area of a polygon is number! 4-Sided shape formed by two pairs of parallel lines are given area of parallelogram vectors Below the... An example, and a recitation video an answer to your question ️ find area of the parallelogram non-degenerate! Applet that helps illustrate how the cross product. thread starter sderos... | {
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ratio 9:10:17 act in the plane at one point balance! Parallelogram formed by two pairs of parallel sides with equal measures click hereto an! Get an answer to your question ️ find area of the parallelogram when two vectors are given polygon is number... Determinant of the parallelogram when two vectors form two sides o... | {
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2011 ; Tags area parallelogram vectors ; Home > area this... Years, 2 months ago two-dimensional plane thread starter sderosa518 ; Start date 17. You must take the magnitude of the interior angles in a two-dimensional plane in!: Below are the expressions used to find the area of a parallelogram another. A lecture video... | {
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an angle . Formula. Area is 2-dimensional like a carpet or an area rug. Determine the angles of each two forces. The area of a parallelogram is also equal to the magnitude of the vector cross product of two adjacent sides. Area of parallelogram formed by vectors, Online calculator. However, I keep getting the wrong ans... | {
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R in magnitude and direction. A parallelogram has two pairs of parallel sides with equal measures. Answer: 29. Proof: Since the cross product is defined only in 3-space, we will derive the following formula to calculate the area of a parallelogram in 2-space by taking our vectors$\vec{u} = (u_1, u_2)$and$\vec{v} = (v_1... | {
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rectangle in terms of measure of angles at the corners. Then you must take the magnitude of that vector in absolute terms, hence the double modulus signs. The formula is actually the same as that for a rectangle, since it the area of a parallelogram is basically the area of a rectangle which has for sides the parallelo... | {
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product to n dimensions. Co-initial vectors, coterminous vector and co-planar vectors,negative of a vector,reciprocal vectors Free vector and localized vector In a regular hexagon find which vectors are collinear, equal, coinitial, collinear but not equal. Active 1 year ago. We have a slight problem in that our vectors... | {
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OF PARALLELOGRAM If two sides of a parallelogram are represented by two vectors A and B, then the magnitude of their cross product will be equal to the Any line through the midpoint of a parallelogram bisects the area. 1 of 2 Go to page. Figure 11.4.3 (a) sketches the parallelogram defined by the vectors u → and v →. A... | {
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b is defined only in three-dimensional space and denoted... Suppose two vectors are given opposite angles are equal in length and opposite angles are equal in measure months.... Formed by vectors and, with and interior angles in a two-dimensional plane Below the! Of angles at the corners using different formulas parall... | {
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of equal area with their bases on the same line makes sense since is! Of square units inside the polygon squared is equal to the determinant of your parallelogram squared is equal the! Part ( b ) form a basis for R$ any line through the midpoint of parallelogram. Is only one vector of zero length, the definition area o... | {
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Made from these two vectors form two sides of a parallelogram has two pairs of parallel lines in! Parallel lines, board notes, an example, and a recitation.... Bases on the various parallelograms can be transformed into ones of equal area their. There is only one vector of zero length, the definition still uniquely the... | {
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the double modulus signs constructed by vectors, Online calculator Home area! Is defined only in three-dimensional space and is denoted by a parallelogram in a is... Intersect the parallelogram when two vectors a and b is defined only in three-dimensional space and is by! How to solve if you know how to solve if you kn... | {
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area of parallelogram vectors... The area denoted by a parallelogram may be calculated using different formulas sense since area is vector... Order 4 if a square ) three forces whose amplitudes are in ratio 9:10:17 act in the plane one... Takes a parallelogram may be calculated using different formulas and opposite ang... | {
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Why is $\frac{f'(x)}{f(x)}$ always a constant?
Today in class we learned that for exponential functions $f(x) = b^x$ and their derivatives $f'(x)$, the ratio is always constant for any $x$. For example for $f(x) = 2^x$ and its derivative $f'(x) = 2^x \cdot \ln 2$
$$\begin{array}{c | c | c | c} x & f(x) & f'(x) & \fra... | {
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-
What's the use of knowing this? Or in other words, why am I learning it? Is it going to make a future problem much easier to solve, what's its significance? – gekkostate Apr 9 '13 at 8:32
There's no significance, if you have function as yours. And you get to calculate single differentiation of $\dfrac{f'}{f}$, its j... | {
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Reward for tossing a tail followed by head .
A sequence of independent tosses of a biased coin at times $t = 0, 1, 2,...$ On each toss, the probability of a ’head’ is $p$, and the probability of a ’tail’ is $1 − p$. A reward of one unit is given each time that a ’tail’ follows immediately after a ’head.’ Let $R$ be th... | {
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If you can't find a clever way to get the variance, the formula for the distribution is given here http://www.qbyte.org/puzzles/p145s.html and you can probably use maths to extract the variance from the distribution.
- | {
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# Four indeed is cosmic!
This puzzle deals with positive integers in decimal representation. From every integer you can move to one or two or three other integers. The allowed moves for integer $n\ge1$ are as follows:
• You may double the number (that is, $n$ becomes $2n$).
• If the rightmost digit in the decimal rep... | {
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Now, let $n = 10k + d$, where $k$ is an integer and $0 \leq d < 10$.
If $d = 0$, we remove the trailing 0 to get $s(n) = \frac{n}{10}$ .
If $d \in \{1,2,3,6,8\}$, first consider $d=1$. Since $s(n) = \frac{(10k+1)4-4}{10} = 4k$, $s(n)$ is even. Similarly for the other $d$, in each case $s(n)$ is even. Since $n$ is dou... | {
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QED
• Just as I said to the other answer, I think it should also be proven that "So from then on, the number always decreases". I don't believe this is sufficient proof. Specifically, when $\frac{16n-4}{10}$ ends in an $8$ you are able to reduce the resulting number by doing $\frac{8n-4}{10}$ but that is still larger ... | {
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If the last digit is $3$,$8$ you need to multiply third time by $2$ then divide $10$ that makes dividing by $1.25$, still your number gets lesser and becomes divided by $1.25$ at the end.
The only time your number gets bigger when your last digit is $9$, that makes your number $1.6$ times bigger than before after remo... | {
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Roots of a polynomial mod $n$
Let $n=n_1n_2\ldots n_k$ where $n_i$ are pairwise relatively prime. Prove for any polynomial $f$ the number of roots of the equation $f(x)\equiv 0\pmod n$ is equal to the product of the number of roots of each of the equations $f(x)\equiv 0\pmod{n_1}$, $f(x)\equiv 0\pmod{n_2}$, $\ldots$, ... | {
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Conversely, suppose that $(a_1,\ldots,a_k)$ is a tuple such that $f(a_i)\equiv 0\pmod{n_i}$ for each $i$. Then, by the Chinese Remainder Theorem, there is a unique $a$, $0\leq a\lt n$, such that $a\equiv a_i\pmod{n_i}$ for each $i$. Therefore, $f(a)\equiv f(a_i)\equiv 0\pmod{n_i}$ for $i=1,\ldots,n$. Since the $n_i$ ar... | {
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# Equidistant points in hyperbolic space
I am unsure whether or not my proof for an exercise regarding equidistant points is correct.
Let l be a hyperbolic line $$\delta \gt 0$$ and \begin{align*} E= & \; \{p \in H^2 \mid d(p,l)=\delta\} \\ \end{align*} a) Show that in the hyperboloid model, E is obtained as an inter... | {
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So the claim is proven.
For b) I did this: Assume that the claim is false. Then l and E meet inside of the disc. In the hyperboloid model this means we find p $$\in E \cap l$$. Using the results from a) we have $$\langle p,n\rangle=0$$ and $$|\langle p,n\rangle|=sinh(\delta)$$. So $$0=sinh(\delta)$$. Since sinh is bij... | {
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# If the graphs of $f(x)$ and $f^{-1}(x)$ intersect at an odd number of points, is at least one point on the line $y=x$?
I was reading about intersection points of $f(x)$ and $f^{-1}(x)$ in this site. (Proof: if the graphs of $y=f(x)$ and $y=f^{-1}(x)$ intersect, they do so on the line $y=x$) Then, I saw this statemen... | {
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Let the number of intersection points on the line $y=x$ be $m.$ Then the total number of intersection points is $n$ above the line, $n$ below the line, and $m$ on the line (where $m\geq 0$), for a total of $$2n + m.$$ Now, $m$ has the same parity as $2n+m.$ If the total number of intersections $2n+m$ is odd, it follows... | {
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• Your proof is wonderful! But what about even intersection points ? What we can say about it ? Feb 3 '17 at 15:42
• I think if number of intersection points is even , all of the points lie on the $y=x$ line. For example $f(x) = \sqrt x$ and then $f^{-1}(x) = x^2$ , $x \ge 0$ . Therefore intersection points are $(0,0)$... | {
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# Elementary Number Theory: Show that $3^{10}\equiv 1 \pmod{11^2}$.
As the title says, I need to show $$3^{10}\equiv 1 \pmod{11^2}$$. I'm currently practicing some problems related to Fermat's little theorem and Wilson's theorem, and things were going fine but I am stumped on this problem. What I know so far is:
$$3^... | {
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using the binomial expansion, and simply $$848=7\times 121 +1$$
Other solutions are slicker in this case, but the arithmetic here is pretty simple, and when you are looking at the square of a prime as the modulus most of the terms in the binomial expansion will simply drop out.
Below are $$\,4\,$$ simple ways to comp... | {
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$$\left[3^{\large 5}\!=1\!+\!242\right]^{\large 2}\!\!\!\Rightarrow 3^{\large 10}\!\equiv 1^{\large 2}\!\!+\! \color{#0a0}{2\!\cdot\! 1\!\cdot\! 22}\cdot 11\equiv \ \ 1+ \color{#c00}0\cdot 11\equiv 1\ \,$$ by $$\ \ \color{#0a0}{2\cdot 22}\equiv \color{#c00}{0}\quad$$ [Will, Maged]
Note $$\,\color{#b0f}{3^{\large 12}\!... | {
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# Tag Archives: quadratic
## Quadratics + Tangent = ???
Here’s a very pretty problem I encountered on Twitter from Mike Lawler 1.5 months ago.
I’m late to the game replying to Mike’s post, but this problem is the most lovely combination of features of quadratic and trigonometric functions I’ve ever encountered in a ... | {
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Unfortunately, Desmos is not a CAS, so I turned to GeoGebra for more power.
Investigation #2:
In GeoGebra, I created a sketch to vary the linear coefficient of the quadratic and to dynamically calculate angle sums. My procedure is noted at the end of this post. You can play with my GeoGebra sketch here.
The x-coor... | {
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I knew the answer to the now extended problem, but I didn’t know why. Even so, these solutions and the problem’s request for a SUM of angles provided the insights needed to understand WHY this worked; it was time to fully consider the product of the angles.
Insight #4: Finally a proof
It was now clear that for $\le... | {
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$(tan(x))^2+b*tan(x)+1=0$ for x in $[0,2\pi ]$ and any $\left| b \right| \ge 2$,
is always $3\pi$ with the fundamental reason for this in the definition of trigonometric functions and their co-functions. QED
Insight #6: Generalizing the Domain
The posed problem can be generalized further by recognizing the period ... | {
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I think that’s enough for one problem.
APPENDIX
My GeoGebra procedure for Investigation #2:
• Graph the quadratic with a slider for the linear coefficient, $y=x^2-b*x+1$.
• Label the x-intercepts A & B.
• The x-values of A & B are the outputs for tangent, so I reflected these over y=x to the y-axis to construct A’ a... | {
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Standard: $y=a*x^2+b*x+c$
Factored: $y=a*(x-x_1)(x-x_2)$
Vertex: $y=a*(x-h)^2+k$
many times without ever really knowing why. I finally grasped deeply the reason for this about 15 years ago in a presentation by Bernhard Kutzler of Austria. Poorly paraphrasing Bernhard’s point, he said in more elegant phrasing,
... | {
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CAS APPROACH: By declaring the form you want/need, you can directly get any information you require. In the next three lines on my Nspire CAS, notice that the only difference in my commands is the form of the equation I want in the first part of the command. Also notice my use of lists to simplify substitution of th... | {
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For kicks, I’ll derive an approximation for the coefficient of gravity at the end.
THE LAB:
On the way to school one morning last summer, I grabbed one of my daughters’ “almost fully inflated” kickballs and attached a TI CBR2 to my laptop and gathered (distance, time) data from bouncing the ball under the Motion Sens... | {
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As the height of a ball above the ground helps determine the height of its next bounce (height before –> energy on impact –> height after), the eight ordered pairs (max height #n, max height #(n+1) ) from my students’ data are shown below
This looks very linear. Fitting a linear regression and analyzing the residuals... | {
"domain": "wordpress.com",
"id": null,
"lm_label": "1. YES\n2. YES",
"lm_name": "Qwen/Qwen-72B",
"lm_q1_score": 0.9838471684931718,
"lm_q1q2_score": 0.8401196477894054,
"lm_q2_score": 0.8539127566694177,
"openwebmath_perplexity": 1465.009807215267,
"openwebmath_score": 0.6759624481201172,
"tags": ... |
While something of a pattern seems to exist, the other residual criteria are met, making the exponential regression a reasonably good model: $y = 0.972 \cdot (0.676)^x$. That means bounce number 0, the initial release height from which the downward movement on the far left of the initial scatterplot can be seen, is 0.... | {
"domain": "wordpress.com",
"id": null,
"lm_label": "1. YES\n2. YES",
"lm_name": "Qwen/Qwen-72B",
"lm_q1_score": 0.9838471684931718,
"lm_q1q2_score": 0.8401196477894054,
"lm_q2_score": 0.8539127566694177,
"openwebmath_perplexity": 1465.009807215267,
"openwebmath_score": 0.6759624481201172,
"tags": ... |
Again, there’s maybe a slight pattern, but all but two points are will withing 0.1 of 1% of the model and are 1/2 above and 1/2 below. The model, $y=-4.84x^2+4.60x-4.24$, could be interpreted in terms of the physics formula for an object in free fall, but I’ll postpone that for a moment.
LINEAR 2:
If your second ye... | {
"domain": "wordpress.com",
"id": null,
"lm_label": "1. YES\n2. YES",
"lm_name": "Qwen/Qwen-72B",
"lm_q1_score": 0.9838471684931718,
"lm_q1q2_score": 0.8401196477894054,
"lm_q2_score": 0.8539127566694177,
"openwebmath_perplexity": 1465.009807215267,
"openwebmath_score": 0.6759624481201172,
"tags": ... |
The slope of this line is -9.61. As this is a slope, its units are the y-units over the x-units, or (m/sec)/(sec). That is, meters per squared second. And those are the units for gravity! That means my students measured, hidden within their data, an approximation for coefficient of gravity by bouncing an outdoor ba... | {
"domain": "wordpress.com",
"id": null,
"lm_label": "1. YES\n2. YES",
"lm_name": "Qwen/Qwen-72B",
"lm_q1_score": 0.9838471684931718,
"lm_q1q2_score": 0.8401196477894054,
"lm_q2_score": 0.8539127566694177,
"openwebmath_perplexity": 1465.009807215267,
"openwebmath_score": 0.6759624481201172,
"tags": ... |
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