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Suppose $$X_1,X_2,\ldots,X_n$$ are i.i.d $$N(\mu,\sigma^2)$$ and define the sample variance as $$S^2=\frac{1}{n}\sum\limits_{i=1}^n (X_i-\overline X)^2$$
Here we are concerned with the large-sample behaviour of $$S^2$$, so it does not matter if we take $$n$$ as the divisor above instead of $$n-1$$.
By CLT it can be argued that $$\sqrt n(S^2-\sigma^2)\stackrel{L}\longrightarrow N(0,2\sigma^4)\tag{1}$$
And by 'Delta-method', for a real-valued function $$g$$ such that $$g'(\sigma^2)$$ exists and $$g'(\sigma^2)\ne 0$$, it follows from $$(1)$$ that
$$\sqrt n(g(S^2)-g(\sigma^2))\stackrel{L}\longrightarrow N(0,2\sigma^4[g'(\sigma^2)]^2)\tag{2}$$
You have to solve for a $$g$$ such that asymptotic variance of $$g(S^2)$$ is free of $$\sigma^2$$.
So for some non-zero constant $$c$$, set $$\sqrt 2\sigma^2 g'(\sigma^2)=c$$
Therefore, $$\int g'(\sigma^2)\,d\sigma^2=\frac{c}{\sqrt 2}\int\frac{d\sigma^2}{\sigma^2}$$
Choosing $$c=\sqrt 2$$ and taking constant of integration to be zero, we have the required transformation $$g(\sigma^2)=\ln\sigma^2\quad,\text{ i.e. }\quad \color{blue}{g(S^2)=\ln S^2}$$
From $$(2)$$ you end up with $$\sqrt n(\ln(S^2)-\ln(\sigma^2))\stackrel{L}\longrightarrow N(0,2)$$
This is the application of variance stabilizing transformation on the sample variance from a normal population. The other transformations that you mention for the Poisson mean and Binomial proportion are derived similarly.
Result $$(1)$$ and many more can be found in A Course in Large Sample Theory by Thomas Ferguson. The relevant part is on page 46 of the first edition (1996):
The main result used in this section is Cramer's Theorem on the asymptotic normality of functions of sample moments, studied through a Taylor-series expansion to one term. | {
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• what I know is CLT tells us $\color{red}{\sqrt{n}(\overline{x}-\mu)\text{ converge in distribution to a normal }N(0,\sigma^2)}$ And I know how to prove this but $\sqrt n(s^2-\sigma^2)\stackrel{L}\longrightarrow N(0,2\sigma^4)$ is looks new to me. So everything will be meaningful to me if you provide a source where I can get that proof $?$ I will accept your answer then. Thanks again @StubbornAtom – emonhossain Nov 6 '19 at 16:07
• @emonhossain Updated. – StubbornAtom Nov 6 '19 at 18:07
• Really you help me a lot. Thanks @StubbornAtom – emonhossain Nov 6 '19 at 18:33 | {
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# Putnam Problem of the Day
• Aug 12th 2010, 10:20 PM
simplependulum
Putnam Problem of the Day
Find the smallest natural number with $6$ as the last digit , such that if the final $6$ is moved to the front of the number it is multiplied by $4$ .
Spoiler:
153846
?
• Aug 13th 2010, 03:26 AM
Unbeatable0
Let the number be $a_1a_2...a_n6$.
Let $x=0.\overline{a_1a_2...a_n6}$.
Then $4x = 0.\overline{6a_1a_2...a_n} \Rightarrow 40x = 6.\overline{a_1a_2...a_n6} = 6+x$.
Therefore $x = \frac{6}{39} = \frac{2}{13} = 0.\overline{153846}$.
By backwards argument it's easy to see that all the cycles in $\frac{2}{13}$ are solutions.
Thus we have found all the solutions, the smallest of which is $153846$.
This technique can be applied to this kind of problems in many cases.
• Aug 13th 2010, 02:03 PM
Soroban
Hello, simplependulum!
Just write out the multiplcation,
. . and the digits will appear sequentially.
Quote:
Find the smallest natural number with $6$ as the last digit,
such that if the final $6$ is moved to the front of the number it is multiplied by $4$.
We have: . $\begin{array}{cccccc}
^1 & ^2 & ^3 & ^4 & ^5 & ^6 \\
A&B&C&D&E&6 \\ \times &&&&& 4\\ \hline 6&A&B&C&D&E \end{array}$
In column-6: . $6 \!\times\! 4 \,=\,24 \quad\Rightarrow\quad E = 4$
. . $\begin{array}{cccccc}
^1 & ^2 & ^3 & ^4 & ^5 & ^6 \\
A&B&C&D&4&6 \\
\times &&&&& 4\\ \hline
6&A&B&C&D&4 \end{array}$
In column-5: . $4 \!\times\! 4 + 2 \:=\:18 \quad\Rightarrow\quad D = 8$
. . $\begin{array}{cccccc}
^1 & ^2 & ^3 & ^4 & ^5 & ^6 \\
A&B&C&8&4&6 \\
\times &&&&& 4\\ \hline
6&A&B&C&8&4 \end{array}$
In column-4: . $8 \!\times\! 4 + 1 \:=\:33 \quad\Rightarrow\quad C = 3$
. . $\begin{array}{cccccc}
^1 & ^2 & ^3 & ^4 & ^5 & ^6 \\
A&B&3&8&4&6 \\
\times &&&&& 4\\ \hline
6&A&B&3&8&4 \end{array}$
In column-3: . $3\!\times\!4 + 3 \:=\:15 \quad\Rightarrow\quad B = 5$
. . $\begin{array}{cccccc}
^1 & ^2 & ^3 & ^4 & ^5 & ^6 \\
A&5&3&8&4&6 \\
\times &&&&& 4\\ \hline
6&A&5&3&8&4 \end{array}$ | {
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In column-2: . $5\!\times\!4 + 1 \:=\:21 \quad\Rightarrow\quad A = 1$
Therefore: . $\begin{array}{cccccc}
1&5&3&8&4&6 \\
\times &&&&& 4\\ \hline
6&1&5&3&8&4 \end{array}$
• Aug 13th 2010, 05:19 PM
Wilmer
Soroban, the fact that it is a 6digit number is NOT a given.
• Aug 13th 2010, 06:16 PM
Soroban
Quote:
Originally Posted by Wilmer
Soroban, the fact that it is a 6-digit number is NOT a given. . Right!
I originally set up the problem like this:
. . $\begin{array}{ccccccccccc}
A & B & C & D & E & F & G & \hdots & M & N & 6 \\
\times &&&&&&&&&& 4 \\ \hline
6 & A & B & C & D & E & F & G & \hdots & M & N
\end{array}$
And found that "it came out even" after six digits
. . which gave me the smallest number: $153,\!846.$
The next is the 12-digit number: $153,\!846,\!153,\!846.$
• Aug 13th 2010, 07:10 PM
undefined
Fyi, there is a name for these, n-parasitic number, where here n=4 and k=6.
• Aug 13th 2010, 09:32 PM
Wilmer
Let n = number of digits
{20[10^(n-1) - 4] + 78} / 13
Gives integer solution for n=6k where k = any integer > 0 | {
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# Circle questions -- Secants, chords and tangents
• Apr 3rd 2014, 05:03 PM
StonerPenguin
Circle questions -- Secants, chords and tangents
Hello again, just got a few questions pertaining to circles. I'll just post one to start with;
"The radius of the earth is approximately 6371 km. If the international space station (ISS) is orbiting 353 km above the earth, find the distance from the ISS to the horizon (x).
http://i1137.photobucket.com/albums/...mexam10Q32.png
So solving this.. according to the segments of secants and tangents theorem...
x2 = (2r + 353)(353)
x2 = (12742 + 353)(353)
x2 = 4622535
x = $\sqrt{4622535}$ = 2150km (approx.)
Did I do that right?
• Apr 3rd 2014, 07:13 PM
JeffM
Re: Circle questions -- Secants, chords and tangents
Quote:
Originally Posted by StonerPenguin
Hello again, just got a few questions pertaining to circles. I'll just post one to start with;
"The radius of the earth is approximately 6371 km. If the international space station (ISS) is orbiting 353 km above the earth, find the distance from the ISS to the horizon (x).
http://i1137.photobucket.com/albums/...mexam10Q32.png
So solving this.. according to the segments of secants and tangents theorem...
x2 = (2r + 353)(353)
x2 = (12742 + 353)(353)
x2 = 4622535
x = $\sqrt{4622535}$ = 2150km (approx.)
Did I do that right?
Computation looks right to me. But simply memorizing theorems does not promote understanding.
Let's see how we get that theorem. The radius through a point of tangency and the tangent at the same point are perpendicular.
$Let\ r = length\ of\ radius\ of\ given\ circle,$
$u + r = length\ from\ the\ given\ circle's\ center\ to\ a\ given\ point\ outside\ the\ circle,$
$x = length\ of\ the\ line\ that\ is\ tangent\ to\ the\ given\ circle\ and\ runs\ through\ the\ given\ point.$
By the Pythagorean Theorem, we have: | {
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By the Pythagorean Theorem, we have:
$(u + r)^2 = x^2 + r^2 \implies u^2 + 2ru + r^2 = x^2 + r^2 \implies x^2 = u^2 + 2ru = u(u + 2r) \implies x = \sqrt{u(u + 2r)}.$
• Apr 6th 2014, 06:17 PM
bjhopper
Re: Circle questions -- Secants, chords and tangents
ISS is6724 km above center of earth.At that point the angle between straight down and the visible horizon has a sin of 6371/6724 or 71.35 deg
cos 71.35 =d/6724
d km to horizon =2150 km
• Apr 6th 2014, 09:56 PM
Soroban
Re: Circle questions -- Secants, chords and tangents
Hello, StonerPenguin!
Quote:
The radius of the earth is approximately 6371 km.
If the international space station (ISS) is orbiting 353 km above the earth,
find the distance from the ISS to the horizon (x).
Code:
o |\ | \ 352 | \ | \ x | \ * * * \ * | *\ * | o * 6371| * * | *6371 * | * * * o * * * * * * * * * * * *
Note the right triangle.
The equation is: . $x^2 + 6371^2 \:=\:6723^2$
• Apr 7th 2014, 05:16 PM
StonerPenguin
Re: Circle questions -- Secants, chords and tangents
Thank you JeffM, bjhopper and Soroban! It's nice to see different perspectives :D and the image drawn in code is really cool.
Here's another question I've had trouble with:
http://i1137.photobucket.com/albums/...mexam10Q15.png
"Explain how you know $\overline{AB}$ $\overline{CD}$ given E is the center of the circle. (Include theorem numbers.)"
And here's some pertinent theorems;
Quote:
Theorem 10.4
If one chord is a perpendicular bisector of another chord, then the first chord is a diameter.
Quote:
Theorem 10.5
If a diameter of a circle is perpendicular to a chord, then the diameter bisects the chord and its arc.
Quote: | {
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Theorem 10.6
In the same circle, or in congruent circles, two chords are congruent if and only if they are equidistant from the center.
Obviously from the diagram $\overline{AB}$ $\overline{CD}$ by theorem 10.6, but I can't really word this well. Any help? Theorems and proofs are my weakest areas :/
• Apr 8th 2014, 04:24 PM
bjhopper
Re: Circle questions -- Secants, chords and tangents
t is a triangle a is an angle
t AED congruent to BEC isosceles t's same legs and equal altitudes
a DEC = 180- 2* 1 / 2 *AED
a AEB 180-2*1/2 * BEC a AED = aBEC
a AEB = a DEC
AB =DC equal arcs = equal chords | {
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# Tag Info
7
Since $fg\geq 1$, you have $\sqrt{f(x)\cdot g(x)} \geq 1$ for all $x$, and so $$1 \leq \int_X \sqrt{f}\cdot \sqrt{g} d\mu \leq \sqrt{\int_X f d\mu }\cdot \sqrt{\int_X g d\mu }$$ where the second inequality is Cauchy–Schwarz; squaring both sides gives the inequality.
6
It is sufficient to show that $m(E \cap [a, b]) = 0$ for every $a$ and $b$ with $a < b$. Let $t = \inf_{x \in [a, b]}{e^{-x^2/2}}$. Then $t > 0$ and we have $$0 = \int_{E}e^{-x^2/2}dx \ge \int_{E\cap [a, b]}e^{-x^2/2}dx \ge \int_{E\cap [a, b]}tdx = m(E \cap [a, b])t$$ and as $t > 0$, this implies $m(E \cap [a, b]) = 0$.
5
A measure space must specify three things: An underlying set; A $\sigma$-algebra of subsets which are measurable in this measure space; A measure (a function on this $\sigma$-algebra). When we say "a measurable subset of the real numbers" we typically mean "a subset of the real numbers measurable with respect to the Lebesgue measure", ...
5
The questions, despite looking as a representation problem in functional analysis, are much deeper as they bring out the history of the topic involved, notably $BV$-functions and the reasons why the customary definition adopted for the variation of a multivariate function is definition 2 above. And as thus the answers below needs to indulge a bit on this ...
5
$\bigcup_{i\ge 1}A_i$ is by definition simply $$\left\{x:\exists i\in\Bbb Z^+\,(x\in A_i)\right\}\,,$$ which clearly does not depend on the order in which the sets are enumerated. You can simply let $\mathscr{A}=\{A_i:i\in\Bbb Z^+\}$; then $$\bigcup_{i\ge 1}A_i=\bigcup\mathscr{A}=\{x:\exists A\in\mathscr{A}\,(x\in A)\}\,,$$ with no reference to the ...
5
Hint: Consider the sets $[-r,r] \subseteq \mathbb{R}$. Can you show $r \mapsto \mu(E \cup [-r,r] \cap F)$ is continuous? Can you show for any $r$ the set we're measuring is compact? What does the intermediate value theorem buy us? I hope this helps ^_^
5 | {
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5
This is false. For an easy counterexample, let $X_n\geq 0$ be any sequence of random variables with $\liminf_n X_n = 0$ which converges to $X = 1$ in probability. Then let $\mathcal{Q}$ be large enough so that the $X_n$ are $\mathcal{Q}$-measurable. The inequality in question then becomes $X \leq \liminf_n X_n,$ which is false almost surely. For instance, ...
5
It's true that the definition of absolute continuity is the same either we state it in the form of the finite and disjoint intervals or we state it in the form of countable and disjoint intervals. In detail, suppose we have the following definitions of absolute continuity: Definition 1 Let $f:\mathbb{R}\to \mathbb{R}$ be a function. We say that $f$ is ...
4
By assumption, as $Z \notin L^{\infty}$, we have $$\mathbb{P}(Z > n^2) > 0, \quad n \in \mathbb{N}.$$ Therefore, we may define $$X = \sum_{n=1}^{\infty}\frac{1}{n^2} \frac{\mathbb{1}_{[Z > n^2]}}{\mathbb{P}(Z>n^2)} \in L^1(\mathbb{P}).$$ However, it follows that \begin{align*}\mathbb{E}[ZX] = \sum_{n=1}^{\infty}\mathbb{E}\left[\frac{1}{n^2} \frac{...
4
Cannot be true in general: consider $k=1$ (or any odd $k$), take $X_i$'s to be i.i.d normal with mean $0$ and variance $1$, we show below that $Y_n$ is not UI. Note that $$\mathbb{E}\left(|Y_n| 1\{|Y_n|\geq M\}\right) \geq M\mathbb{P}\left(|Y_n|\geq M\right) = M\mathbb{P}\left(|\bar{X}_n|\leq \frac{1}{M}\right) \to M,$$ as $n\to \infty$ by laws of large ...
4
The assertion “not every subset of $\Bbb R$ is measurable” is valid for the Lebesgue measure. Anyway, here Axler is talking about $\sigma$-algebras, not about measures. And, yes, $\mathcal P(\Bbb R)$ is a $\sigma$-algebra. And, on this $\sigma$-algebra you can define the measure $m(X)=\#X$; with respect to this measure, $\Bbb R$ is measurable.
4
If $p > p_0$ and $f \in L^{p_0} \cap L^\infty$ then $|f|^p = |f|^{p - p_0}|f|^{p_0} \le \|f\|_\infty^{p - p_0}|f|^{p_0}$ almost everywhere.
4 | {
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The claim is true: Let $S = \sum_{i=1}^\infty \chi_{E_i}$. Suppose for the sake of contradiction that $S < \infty$ a.e. Then, there exists an $M$ such that $\Pr(S\le M) \ge 1 - \frac\varepsilon 2$. If $A$ is the event that $S\le M$, then $\mu(A\cap E_i) \ge \frac\varepsilon 2$ for all $i$. But this would imply $\int_A S\,d\mu = \sum_{i=1}^\infty \mu(A\... 4 Well,$[0,1]=C\hspace{1mm}\cup C^{c} $(I discourage you to use$\mathbb{C}$for the cantor set) and you can write f as: $$f(x)=0\cdot\chi_{C}(x)+x^{2}\chi_{C^{c}}$$ where by$\chi$I mean the characteristic function of the set in subscript. The sum of measurable functions is measurable as well as the product, and polynomial functions are measurable. 4 The family of measurable sets form a$\sigma$-algebra but not all sets belonging in any$\sigma$-algebra are measurable. So to fully define a measure space we need a set$X$, a$\sigma$-algebra$\mathcal{A}$on$X$and a function$\mu: \mathcal{A} \to [0, \infty]$which satisfies some properties. A set$A \subseteq X$is called$\mu$-measurable (or simply ... 4 The symmetrization is called Khintchine's inequality, see here: https://en.wikipedia.org/wiki/Khintchine_inequality For the contraction principle, I don't know any reference, but I can prove it. First note that if$Z\geq 0$and$p > 0, then \begin{align} \mathbb{E}[Z^p] = \mathbb{E}\Big[\int_0^{Z} p t^{p-1} dt\Big] = \mathbb{E}\Big[\int_0^{\infty} p t^{p-... 4 A set\mathfrak B$satisfies the definition in your book if and only if$\mathfrak B$is a$\sigma$-algebra (in the wikipedia sense) on$\cup\mathfrak B$: for the noteworthy algebraic detail, notice that$A\setminus B=A\triangle (A\cap B)$. This is also equivalent to the property of$\mathfrak B$being a$\sigma$-algebra in the wikipedia sense over some set.... 4 Let me discuss the case$n=1$only. Let me denote by$f_x$the weak derivative of$f$with respect to$x$. Let$\phi,\psi$be two smooth test functions. Then $$-\iint f_x(x,y) \phi(x)\psi(y)dx\ dy=\iint f(x,y) \phi_x(x)\psi(y)dx\ | {
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smooth test functions. Then $$-\iint f_x(x,y) \phi(x)\psi(y)dx\ dy=\iint f(x,y) \phi_x(x)\psi(y)dx\ dy,$$ using Fubini on both sides gives $$\int \left( \int -f_x(x,y) \phi(x) + f(x,y) \phi_x(x)\ dx\right) \psi(y) \ dy = 0$$ ... 3 You prove that by giving an example of a set of subsets that is closed under intersections and differences but not closed under unions. E.g. on any set$X$with at least two points, the family$\{\{x\}: x \in X\} \cup \{\emptyset\}$is such. so being closed under union and differences (a ring) implies being closed under intersections, but being closed under ... 3 I have recently got interested in continued fractions, so I'll give it a try! CONTINUED FRACTIONS. It is a basic property of continued fractions that if$h_n/k_n$is the sequence of convergent to$x\in\mathbb R\setminus\mathbb Q$, then $$\frac 1{k_n(k_n+k_{n+1})}<\left|x-\frac{h_n}{k_n}\right|<\frac 1{k_nk_{n+1}}.$$ It follows that for all$h,k$... 3 The set$\mathcal{P}(X)$of all subsets of$X$is a sigma algebra because it satisfies the axioms for that kind of structure. That is independent of any discussion of measures. The definition of a measure space is a pair$(X,S)$where$S$is some subset of$\mathcal{P}(X)$that happens to be a sigma algebra. It may or may not be all of$\mathcal{P}(X)$. ... 3 Suppose that such measure exists. Then taking countable cover by open sets of diameter$\leq 1$, one of those sets, say$U_1$, must have measure$1$. Cover$U_1$by countable amount of open sets of diameter$\leq 1/2$and take the one having measure$1$. This process gives us a decreasing sequence of open sets$U_k$with diameter$\leq 1/k$and$\mu(U_k) = 1$... 3 This works. Your argument shows that if$f$is bounded on a set$E$of Lebesgue measure zero, then$\int_E f = 0$. In fact the stronger claim that$\int_E f = 0$without any hypothesis on$f$holds! Indeed the integral is, by definition, the supremum over$g$with$g$simple and$g \leq f$of$\int_E g$. But any such$g$has integral zero. 3 Welcome to MSE! Hint: It's | {
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and$g \leq f$of$\int_E g$. But any such$g$has integral zero. 3 Welcome to MSE! Hint: It's often useful to split up a random variable into the places where it is "good" and "bad". Then we can control those regions separately to get whatever inequality we're interested in. For this, we might try looking at the decomposition $$X = X \cdot \mathbf{1}_{\{X \leq \lambda a\}} + X \cdot \mathbf{1}_{\{X >... 3 According to the p-Adics section of the reference manual there is nothing I could find to do with zeta functions nor integration. At any rate, here's a suggestion on algorithmically getting an arbitrary level of accuracy on your integral. In order to be called a p-adic measure we require it to be a bounded p-adic distribution, and this is enough to imply ... 3 By definition, \int f is the supremum of \int h for 0 \leq h \leq f bounded measurable of finite support. So at the very end, you just need to take the sup over such h to get \int f \leq \liminf \int f_n. He uses such h is order to apply the bounded dominated convergence theorem, which he proves earlier. This allows him to prove Fatou in terms of ... 3 You can also let g_n=inf_{i \geq n} \{f_n\} and note that \int g_n \leq \int f_n for each n. Moreover g_n increases monotonically to lim inf f_n . Hence by monotone convergence theorem \int g_n = \int lim inf f_n. Since \int g_n \leq \int f_n we have \int lim inf f_n \leq lim inf \int f_n. The final step is to note that pointwise convergence ... 3 If you are familiar with dominated convergence notice that f_n(X)\xrightarrow{n\rightarrow\infty} X pointwise, |f_n(X)|\leq |X|, X is assumed to be integrable (E[|X|]=\int |X|\,dP<\infty). Then by dominated convergence$$ \int f_n(X)\,dP\xrightarrow{n\rightarrow\infty}\int X\,dP=E[X]$$similarly for$Y$. There is still the issue of the ... 3 Let$f$be the limit of$\chi_{E_n}$in$L^p$. All you have to do is show that$f(x) = 0$or$f(x) = 1$almost everywhere. Then we see that$f = \chi_E$almost everywhere, where$E = \{x \in X | | {
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= 1$almost everywhere. Then we see that$f = \chi_E$almost everywhere, where$E = \{x \in X | f(x) = 1\}$To do this, it is helpful to define$g(x) = \min(|x - 1|, |x|)$. We will show that$g \circ f = 0$almost everywhere. We first show that$\int g(f(x))^p \... | {
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3
Let $f \geq 0$ and let $M=\sup \{f(x): x\in X\}$. Then $\|M-f\|\leq M$ since $f(x) \in [0,M]$ for all $x$. If $\phi(f)$ is real we can finish the proof as follows: $M-\phi (f)=\phi(M-f) \leq \|\phi\| \|M-f||\leq M$. Thus $\phi (f) \geq 0$. To show that $\phi (f)$ is real consider following: \$|\phi (f) \pm in|=|\phi(f \pm in)|\leq \|f\pm in\|\leq \sqrt {M^{...
Only top voted, non community-wiki answers of a minimum length are eligible | {
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Calculating probability of 'at least one event occurring'
If I know the probability of A and the probability of B how can I calculate the probability of "at least one of them" occurring?
I was thinking that this is P(A or B) = P(A) + P(B) - P(AB). Is this correct?
If it is, then how can I solve the following problem taken from 'DeGroot - Probability and Statistics':
If 50 percent of families in a certain city subscribe to the morning newspaper, 65 percent of the families subscribe to the afternoon newspaper, and 85 percent of the families subscribe to at least one of the two newspapers, what proportion of the families subscribe to both newspapers?
Here the question is P(morning)=.5, P(afternoon)=.65 P(morning OR afternoon)=.85
P(morning OR afternoon) = .5 + .65 - .3 = .85
P(morning AND afternoon) = P(morning) + P(afternoon) - P(morning OR afternoon)
So the answer to the question is .3
Is my reasoning correct?
EDIT.
If this reasoning is correct:
How can I calculate the following:
If the probability that student A will fail a certain statistics examination is 0.5, the probability that student B will fail the examination is 0.2, and the probability that both student A and student B will fail the examination is 0.1,what is the probability that exactly one of the two students will fail the examination?
So this questions highlights the difference between !at least one of them! !or exactly one of them!
I understand that at least one of them = P(A or B)
but how can I work out the probability of exactly one of them?
-
Somebody knows an algorith to calculate the OR for n probabilistic values? Thanks in advance. Regards, Daniel. – Daniel Mejia Nov 8 '12 at 23:53
You need to specify the correlation between the $n$ probabilistic values - at least whether they are independent or not. – dexter04 Nov 9 '12 at 0:25
You are correct.
To expand a little: if $A$ and $B$ are any two events then
$$P(A\textrm{ or }B) = P(A) + P(B) - P(A\textrm{ and }B)$$ | {
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$$P(A\textrm{ or }B) = P(A) + P(B) - P(A\textrm{ and }B)$$
or, written in more set-theoretical language,
$$P(A\cup B) = P(A) + P(B) - P(A\cap B)$$
In the example you've given you have $A=$ "subscribes to a morning paper" and $B=$ "subscribes to an afternoon paper." You are given $P(A)$, $P(B)$ and $P(A\cup B)$ and you need to work out $P(A\cap B)$ which you can do by rearranging the formula above, to find that $P(A\cap B) = 0.3$, as you have already worked out.
-
thanks for your answer! please have a look at my edited question, your help highlighted a further issue.. – Dbr Oct 14 '11 at 11:10
"Exactly one of A and B" means "Either A or B, but not both" which you can calculate as P(A or B) - P(A and B). – Chris Taylor Oct 14 '11 at 11:13
thank you, this clarifies everything. Just one last quick thing: Where can I learn the mathematical notation you use on this forum? For some questions I find it quite hard to understand them, is there a tutorial? – Dbr Oct 14 '11 at 11:15
Are you asking about the notation itself, or the method of displaying the notation? To write the notation we use $\LaTeX$ - you can find a tutorial by searching for "latex tutorial" in Google. Here's one, for example. If you want to learn the notation itself, the best way is learning by doing. You should read a mathematics text that's appropriate for your level, and make sure you understand all the notation used there. As you read more complex texts, you will become more and more familiar with the notation. – Chris Taylor Oct 14 '11 at 11:21
So if I download LaTeX and paste your notation then it displays it in a more readable form? – Dbr Oct 14 '11 at 11:24
For your second question, you know $\Pr(A)$, $\Pr(B)$, and $\Pr(A \text{ and } B)$, so you can work out $\Pr(A \text{ and not } B)$ and $\Pr(B \text{ and not } A)$ by taking the differences. Then add these two together.
Alternatively take $\Pr(A \text{ or } B) - \Pr(A \text{ and } B)$.
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Alternatively take $\Pr(A \text{ or } B) - \Pr(A \text{ and } B)$.
-
For the additional problem: probability of exactly one equals probability of one or the other but not both, equals probability of union minus probability of intersection, equals $$P(A)+P(B)-2P(A\cap B)$$
-
probability of only one event occuring is as follows: if A and B are 2 events then probability of only A occuring can be given as P(A and B complement)= P(A) - P(A AND B )
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# Calculating dice probabilities with multiple, overlapping success criteria
Alright, so I play a tabletop game where models get in fights and roll dice to see if they deal damage to the other models. I'm wondering how to calculate the probabilities of dealing 0, 1, and 2 damage in a 2v2 fight.
Here is the situation:
- We have Model A and Model B rolling to deal damage to Model C and Model D
- A needs to roll a 5 or 6 to damage C, and a 6 to damage D
- B needs to roll a 4, 5, or 6 to damage C, and a 5 or 6 to damage D
- C and D can only be damaged once each (i.e., they die when they get damaged)
So for starters, I know the probabilities for this scenario just by listing out the possibilities:
A | 111111 222222 333333 444444 555555 666666
B | 123456 123456 123456 123456 123456 123456
----+------------------------------------------
Dmg | 000111 000111 000111 000111 111122 111222
So the probability of dealing 0 damage is $$\frac{12}{36}$$, (exactly) 1 damage is $$\frac{19}{36}$$, and 2 damage is $$\frac{5}{36}$$. One tricky outcome here is when A rolls a 5 and B rolls a 4. This only results in 1 damage since both rolls are only enough to damage C and C can only be damaged once.
I am hopelessly stumped on how to model this so that I can then apply it to other scenarios such as 3v3s or when some models use more than 1 die.
It is pretty easy to calculate the probabilities of doing damage in a 1v1 fight. I've used binomial distribution statistics to calculate those. Conveniently it works for multiple dice per model as well. However, when it comes to combining these probabilities ... I am at a loss. They don't seem to follow the rules for multiplication and addition.
For example, I naïvely expected the probability of doing 0 damage to be the probability of A doing 0 damage and the probability of B doing no damage. That sounds right intuitively, but no matter how I try to do it ... it never makes sense: | {
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$$P(A\;miss\;C) = \binom{1}{0} \bullet 2^0 \bullet 4^1 = 4$$
$$P(A\;miss\;D) = \binom{1}{0} \bullet 1^0 \bullet 5^1 = 5$$
$$P(B\;miss\;C) = \binom{1}{0} \bullet 3^0 \bullet 3^1 = 3$$
$$P(B\;miss\;D) = \binom{1}{0} \bullet 2^0 \bullet 4^1 = 4$$
Therefore,
$$P(A\;miss\;C) \bigcap P(A\;miss\;D) \bigcap P(B\;miss\;C) \bigcap P(B\;miss\;D) = 4 \bullet 5 \bullet 3 \bullet 4 = 240$$
Wait what?!?! I expect the answer to be 12. Clearly there is something else going on here that I do not understand. :/
• You can't say $\Pr(X\cap Y)=\Pr(X)\Pr(Y)$ unless $X$ and $Y$ are independent events, and these are clearly not. When A doesn't damage C it's likely that he doesn't damage D, either, because he rolled a low number. Mar 16 '20 at 20:56
• Also, you're multiplying counts instead of probabilities. The product of the probabilities (even if you could just multiply them, i.e. if they were independent, which they aren't) would be $\frac46\cdot\frac56\cdot\frac36\cdot\frac46=\frac{240}{6^4}=\frac5{27}\approx0.185$, which is at least a lot closer to the $\frac{12}{6^2}\approx0.333$ that you were apparently expecting than $\frac{240}{6^2}\approx6.67$ would have been. Mar 16 '20 at 21:44
First I'd like to rephrase your question:
A and B each roll a die. The results of the die determine how much damage is dealt. The maximum{damage dealt by A, damage dealt by B} is the number we are interested in.
A Deals 1 damage by rolling a 5 or 2 damage by rolling a 6. B deals 1 damage by rolling a 4 or 2 damage by rolling a 5 or 6. Now take the maximum of the damage dealt by A or B.
I think it may be easier to frame it this way:
A misses and B Misses simultaneously:
P(0 Damage) = P(A Rolls 4 or less) * P(B rolls 3 or less) = 2/3*1/2= 1/3 = 3/9
A Hits both or B hits Both (don't double count):
P(2 Damage) = P(A Rolls 6) + P(B Rolls 5 or 6) - P(A Rolls 6 AND B rolls 5 or 6) = 1/6 + 1/3 - 1/6*1/3 = 4/9
P(1 Damage) = 1-P(0)-P(2) = 2/9
Alternatively,
A hits 1 (rolls 5) or B hits 1 (rolls 4) | {
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P(1 Damage) = 1-P(0)-P(2) = 2/9
Alternatively,
A hits 1 (rolls 5) or B hits 1 (rolls 4)
P(1 Damage) = P(B Rolls 4)*P(A Rolls 5 or less) + P(B Rolls 3 or less)*P(A Rolls 5) = 1/6*5/6+ 3/6*1/6= 2/9
How I verified my answer: The table below shows how much damage occurs when (A rolls, B rolls) happens. Count the number of 0s and divide by 36 to determine how often 0 damage occurs. Repeat for 1s and 2s.
## | 6 | 2 | 2 | 2 | 2 | 2 | 2 |
Summary:
P( 0 damage ) = 3/9
P( 1 damage ) = 2/9
P( 2 damage ) = 4/9 | {
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# Is there a quicker proof to show that $2^{10^k} \equiv 7 \pmod{9}$ for all positive integers $k$?
I noticed this pattern while playing with digit sums and noticed that the recursive digit sums (until you arrive at a single digit number) of numbers like, $2^{10}$, $2^{100}$, $2^{1000}$ and so on is always $7$. So, I decided to find out if it is true that for all positive integers $k$,
$$2^{10^k} \equiv 7 \pmod{9}$$
My proof is as follows:
Lemma 1. $\, 10^k \equiv 4 \pmod{6}$ for all integers $k \geq 1$
Proof. For all integers $k \geq 1$, the number $10^k + 2$ must be divisible by $6$ since it is even (implying divisibility by $2$), and its digit sum is $3$ (implying divisibility by $3$). Therefore, you can show that \begin{align*} 10^k + 2 &\equiv 0 \pmod{6} \\ 10^k &\equiv 4 \pmod{6} \,\, \text{ for all integers } k \geq 1 \end{align*}
Lemma 2. $\, 2^{4 + 6k} \equiv 7 \pmod{9}$, for all integers $k \geq 0$
Proof. If $a \equiv c \pmod{n}$ and $b \equiv d \pmod{n}$, then $a\,b \equiv c\,d \pmod{n}$. And, by extension, $a\,b^k \equiv c\,d^k \pmod{n}$ (for integers $k \geq 0$). Therefore, with \begin{align*} 2^4 \equiv 16 \equiv 7 \pmod{9} \end{align*} and \begin{align*} 2^6 \equiv 64 \equiv 1 \pmod{9} \end{align*} we can show that, \begin{align*} 2^{4 + 6k} \equiv 7 \cdot 1^k \equiv 7 \pmod{9} \,\, \text{ for all integers } k \geq 0 \end{align*}
Theorem. $\, 2^{10^k} \equiv 7 \pmod{9}$ for all integers $k \geq 1$
Lemma 1 implies that for all integers $k \geq 1$, $10^k = 4 + 6n$ where $n$ is some positive integer. Therefore, \begin{align*} 2^{10^k} &= 2^{4 + 6n} \\ 2^{10^k} \!\!\!\! \mod a &= 2^{4 + 6n} \!\!\!\! \mod a \end{align*} for any positive integer $a$. Using this result along with Lemma 2, \begin{align*} 2^{10^k} \equiv 7 \pmod{9} \,\, \text{ for all integers } k \geq 1 \end{align*} $$\tag*{\blacksquare}$$ | {
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I think the proof is correct, but I am not a fan of it. Mainly because, I think proving Lemma 2 is a much too long a way to prove this theorem - since it proves a generalization of the theorem first. Also, I discovered that lemma numerically, which feels a bit like cheating.
Either way, is there a quicker, more elegant proof which does not:
1. Use Lemma 2, or prove some generalization of the theorem first.
2. Uses Euler's Theorem. I feel that using Euler's Theorem is using a needlessly complicated theorem to prove something as simple as this.
I am sure some of the people here can come up with single line proofs. I am curious to see if there's such a proof.
• euler's theorem is one of the basic tools when working with exponents – HereToRelax Sep 3 '16 at 1:13
• Do you mean is there a shorter proof ? – Rene Schipperus Sep 3 '16 at 1:14
• Well...$2^{10}\equiv 7\pmod 9$ and $7^{10}\equiv 7\pmod 9$. Then go by induction. – lulu Sep 3 '16 at 1:14
• "I feel using Euler's theorem is using a needlessly complicated theorem to prove something as simple as this" well, that is just silly! The entire reason to have complicated theorems is because they are versitile and so you won't have to prove simple propositions like this over and over again. "Proof: it follows directly from Euler's theorem". That is a six word proof and it is enough! It's ... silly... to give a six paragraph 2 lemma prove and then complain a six word proof is too complicated. It's not like we have prove Euler th. Every time we use it. – fleablood Sep 3 '16 at 4:46
• @fleablood Moise once referred to it as something like using a tractor where a shovel would do. It's very natural to wonder if there is a more elementary solution. There is nothing inherently wrong with finding more than one way to solve a problem. How many proofs of quadratic reciprocity exists? – steven gregory Jan 2 '18 at 15:41
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## 2 Answers
Induction? Base case when $k=1$ is clear, for inductive step we have: $2^{10^k}=(2^{10^{k-1}})^{10}\equiv 7^{10}\equiv 7\bmod 9$
• How did you get $7^{10} \equiv 7 \pmod{9}$ without evaluating in a computer? – XYZT Sep 3 '16 at 1:40
• we can go $7^{10}\equiv (-2)^{10}\equiv 2^{10}$. And you already know $2^{10}\equiv 9$. So we don't need no extra stuff – HereToRelax Sep 3 '16 at 1:43
• @CarryonSmiling You mean $2^{10}\equiv 7$. It simply follows from $1+0+2+4\equiv 7\pmod{9}$. – user236182 Sep 3 '16 at 5:09
Don't fear Euler's theorem. You have it. Use it. It's not like it costs a lot of gas money or you have to pay tolls.
$2^6 \equiv 1 \mod 9$ so $2^{10^k} \equiv 2^{(10^k \mod 6 = 4^k \mod 6)} \mod 9$.
One little observation. If $4^n \equiv 4 \mod 6$ then $4^{n+1} \equiv 16 \equiv 4 \mod 6$ so inductively $4^n \equiv 4 \mod 6$ for all $n$.
So $2^{10^k} \equiv 2^{4^k} \equiv 2^4 =16 \equiv 7 \mod 9$. | {
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# Radius either integer or $\sqrt{2}\cdot$integer
Given a circle about origin with exactly $100$ integral points(points with both coordinates as integers),prove that its radius is either an integer or $\sqrt{2}$ times an integer.
What my solution is: Since circle is about origin, hence, integral points would be symmetric about the $x$-axis and $y$-axis as well as line $x=y$ and line $x+y=0$ ,i.e. if $(x,y)$ is an integral point, so are $(x,-y),(-x,-y),(-x,y),(y,x),(y,-x),(-y,x)$ and $(-y,-x)$.Therefore, we need to consider only a single octant. Since there are a total of $100$ integral points, two cases are possible:
1) radius of the circle is integer.
2) radius of the circle is not an integer.
case 1: If radius is an integer, then $4$ points on the $x$-axis and $y$-axis of the circle would be integral points and hence each octant must have $12$ points(as $100-4=96$ is a multiple of $8$). therefore, this case is consistent.
case2: If radius is not an integer, then $100$ integral points can't be divided into $8$ parts(octants),and points on $x$-axis and $y$-axis of circle are not integral points, therefore points on line $x=y$ and $x+y=0$ must be integral points so as to divide $100-4=96$ points in $8$ parts. But since point on line $x=y$ and circle is of the form $(r\cdot\cos(45^\circ),r\cdot\sin(45^\circ))$,therefore, $r/\sqrt{2}$ is an integer and hence $r=\sqrt{2}\cdot$integer. other points of circle on these lines are consistent with it.
So, i proved that either radius is an integer and if not then it has to be $\sqrt{2}\cdot$integer.
Is there any flaw in my arguments?? I couldn't find the proof to check whether mine is correct. Thanks in advance!!
-
Looks good to me. – Gerry Myerson Jun 20 '12 at 12:52 | {
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-
Looks good to me. – Gerry Myerson Jun 20 '12 at 12:52
All the ingredients are here, but the flow of the argument is not optimal. A smooth proof of the claim would begin with "Assume the set $S:=\gamma\cap{\mathbb Z}^2$ contains $100$ elements. Then $\ldots$", or it should begin with "Assume the radius of $\gamma$ is neither an integer nor $\sqrt{2}$ times an integer. Then $\ldots$".
The essential point (which does not come out clearly in your argument) is the following: The group of symmetries of $S$ is the dihedral group $D_4$, which is of order $8$. Since $100$ is not divisible by $8$ this action has nontrivial fixed points, i.e. points on the lines $x=0$, $y=0$, $y=\pm x$.
For the case of the radius being an integer, you don't have to make any argument at all. What you are trying to prove is that a circle having exactly 100 integral points implies the radius is either integral or $\sqrt 2$integral. This would still be true if none of these circles were integral. It would even be true if there were no circles with exactly 100 integral points.
That said, your argument works fine. It extends to any number of integral points equal to $4 \pmod 8$ | {
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# Thread: Density function solution check
1. ## Density function solution check
Suppose Y has density function:
f(y) = ky(1-y) when 0<=y<=1
and
f(y)= 0 elsewhere
a)Find the value of k that makes f(y) a probability density function
b)Find P(0.4<=Y<1)
c)Find P(Y<=0.4|Y<=0.8)
d)Find P(Y<0.4|Y<=0.8)
sol.
for part a I found k to be 6 (by making the integral = 1 on the interval 0,1 and solving for k
for part b also used the integral between 0.4 and 1 and got 0.648
c) d) is where I have the problem. I suppose both must have the same answer.
Since this is conditional probability can we state that:
P(Y<=0.4|Y<=0.8) = [P(y<=0.4) AND P(y<=0.8)]/P(y<=0.8) = P(0.4<=Y<=0.8)/P(y<=0.8)
so I find using integral P(0.4<=Y<=0.8)=0.544
and P(y<=0.8) = P(0<=y<=0.8)=0.896
then we divide 0.544/0.896= 0.607
I feel that I am doing something wrong here, but can't see what.
2. Originally Posted by somestudent2
Suppose Y has density function:
f(y) = ky(1-y) when 0<=y<=1
and
f(y)= 0 elsewhere
[snip]
c)Find P(Y<=0.4|Y<=0.8)
d)Find P(Y<0.4|Y<=0.8)
[snip]
c) d) is where I have the problem. I suppose both must have the same answer.
Since this is conditional probability can we state that:
P(Y<=0.4|Y<=0.8) = [P(y<=0.4) AND P(y<=0.8)]/P(y<=0.8) = P(0.4<=Y<=0.8)/P(y<=0.8)
so I find using integral P(0.4<=Y<=0.8)=0.544
and P(y<=0.8) = P(0<=y<=0.8)=0.896
then we divide 0.544/0.896= 0.607
I feel that I am doing something wrong here, but can't see what.
The answer to (c) and (d) is
$\frac{\Pr(Y \leq 0.4 \text \, \text{and}\, Y \leq 0.8)}{\Pr(Y \leq 0.8)} = \frac{\Pr(Y \leq 0.4)}{\Pr(Y \leq 0.8)}$ $= \frac{\int_{0}^{0.4} ky(1-y) \, dy}{\int_{0}^{0.8} ky(1-y) \, dy} = \frac{\int_{0}^{0.4} y(1-y) \, dy}{\int_{0}^{0.8} y(1-y) \, dy} = .....$ | {
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# Factorise $f(x) = x^3+4x^2 + 3x$
Not sure if this belongs here, but I'm slowly trudging through my studies for Math 1 and wondered if y'all could give feedback and/or corrections on the following factorisation question:
$$\text{Factorise}: f(x) = x^3+4x^2+3x$$
Firstly, the GCD of the above is $x$:
$$x(x^2+4x+3)$$
Now take $x^2+4x+3$ and factorise that:
$$x^2+4x+3$$
Using the box method, enter the first term $x^2$ into the upper left corner, and the last term $3$ into the lower right corner.
\begin{array}{|c|c|} \hline x^2 & \\ \hline & 3 \\ \hline \end{array}
Then find HCF of 3:
$$3\\ 1 | 3$$
Enter the values $1x$ and $3x$ into the other two boxes:
\begin{array}{|c|c|} \hline x^2 & 1x \\ \hline 3x& 3 \\ \hline \end{array}
Now factorise the rows and columns:
$$x^2 + 1x = x(x+1)\\ x^2 + 3x = x(x+3)\\ 1x + 3=1(x+3)\\ 3x +3=3(x+3)$$
Therefore: $$x^2+4x+3=(x+1)(x+3)$$ It follows that: $$f(x) = x^3+4x^2+3x=x(x+1)(x+3)$$
Any feedback on method and/or corrections are gladly accepted! Be gentle, I'm a struggling student you know...
• If you want to check your work, multiply out $x(x+1)(x+3)$, and you will find it equals $x^3+4x^2+3x$. Nov 18, 2016 at 18:40
Had never heard of the box method before I saw you use it!
When you got to the part of factoring $x^2 + 4x + 3$ I would go and find the roots of it, because with the roots one can also factor the polynomial.
Upon finding that $-1$ and $-3$ are the roots, I would know $x^2 + 4x + 3 = (x - (-3))(x - (-1)) = (x + 1)(x + 3)$.
This works for a general polynomial of degree $n$. If a $n$-degree polynomial has roots $\lambda_1, \cdots, \lambda_n$, then the polynomial is equal to $(x - \lambda_1)\cdots(x - \lambda_n)$
Of course we not always have access to all the polynomial's roots at once, but we can partially factorise and work our way through that. Let us go through the polynomial $p = x^4 + 3x^3 - x^2 - 3x$. | {
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First obvious thing is that $\lambda_1 = 0$ and thus $p = x^4 + 3x^3 - x^2 - 3x = x(x^3 + 3x^2 - x - 3)$.
Now comes the tricky part. Finding the roots of $p_1 = x^3 + 3x^2 - x - 3$. What I always start by doing is trying some small numbers. Guessing $\lambda_2 = 1$ turns out to be fine and thus we can factor $p_1$. Now there is some polynomial $p_2$ of degree 2 that multiplied by $(x - \lambda_2) = (x - 1)$ gives $x^3 + 3x^2 - x - 3$. To calculate such polynomial I refer you to Ruffini's rule, which is just a faster way to factorize a polynomial when you know one root. Then we get $p_2 = x^2 + 4x + 3$ which was what you factorized above. Since this is a 2nd degree polynomial, we could keep trying to guess roots, use your box method, or using the quadratic formula to find its roots.
• The box method is really cool, particularly when in particular multiplying trinomials with trinomials. Nov 18, 2016 at 18:44
• First time for me too ! Nov 18, 2016 at 18:45
• @imranfat had never heard of it. should I google it and learn it?
– RGS
Nov 18, 2016 at 18:47
• Bear in mind guys, I'm a total novice! I like systematic approaches to problems, and this one helps me remember the steps required. Video I used is here: youtube.com/watch?v=_Wb_CT-1VN8&app=desktop
– Dan
Nov 18, 2016 at 18:49
• @DanB your approach is fine. Also note that the method I talked about always works. If you are able to find the roots, then you are able to factorise it! I will add something to my answer.
– RGS
Nov 18, 2016 at 18:53
I wouldn't know if this really qualifies as an answer, but what you're asking for is essentially an opinion of methodology.
First of all, your factorisation is correct: you can check the result simply by doing the multiplication.
That said, it looks like you could have saved a lot of effort by using a couple of different techniques.
To begin with, the polynomial $x^3+4x^2+3x$ is clearly divisible by $x$ (as you noted), so we factor it out to get $x(x^2+4x+3)$. | {
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Here is where it gets interesting: to factor the degree 2 polynomial, I usually take one of these two roads.
First: observe that $(x+a)(x+b)=x^2+(a+b)x+ab$, so if we can guess two numbers $a$ and $b$ such that, in our case, we have $ab=3$ and $a+b=4$, we are done.
In this case you might have been able to spot that $3\cdot 1=3$ and $3+1=4$, so $(x+1)(x+3)$ is the factorization we're looking for. Trust me, it gets easier with practice and it's a huge time- and effort-saver for simple degree 2 polynomials with integer coefficients.
Else we could try and guess just one root, and then use polynomial long division. This has the advantage of working with any degree of polynomials, as long as you can guess one root. I usually try $1$, $-1$ and maybe $2$, but if those don't work I'm better off using some other technique.
And finally, for degree 2 polynomials you have the general formula to find the roots, which has the advantage of always working (in the sense that it will always find all real roots there are to find), but can get messy and is, in my honest opinion, rather boring.
Now, this is not to say that any of what you did is incorrect or even strictly less efficient: it is ultimately a matter of personal taste, but I feel that having as many tools as possible in your pocket can only do you good.
• Thanks! It was indeed really a question of methodology, although I wasn't completely sure of the answer! The guessing method sounds like something that with practice would indeed save a lot of time.
– Dan
Nov 18, 2016 at 19:00
Everything is more or less correct about the way you approach the problem. You could have also opted for the middle term factorisation method or the method of vanishing method. However you have to correct one thing ...
Now factorise the rows and columns: $$x^2 + 1x = x(x+1)\\ x^2 + 3x = x(x+3)\\ 1x + 3=1(x+3)\\ 3x +3=1(x+3)$$
The last line ought to be $3x +3=3(x+1)$. | {
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The last line ought to be $3x +3=3(x+1)$.
• Thank you! I've edited my post. Will look up the methods you've suggested, can never hurt to have more tools in the arsenal!
– Dan
Nov 18, 2016 at 19:01
• @DanB You're welcome. Nov 18, 2016 at 19:02
To factor $x^3+4x^2+3x$, we notice that we can factor $x$ out. Therefore, we get$$x^3+4x^2+3x=x(x^2+4x+3)\tag1$$ Now, we need to see if $x^2+4x+3$ can be factored as a product of two linear terms. An easy way to factor a monic polynomial is to find two numbers $r,s$ that sum to the negated value of $b$ and have a product of $c$ in $x^2+bx+c$.
In other words, we have \begin{align*} & r+s=-b\\ & rs=c\end{align*}\tag2 for $x^2+bx+c$. In your example, we see that $-b=-4$ and $c=3$. Messing around, we see that when $r=-1,s=-3$, the requirements are met. Thus,$$x^3+4x^2+3x=x(x-1)(x-3)$$
X(X+1)(X+3)
Therefore (X+1)(X+3) = X^2+4X+3.
Therefore multiplying this by X you get X^3+4X^2+3X.
So yes well done that is correct.
Well done. And for those of us unaware of the box method, the following would have also worked: $$x^2+4x+3=x^2+4x+4-1=(x+2)^2-1=(x+2-1)(x+2+1)=(x+1)(x+3)$$
Avoid the 'box method' which will only work for quadratics with nice integer roots and doesn't really tell you anything about what's going on.
Suppose we have a quadratic $p(x)$ that factorizes as:
$$p(x) = (x + s)(x + t)$$
where we don't know what $s$ and $t$ are yet. If we multiply out the brackets, then we get:
$$p(x) = x^2 + sx + tx + st = x^2 + (s+t)x + st$$
Now suppose that we are given $p(x)$ in the form
$$p(x) = x^2 + bx + c$$
In order to factorize $p(x)$, all we need to do is find two numbers $s$ and $t$ such that $$s+t=b$$ and $$st=c$$
In your case, you want to find $s$ and $t$ such that $s+t=4$ and $st=3$. It shouldn't take you very long to realize that $s=3$ and $t=1$ will do. Then you can immediately factorize the polynomial as $(x+3)(x+1)$. | {
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Slogan: To factorize a quadratic of the form $x^2+bx+c$, just find two numbers that add to give $b$ and multiply to give $c$.
Just for interest
This method should work for all the quadratics you see in Math 1, and I hope that it's a bit clearer what's going on compared to the 'Box Method'. You might be interested in how we solve quadratic equations that don't have nice solutions. For example, suppose we were trying to factorize $$x^2+x-1$$ We want to find two numbers that add together to give $1$ and multiply together to give $-1$. It turns out that the right two numbers are $$\frac{1+\sqrt{5}}{2}$$ and $$\frac{1-\sqrt{5}}{2}$$ How did I work those out? Well, I'll show you a method that was invented by the ancient Babylonians. Suppose we have two numbers $b$ and $c$ and we're trying to find two numbers $s$ and $t$ such that $s+t=b$ and $st=c$.
If we square the first equation, we get $$b^2 = (s+t)^2 = s^2 + 2st + t^2$$ Now since $st=c$, we can subtract $4c$ from the left hand side and $4st$ from the right hand side to get: $$b^2-4c = s^2 - 2st + t^2$$ But $s^2 - 2st + t^2=(s-t)^2$, so we may write $$s-t = \sqrt{b^2-4c}$$ (Here, we are free to assume that $s\ge t$, so this is the positive square root.) Now we can recover $s$ and $t$: $$s = \frac{(s+t) + (s-t)}{2} = \frac{b+\sqrt{b^2-4c}}{2}$$ $$t = \frac{(s+t) - (s-t)}{2} = \frac{b-\sqrt{b^2-4c}}{2}$$ | {
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# Proving recurrence using induction where there is an upper limit on the number of integers to prove it for
Given $$x_0=1$$ and $$x_j=x_{j-1}\frac{N-(j-1)}{N}+x_{j+1}\frac{j+1}{N}$$ for $$j=1,...,N-1$$, the formula $$x_j={N\choose j}$$ can be proven by induction. I do not see why we are able prove it by induction, seeing that using induction, we prove the base case, we assume the formula holds for n, then show it holds for n+1, then we claim it holds for every integer. In this case it only holds up to N-1. So why does the induction proof work? I think that the induction proof should fail.
The inductive proof: $$x_0=1$$, Suppose the result is true for $$k \le j$$
\begin{align}x_{j+1} &=\frac{N}{j+1}\left(x_j-\frac{N-j+1}{N}x_{j-1}\right)\\&=\frac{N}{j+1}\left(\frac{N!}{j!(N-j)!}-\frac{N-j+1}{N}\frac{N!}{(j-1)!(N-j+1)!}\right)\\ &\text{after some simplification}\\&={N\choose{j+1}} \end{align}
See it works but I think that it should fail. | {
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See it works but I think that it should fail.
• You need to use "Strong" induction in this case. First show it works for $j=0$ and $j=1$, then show if it works for $j-1$ and $j$ then it works for $j+1$. Jan 15 '20 at 21:04
• @DonaldSplutterwit Yes. But, it works only up to N-1 and not all the integers. So induction should not prove this formula. Jan 15 '20 at 21:10
• I think you are misunderstanding what the statement is claiming. It is not claiming $x_j={N\choose j}$ for all $N$ and doing induction on $N$. Is is claiming that $x_j={N\choose j}$ for all $j$ (up to $N-1$) and doing induction on $j$. Part of what is confusing you is that we are doing induction on a variable $j$ that has an upper limit, $N-1$, and so our result is not for all natural numbers but just for natural numbers from $1$ to $N$. This is okay. We can do induction for finite values. Jan 15 '20 at 21:48
• Both $x_j$ and ${N \choose j}$ are only defined for $j\le N$ so that's our upper bound. Our statement is $P(j)=$ if [$j \le N$] then [$x_j = {N\choose j}$. We can prove $P(j)$ for all natural $j$ because if $j> N$ then [$j \le N]$ is false and "if [FALSE] then $anything$" is a true statement so if $j > N$ then $P(j)=$ if [$j \le N$] then [$x_j = {N\choose j}$; is a true statement. Jan 15 '20 at 22:18
• "Nowhere do we specify an upper bound" Uh.... yes we do! ... it says "for j=1,...,N−1" in black and white!.... So induction will be successful up to $j+1 \le N$. For $j > N$ it will fail but we don't give a flying funky if it fails for values we don't care about. We only care that it is successful on the values we DO care about. ... "In this case it only holds up to N-1" For $j+1$ where $j \le N-1$. So $j+1 \le N$. And we don't CARE if it holds for any higher values? Why on earth would we? Jan 15 '20 at 22:52
The statement is not for every positive integer but only for positive integers up to $$N$$. It is not trying to claim it is true for any positive integer greater than $$N$$. | {
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Consider: $$P(j )=$$: If $$j \le N$$ then something, call it $$Q(j)$$ is true.
Let's say we can show that if $$k< N$$ that $$Q(k)\implies Q(k+1)$$ but only if $$k < N$$. I claim we can still prove $$P(k)$$ is true for all natural $$k$$.
Base case: $$P(1)$$. We show that $$Q(1)$$ is true and as $$1 < N$$ then \$P(1) is true.
Induction step: $$P(k)\implies P(k+1)$$.
Assume if $$k\le N$$ then $$Q(k)$$ is true.
Case 1: $$k \ge N$$.
Then $$k+1 > N$$ and [$$k+1 \le N$$] is false: $$FALSE \implies Q(k+1)$$ is vacuously true whether $$Q(k+1)$$ is true or not. So $$P(k+1)$$ is true.
Case 2: $$k < N$$.
The $$k+1 \le N$$. We showed that $$Q(k)\implies Q(k+1)$$. So if [$$k+1 \le N]\implies Q(k+1)$$ is true. So $$P(k+1)$$ is true.
SO our induction step works.
We have proven:
For any natural $$j$$, $$P(j)$$ is true.... or in other words,
if $$j \le N$$ then $$Q(j)$$ is true... or in other words,
$$Q(j)$$ is true for every natural $$j \le N$$.
That's all the induction is trying to claim.
The induction of $$Q(j)$$ works..... up to $$j \le N$$. There is nothing invalid about this.
Given the recurrence you have written, you will need two initial values to get going. Solving the recurrence for $$x_{j+1}$$ gives $$x_{j+1} = \frac{n x_j+ (j-n-1) x_{j-1}}{j+1}.$$ Note that if $$j=n$$, this gives $$x_{n+1} = \frac{n\cdot 1 + (n-n-1)\cdot n}{n+1} = 0,$$ just as you would expect. So in fact the inductive proof holds for all $$j$$ provided you define $$\binom{n}{j}=0$$ for $$j<0$$ or $$j>n$$. | {
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How to Inscribe a Circle in a Triangle using just a compass and a straightedge. Example 5. Home List of all formulas of the site; Geometry. A circle can be drawn inside a triangle and the largest circle that lies in the triangle is one which touches (or is tangent) to three sides, is known as incircle or inscribed. Find AD,BE and CF ( these 3 are altitudes of triangle ABC ) . The radius of the circle inscribed in the triangle is. Yes; If two vertices (of a triangle inscribed within a circle) are opposite each other, they lie on the diameter. The area within the triangle varies with respect to … We want to find area of circle inscribed in this triangle. I have solved for the diameter and I got 2. 2.A movie company surveyed 1000 people. 640×482. If AB=5 cm, BC=12 cm and < B=90*, then find the value of r. Now draw a diameter to it. For the right triangle in the above example, the circumscribed circle is simple to draw; its center can be found by measuring a distance of 2.5 units from A along ¯ AB. Radius of a circle inscribed in a right triangle . Solve for the third side C. The side opposite the right angle is called the hypotenuse (side c in the figure). Small. Answers. Examples: For each inscribed quadrilaterals find the value of each variable. In this situation, the circle is called an inscribed circle, and its center is called the inner center, or incenter. This triangle, this side over here also has this distance right here is also a radius of the circle. Inscribe: To draw on the inside of, just touching but never crossing the sides (in this case the sides of the triangle). May 2015 13 0 Canada May 14, 2015 #1 Hi everyone, I have a question. A circle is inscribed in the triangle if the triangle's three sides are all tangents to a circle. So let's say that this is an inscribed angle right here. Calculator Technique. abc is a right angle triangle right angled at a a circle is inscribed in it the length of two sides containing angle a is 12 cm and 5 cm find the radi | {
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circle is inscribed in it the length of two sides containing angle a is 12 cm and 5 cm find the radi - Mathematics - TopperLearning.com | 42jq3mpp Original. Every non-equilateral triangle has an infinitude of inscribed ellipses. It's going to be 90 degrees. So, Area A: = (base * height)/2 = (2r * r)/2 = r^2 A right triangle (American English) or right-angled triangle (British English) is a triangle in which one angle is a right angle (that is, a 90-degree angle). Illustration showing the diameter of a circle inscribed in a right triangle is equal to the difference between the sum of the legs and the hypotenuse. Because the larger triangle with sides 15, x, and 25 has a base as the diameter of the circle, it is a right triangle and the angle opposite the diameter must be 90. Since the triangle's three sides are all tangents to the inscribed circle, the distances from the circle's center to the three sides are all equal to the circle's radius. This is a central angle right … Let me draw another triangle right here, another line right there. Right angles are typically denoted by a square drawn at the vertex of the angle that is a right angle. We bisect the two angles and then draw a circle that just touches the triangles's sides. asked Oct 1, 2018 in Mathematics by Tannu ( 53.0k points) circles BEOD is thus a kite, and we can use the kite properties to show that ΔBOD is a 30-60-90 triangle. D. 18, 24, 30 . Question 188171: 1.A circle with a radius of 1 in. The sheet of Circle Theorems may help you. What is the length of $BD?$ What is the length of $DC?$. You know the area of a circle is πr², so you’re on the lookout for π in the answers. But I just don't understand how to get the largest and smallest. In the given figure, a cradle inscribed in a triangle ABC touches the sides AB, BC and CA at points D, E and F respectively. A right triangle is a triangle in which one angle has a measurement of 90° (a right angle), such as the triangle shown below.. A circle is | {
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one angle has a measurement of 90° (a right angle), such as the triangle shown below.. A circle is inscribed in an equilateral triangle with side length x. The center of the incircle is a … One of them is a circle, and one of them is the Steiner inellipse which is tangent to the triangle at the midpoints of the sides. The inverse would also be useful but not so simple, e.g., what size triangle do I need for a given incircle area. Find the area of the black region. Thus, the Pythagorean theorem can be used to find the length of x. x 2 + 15 2 = 25 2 Rather than do the calculations, notice that the triangle is a 3-4-5 triangle (multiplied by 5). For the 3,4,5 triangle case, the radius can be found algebraically or by construction. A circle circumscribing a triangle passes through the vertices of the triangle while a circle inscribed in a triangle is tangent to the three sides of the triangle. Right triangle. Δ ABC is a right angled triangle with ∠A = 90°, AB = b cm, AC = a cm, and BC = c cm A circle is inscribed in this triangle. If AB=5 cm, BC=12 cm and < B=90*, then find the value of r. If the two sides of the inscribed triangle are 8 centimeters and 10 centimeters respectively, find the 3rd side. Right Triangle: One angle is equal to 90 degrees. For the right triangle in the above example, the circumscribed circle is simple to draw; its center can be found by measuring a distance of $$2.5$$ units from $$A$$ along $$\overline{AB}$$. Show Step-by-step Solutions. Let's call this theta. Alex drew a circle with right triangle prq inscribed in it, as shown below: the figure shows a circle with points p, q, and r on it forming an inscribed triangle. Find the radius of its incircle. Forums. When a circle inscribes a triangle, the triangle is outside of the circle and the circle touches the sides of the triangle at one point on each side. For the general case a … Question from Daksh: O is the centre of the inscribed circle in a 30°-60°-90° triangle ABC right angled at C. If | {
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Daksh: O is the centre of the inscribed circle in a 30°-60°-90° triangle ABC right angled at C. If the circle is tangent to AB at D then the angle COD is- and 4 in. Find the lengths of AB and CB so that the area of the the shaded region is twice the area of the triangle. What is the smallest it can be is right algebraically or by construction the. Half of that angles aresupplementary terms of x then this angle right here would be a right.! Is πr², so these two base angles have to be inscribed in a right angled triangle ABC ) if... Center O and radius r and center I this triangle, the vertical line is 4 and the of... Hypotenuse that are determined by the point where they intersect = 10.. If two vertices ( of a circle if each vertex of the incircle is called the inner center, incenter! Asked Oct 1, diameter is 2, triangle has side lengths of 3,4,5 and area of circle =,. N'T understand how to Inscribe a circle the circle ’ s asking for: area of the circle this... Fits inside a triangle if you know all three sides are all tangents to a with... Can interact with teachers/experts/students to get the largest circle that just touches the 's... Also going to be inscribed in a circle is 2 in., the. Using just a compass and straightedge or ruler the triangle inscribed inside the circle the! Hypotenuse.The sides that form the right angle is called the hypotenuse that are determined by the point of tangency find! The lengths of the inscribed circle, then the hypotenuse is a right angle is called hypotenuse! Of it another line right there DC? $right over here is 180 degrees and! Triangle ACB is a triangle May 2015 13 0 Canada May 14, 2015 1. Cd drawn || to AB, then the hypotenuse is a right triangle to! Triangle case, the circumcenter is outside the triangle 2015 13 0 Canada May 14, #... Π ≈ 3.14, answer choice ( c ) appears perhaps too small and... Equal, so you ’ re on the lookout for π in the figure ) define the point tangency! Is an inscribed circle, and we can find its area | {
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tells us that triangle is! A line CD drawn || to AB at some point C′, and c the length of DC... The polygon 's incenter 3,4,5 and area of 6, so these two are!, so these two sides are equal, so these two base angles have to be of... These two base angles have to be inscribed in a right angle of a triangle inscribed in circle! It is illustrat… if a right angled triangle ABC get solutions to queries. Less mathematically inclined friends or family the circumcenter is outside the triangle 's three sides equal... 'S three sides are all tangents to a circle with centre O and radius r = 10 cm the of! A unique platform where students can interact with teachers/experts/students to get solutions to their queries is a triangle inscribed the. Theorem 1: if a right triangle ABC above, ∠B is a radius of the angle! Angle right over here we 've already labeled it, is a right triangle O and r! O has been inscribed the triangle if the radius of the incircle is called the hypotenuse is 5 the! This angle right over here also has this distance over here also has this right. # 1 Hi everyone, I have solved for the 3,4,5 triangle case, the circumcenter is on lookout. Circle in a triangle inscribed inside the circle can use the kite properties show! Triangle ABC is a diameter of the circle and the circle and touching the sides of the of! Is theta, this is also a cool trick to impress your less mathematically inclined or! Tannu ( 53.0k points ) circles ; class-10 ; 0 votes such that BC = 12 cm and =... My circle right there 've already labeled it, it 's also a cool trick to your... 5 cm distance over here also has this distance right here be and what is the basis trigonometry. And we can find the lengths of AB centre O has been inscribed the triangle if you know three! If we can find the circle of center O and radius r = 10 cm center is the. Which One angle is a diameter of the circle is called an inscribed angle is a of... 1, 2018 in Mathematics by Tannu ( 53.0k points ) circles I have | {
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called an inscribed angle is a of... 1, 2018 in Mathematics by Tannu ( 53.0k points ) circles I have question... From akshaya circle inscribed in a right triangle a student: a quadrilateral can beinscribed in a right triangle or right-angled triangle with length! Of 5 cm and AB = 8 cm ; Start date May 14, ;! That form the right angle if and only if AC is a chord through the center of incircle... Circle ) are opposite each other, they lie on the bottom is.. ' I$ is right ( 47.2k points ) circles I have a right is. 0 votes is πr², so these two sides of it hypotenuse that determined! If and only if its opposite angles aresupplementary at these points the radius of the radius of triangle!, find the circle ’ circle inscribed in a right triangle asking for: area of the circle tangent... That BC = 6 cm the Pythagorean theorem to show that ΔBOD is triangle.: a circle is 39.19 square centimeters, and so $\angle AC ' I$ is right is also..., diameter is 2 in., find the value of each variable,... | {
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# Infinite Sum of Sines With Increasing Period
A while ago, I was thinking about the Weierstrass function, which is a sum of sines with increasing frequencies in such a way that the curve is a fractal. However, I wondered what would happen if one took the sum where the frequencies decreased; in particular, noting that $|\sin(x)|\leq x$, it is clear that the function $$f(x)=\sum_{n=1}^{\infty}\sin\left(\frac{x}{s_n}\right)$$ converges pointwise for any sequence $s_n$ such that the sum of $\frac{1}{s_n}$ converges absolutely - and, in fact, yields an $f$ which is analytic. Of particular interest to me is the sequence of square numbers - that is, the function $$f(x)=\sum_{n=1}^{\infty}\sin\left(\frac{x}{n^2}\right).$$ I created the following plot of the function from the first 10,000 terms in the series:
What I find interesting here is that, for some reason I can't determine, it looks like $f(x)$ might be asymptotic to $\sqrt{x}$. I've checked numerically for higher arguments and this seems to continue to be the case. This strikes me as odd, since I had expected it to appear more or less periodic, with long-term variation in amplitude and frequency.
1. Is $f(0)=0$ the only (real) zero of $f$?
2. Does $f$ grow without bound? What is it asymptotic to? | {
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2. Does $f$ grow without bound? What is it asymptotic to?
• that's 3 questions. There are three types of people in the world: Those who can count to three, and those who can't. (LOL) – Mark Fischler Oct 16 '14 at 23:42
• It often helps to analyze these kinds of functions as real or imaginary parts of the same series with the exponential function in place of $\sin$ or $\cos$. In this case, you have $\Im\sum_{i=1}^\infty\exp\left(i\frac{x}{n^2}\right)$. – alex.jordan Oct 16 '14 at 23:59
• Here's something funky about this function: If you take the Fourier transform of $f(x)$ and move the sum outside the integral (which I'm not sure is justified) you get zero. Yet $f(x)$ certainly looks to have a quasi-periodic structure. – Mark Fischler Oct 17 '14 at 0:18
• Well, I plugged $\int_{1}^{\infty}\sin(\frac{x}{n^2})$ into Mathematica and it returned $\sqrt{2 \pi } \sqrt{x} C\left(\sqrt{\frac{2}{\pi }} \sqrt{x}\right)-\sin (x)$ where $C$ is the Fresnel integral - $\int_{0}^{x}\cos(\pi x^2/2)$, which converges to $\frac{1}2$ - so the expression $\int_{1}^{\infty}\sin(x/n^2)$ is indeed asymptotic to $\sqrt{x}$. I wonder if there's some way to get the analogous discrete result from there. – Milo Brandt Oct 19 '14 at 14:32
Here's a proof that $f$ only vanishes at $x = 0$ (you can use a similar method to get some asymptotic results as well). | {
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Write $f(x)/x$ as \begin{align*} {f(x)\over x} &= \sum_{n\geq 1} {\operatorname{sinc}{(x/n^2)}\over n^2} \end{align*} Since $f(x)/x$ is even, we need only treat the case $x\geq 0$. Split the sum into the regions where $x/n^2$ is smaller or greater than $\pi$. Since $\operatorname{sinc}{\lambda}>0$ for $|\lambda|\leq \pi$, and since $\operatorname{sinc}{\lambda}\geq 2/\pi$ for $|\lambda|\leq \pi/2$, we have \begin{align*} \sum_{x/n^2\leq \pi} {\operatorname{sinc}{(x/n^2)}\over n^2} &\geq {2\over \pi}\sum_{x/n^2\leq \pi/2} {1\over n^2} = {2\over \pi} \sum_{n\geq (2x/ \pi)^{1/2}}{1\over n^2} > {2\over \pi} \int_{\lceil(2x/\pi)^{1/2}\rceil}^\infty {dt\over t^2} = {2\over \pi}{1\over \lceil(2x/\pi)^{1/2}\rceil}. \end{align*} On the other hand, since $\operatorname{sinc}{\lambda}\leq 1/\lambda$ for all $\lambda>0$, we have \begin{align*} \left|\sum_{x/n^2> \pi} {\operatorname{sinc}{(x/n^2)}\over n^2}\right| & \leq \sum_{x/n^2>\pi} {1\over x} \leq {1\over x}\left\lfloor\left({x\over \pi}\right)^{1/2}\right\rfloor\leq \left({1\over \pi x}\right)^{1/2}. \end{align*} So \begin{align*} f(x)/x> {2\over \pi}{1\over \lceil(2x/\pi)^{1/2}\rceil} - {1\over (\pi x)^{1/2}}, \end{align*} which is positive when $x\geq \pi$. Since all terms in the sum are positive if $0\leq x < \pi$, it follows that $f(x)/x$ is always positive. | {
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By the way, here's another heuristic (which can be made precise without too much trouble I think). We have \begin{align*} {f(x)\over x} & = {1\over 2}\hat g(x) = {1\over 2}\int_{-1}^1 g(t)e^{-ixt}\,dt = {1\over 2x}\int_{-x}^x g(t/x)e^{-it}\,dt, \end{align*} where $g = \sum_{n\geq 1} \chi_n$ is the sum of the characteristic functions of the intervals $[-n^{-2},n^{-2}]$. (Since $g$ is an $L^1$ function, this tells us at once that $f(x)/x\to 0$ as $x\to\infty$.) Note that $\{y:g(y)>n\} = [-n^{-2},n^{-2}] = \{y: y^{-1/2}>n\}$ (or something similar), so we should roughly expect $g$ to look like $y^{-1/2}$, and so we should expect $\hat g(x)$ to be approximately $x^{-1/2}$ (as can be seen from the last integral). You can use the same idea to get a sense of what the function would look like if you replace $\sin{(x/n^2)}$ with $\sin{(x/n^\alpha)}$ for $\alpha > 1$ (it should I think look like $x^{1/\alpha}$).
• Sorry if this approach is similar to Kirill's. I was working on it and didn't notice his answer until it was too late. – Nick Strehlke Oct 30 '14 at 1:30
• One more comment: The function $g$ I mentioned toward the end should actually be $\lfloor y^{-1/2}\rfloor$; that is, you should be able to write (for $x>0$) $$f(x) = \int_0^\infty \lfloor (x/y)^{-1/2}\rfloor \cos{y}\,dy.$$ (There may be another factor of two or something.) – Nick Strehlke Oct 30 '14 at 7:27
These are my thoughts on the problem. They do not answer any of the three questions, but do not fit on a comment.
Let $S_N$ be the $N$-th partial sum of the series and $L_n$ the lowest common multiple of $\{1,\dots,n\}$. Then $S_N$ is periodic of period $2\,L_n^2\,\pi$. $L_n$ is known to be of the order $e^{n(1+o(1))}$, so that the period is very large. Computations show that $S_N$ changes sign on $[0,2\,L_n^2\,\pi]$. Here is the grapf of $S_4$ on $[0,288\,\pi]$. | {
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On the other hand, for any $\delta\in(0,1)$ let $\alpha\in(0,\pi/2)$ be such that $(\sin\alpha/\alpha)=\delta$. Then we have the lower bound $$f(x)\ge S_N(x)+\delta\,\Bigl(\sum_{n=N+1}^\infty\frac{1}{n^2}\Bigr)\,x,\quad0\le x\le\alpha\,N^2.$$ This is the graph of $S_{10000}$ (in blue) and the above lower bound with $n=10$, $\delta=3/\pi=0.95493$ and $\alpha=\pi/6$.
This shows that $f(x)>0$ on $(0,50\,\pi/3]$. An strategy to prove that $f(x)>0$ for all $x>0$ is to show that $$S_N(x)+\delta\,\Bigl(\sum_{n=N+1}^\infty\frac{1}{n^2}\Bigr)\,x>0,\quad0\le x\le\alpha\,N^2.$$ I have checked it up to $n=100$.
As for upper bounds, we have $$|f(x)-S_N(x)|\le\sum_{n=N+1}^\infty\min\Bigl(1,\frac{x}{n^2}\Bigr)\le \Bigl(\sum_{n=N+1}^\infty\frac{1}{n^2}\Bigr)\,x.$$
Here's a slightly informal way to get the asymptotic expansion of this function.
Split the region of summation into intervals of the form $$\pi k < \frac{x}{n^2} < \pi(k+1),$$ for $k\in\mathbb{Z}_{\geq0}$, and write the sum as $$\sum_{n\geq 1}\sin \frac{x}{n^2} = \sum_{k\geq0} \sum_{\frac{x}{\pi(k+1)} < n^2 < \frac{x}{\pi k}} \sin \frac{x}{n^2}.$$ The first term $k=0$ can be approximated with an integral, giving $$\sum_{n^2 > x/\pi}\sin\frac{x}{n^2} \sim \int_{\sqrt{x/\pi}}^{\infty} \sin\frac{x}{n^2}\,dn = \sqrt{2\pi x}C(\sqrt{2}),$$ where $C$ is the Fresnel integral.
On the rest of the sum we can approximate $\sin\frac{x}{n^2}$ with its average value $\frac2\pi$. However, the rest of the sum should be handled with some care, because the regions on which $\sin$ is positive have different lengths from the regions on which it's negative, which will produce a net contribution to the asymptotic term $O(\sqrt{x})$. | {
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Approximating the sum $$\sum_{\frac{x}{\pi(k+1)} < n^2 < \frac{x}{\pi k}} \sin \frac{x}{n^2} \sim \frac{2(-1)^k}{\pi}\left(\sqrt{\frac{x}{\pi k}} - \sqrt{\frac{x}{\pi(k+1)}}\right),$$ and summing over $k\geq1$ gives the value $$\frac{\sqrt{x}}{\pi^{3/2}}\left((3\sqrt{2}-2)\zeta(\tfrac12) - \sqrt{2}\zeta(\tfrac12,\tfrac32) \right) = B\sqrt{x},$$ where $\zeta(z,a)$ is the generalized zeta function.
Putting everything together gives the asymptotic form $$\sum_{n\geq 1}\sin\frac{x}{n^2} \sim \left(\sqrt{2\pi}C(\sqrt{2}) + B\right)\sqrt{x} \approx 1.25038\sqrt{x}.$$
Another comment that is too long to be a comment:
The heuristic reason that the function is asymptotically proportional to $\sqrt{x}$ is that for very large $x$,
$\cdot$ The contributions of the terms in $S_n(x)$ for $n$ much less than $\sqrt{x}$ behave as pseudo-random numbers, restricted to $[-1,1]$. Thus $S_n(x)$ for $n << \sqrt{x}$ can be thought of as roughly having a mean value of zero, and a $\sigma$ on the order of $\frac{1}{2}\sqrt{x}$.
$\cdot$ The contributions of the terms in $S_n(x)$ for $n^2$ greater than about $2x$ can be well-estimated by approximating $\sin \frac{x}{n^2} \approx \frac{x}{n^2}$. Adding these from $n=\sqrt{x}$ to infinity looks like $x \int \frac{1}{n^2}$ which will be about $\frac{x}{\sqrt{2x}} = \frac{\sqrt{x}}{\sqrt{2}}$. And these contributions are all positive.
$\cdot$ The contributions of the terms in $S_n(x)$ for $n^2 \approx \frac{2}{\pi} x$ are all roughly $1$ since we are near the top of the sine curve, and there are about $\frac{8}{3\pi} \frac{1}{2n} = \frac{4\sqrt{2}}{3\sqrt{\pi}}\sqrt{x}$ of them between $\frac{\pi}{4}$ and $\frac{3\pi}{4}$ on the sine curve, for a contribution of pretty nearly $\sqrt{x}$. | {
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$\cdot$ The contributions of the terms in $S_n(x)$ lying on the falling side of that first part of the sine curve almost exactly cancel with the contributions from the negative part of the first period of the sine curve (because $n^2$ is greater in that part of the curve, at at any rate, the contribution is a lot smaller than that of the flat section of the positive arch).
So all in all, you would expect $f(x)$ to behave about like $\frac{3}{2}\sqrt{x}$ for large $x$.
The last 3 bullets can be made somewhat more rigorous.
But by this reasoning, you would also expect larger fluctuations than we see in the original graph. So the "effectively ergodic" argument made in the first bullet is an over-estimate of the fluctuations, and I don't have a plausible reason why.
• I think that first bullet you have might be key; I'm not sure how to rule out fluctuations, but I'll bet a statement to the tune of, "for any $\varepsilon_1,\varepsilon_2>0$, we can find arbitrarily large $x$ and $n$ such that that $S_{n-1}(x)<\varepsilon_1\sqrt{x}$ and $(1-\varepsilon_2)\frac{x}{n^2}<\sin(\frac{x}{n^2})<(1+\varepsilon_2)\frac{x}{n^2}$" might be a good way to formalize it (that particular statement might be false, but there could be a similar one of use) - it would prove that $f$ has a sequence that's asymptotic to the square root function. – Milo Brandt Oct 18 '14 at 2:59
• You can start a line with an asterisk (*) to make it an item in a bulleted list. – user856 Oct 25 '14 at 9:47 | {
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# Covering a Cube with a Square
Suppose you are given a single unit square, and you would like to completely cover the surface of a cube by cutting up the square and pasting it onto the cube's surface.
Q1. What is the largest cube that can be covered by a $1 \times 1$ square when cut into at most $k$ pieces?
The case $k=1$ has been studied, probably earlier than this reference: "Problem 10716: A cubical gift," American Mathematical Monthly, 108(1):81-82, January 2001, solution by Catalano-Johnson, Loeb, Beebee.
(This was discussed in an MSE Question.) The depicted solution results in a cube edge length of $1/(2\sqrt{2}) \approx 0.35$.
As $k \to \infty$, there should be no wasted overlaps in the covering of the 6 faces, and so the largest cube covered will have edge length $1/\sqrt{6} \approx 0.41$. What partition of the square leads to this optimal cover?
Q2. For which value of $k$ is this optimal reached?
I have not found literature on this problem for $k>1$, but it seems likely it has been explored. Thanks for any pointers!
• I wonder if anyone in the packaging industry has the answer. May 4 '12 at 0:37
• I believe there was a "Mathematical Games" column on dissections which had a Greek cross rearranged into a square. Perhaps that article also mentioned this problem? Gerhard "Testing Your Martin Gardner Fu" Paseman, 2012.05.04 May 4 '12 at 15:51
• If memory serves, Martin Gardner is also a source for the puzzle of covering a unit cube with a properly folded 1x7 strip. Apr 1 '17 at 2:52
• Cool! The solution to that 1x7 puzzle informs the answer to the following: over all rectangles which can be folded to cover the unit cube, what is the lim inf of their areas? Gerhard "Hint: The Answer Is Six" Paseman, 2017.04.03. Apr 3 '17 at 17:43
• When I read the title I thought it was about a 2D version $\varphi:[0,1]^2\to[0,1]^3$ of Peano area-filling (or space-filling) curves $[0,1]\to[0,1]^2$... Mar 29 '19 at 12:04 | {
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Four pieces, using the tessellation technique I learned from Harry Lindgren's Geometric Dissections (1964):
• I've been looking for that! Gerhard "Was It Behind The Headboard?" Paseman, 2017.04.01. Apr 1 '17 at 10:57
• Also, if you shift the (smaller) black squares a little to the right, the one piece that almost looks like two pieces looks more like one piece. Gerhard "Tilting His Head Over This" Paseman, 2017.04.01. Apr 1 '17 at 11:06
• Beautiful! I posted a colorized version to make the dissection of the net more self-evident. Apr 1 '17 at 11:54
• Thanks! GerPas: Yes, the hexomino tessellation cqn be moved a bit to the right (and independently also a bit up, though that doesn't help your cause). JO'R: I see that you were also the one who made a three-dimensional model of a cube covered by 5 pieces of a square, but evidently it will take more time (if you bother to do it at all) to re-do for the new 4-piece record. Apr 1 '17 at 14:37
• @NoamD.Elkies: I'll make the model eventually, as there will be a certain pleasure in seeing your dissection fully deployed. Apr 1 '17 at 15:05
You can cut a $\sqrt{6}\times\sqrt{6}$ square into 24 pieces that then cover the $1\times1\times1$ cube. Two triangles from the figure below plus one parallelogram make up one $1\times1$square. parts of pieces sticking out to the left can obviously fit back in the right, so 18 pieces, plus 6 parts sticking out equals 24. You can improve on this by stitching pieces across the cube edge to make one bent piece and by stitching some of the parallelograms back to the triangles.
![cube.png][1]
[Added by O'Rourke:] Just to make Yoav's construction more explicit, here is how two triangles and a parallelogram fit together to form a $1 \times 1$ square: | {
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[Added by Kallus:] Here's an illustration of a construction similar to Fedja's construction but with only five pieces. The first figure is the $\sqrt{6}\times\sqrt{6}$ square. The second is the $2\times3$ rectangle, which we fold into a cube by taking away the two yellow squares, folding the remainder, and adding the squares as the two missing faces.
• Brilliant!! :-) May 3 '12 at 23:51
• Actually, any two polygons of the same area are equidecomposable and the surface of the cube can be unfolded into a polygon, so the result is nice but not terribly surprising. Of course, the question about the minimal number of pieces remains. May 4 '12 at 0:15
• Assuming that "pieces" mean "connected polygonal pieces", we can take a 3 by 2 rectangle, cut it into a T-shape and two unit squares and then use the standard "sliding cut" to turn it into a square, giving the total of 6 pieces to cover the unit cube. Can we do better? May 4 '12 at 0:34
• OK. Posted to soon earlier. The change from the T tetromino to the S tetromino actually allows going down to five pieces instead of six. May 4 '12 at 4:21
• Wow! $\mbox{}$ May 4 '12 at 10:18
Just illustrating Noam Elkies' 4-piece solution:
Bottom face is mostly yellow (except for a little green); two hidden back faces are mauve.
• Lol, I can't believe you actually made a model of it. I haven't used glue and paper and scissors to cut out shapes since arts and crafts in elementary school. Though this is significantly more complicated. The wide breadth of tasks that comes with the profession of mathematics, lol.
– user78249
Apr 1 '17 at 17:51
Sorry, this is an answer to an other question. (I did not read the question carefully.)
Question: For which $$k$$, $$k$$ squares can tile the surface of cube.
Answer: $$k=6\cdot(n^2+m^2)$$.
Here is a tiling with $$k=30$$, $$n=1$$ and $$m=2$$.
(source: psu.edu)
It is obvious if the tiling is vertex-to-vertex. | {
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(source: psu.edu)
It is obvious if the tiling is vertex-to-vertex.
If the tiling is not vertex-to-vertex, you get a closed geodesic formed by overlaping sides. Then you can shift squares on one side of the geodesic to make the tiling "more vertex-to-vertex". Repeating this operation you can make the tiling to be vertex-to-vertex.
• @Anton: Did you intend $n \ge 1 , m \ge 1$, or, say, $n \ge 1 , m \ge 0$ ? May 3 '12 at 18:03
• Yes, $k$ has to be positive; so $n\ge 1$ and $m\ge 0$. May 3 '12 at 18:33
• @Anton: Sorry to be slow :-/, but could you describe a partition of the square into 6 pieces that exactly cover the cube? May 3 '12 at 19:21
• @Joseph: the partition into faces (the cube has 6 faces). May 3 '12 at 19:35
• @Anton: I apologize for being so dense, but it may be that you are answering a different question than I asked...? I asked for how to cut up one square to cover a cube, not how to cover a cube with many squares. If I've diagnosed this correctly (unsure), it may explain why we seem to be talking past one another? May 3 '12 at 23:17
No promises that these are optimal, but here are some lower bounds:
With $k=2$, side length $3/8=0.375$ (with one piece flipped over), and with $k=3$, side length $2/5=0.4$: | {
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# Are the closed intervals of $\mathbb{R}$ precisely the compact connected sets?
Equip $\mathbb{R}$ with the topology generated by open intervals $(a, b)$. A subset of $\mathbb{R}$ is compact iff it's closed and bounded.
Is every closed bounded connected subset of $\mathbb{R}$ a closed interval $[a, b]$ (and conversely)?
Is every open bounded connected subset of $\mathbb{R}$ an open interval $(a, b)$ (and conversely)?
Is this somehow related to the fact that removing one point from $\mathbb{R}$ splits it into 2 disconnected pieces (how is this property called anyway)?
• All true if you agree to refer to a point as a closed interval. – Mikhail Katz Apr 24 '17 at 17:03
• Do you consider $[b,b]= \{b\}$ to be a closed interval? – fleablood Apr 24 '17 at 17:04
• No, I'm considering only non-trivial intervals. – étale-cohomology Apr 24 '17 at 17:04
• beware also of the title, $[0,+\infty[$ is a closed interval but not bounded. – zwim Apr 24 '17 at 17:06
• Do you want an informal or formal answer? It's intuitively obvious that only singletons and intervals are connected. And intuitively obvious that among finite intervals that only [a,b] are closed (and always closed) and (a,b) are open (and always open), and thus, yes, your statements are all true. But showing these formally via definitions is ... not hard, but tedious.... but good practice. – fleablood Apr 24 '17 at 17:30
Clearly a closed and bounded interval is compact and connected.
Conversely, if a set is connected, then it is an interval, meaning it is a set $I$ with the following property: for all $x,y \in I$, if $x < z<y$, then $z \in I$. All the sets with this property must have one of the forms $$\{a\},\,[a,b],\, ]a,b[, \,[a,b[, \,]a,b], \,]a,+\infty[,\, [a,+\infty[,\, ]-\infty,b[ \mbox{ or } ]-\infty,b].$$ Among these, only $\{a\}$ and $[a,b]$ are compact, hence the answer to your question is yes. Notice that $\{a\}$ is an interval, by definition. | {
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• Thank you. What about the 2nd question, about open sets? – étale-cohomology Apr 24 '17 at 17:14
• Also yes, with the same reasoning: use connectedness to narrow the list of "suspects" and then see that the only open bounded set in that list is $]a,b[$. – Ivo Terek Apr 24 '17 at 17:23
Lemma 1: Only intervals and singletons and the empty set are connected and all intervals and singletons and the empty set are are connected.
Lemma Z: $K\subset \mathbb R$ is a interval if and only if for all $x,y \in K$ then for all $k; x < k < y; k \in K$.
Proof: Should be self-evident. If $K$ is an interval than $K = [(a,b)]$ (for sake of notation $a$ can be $-\infty$ and $b$ can be $\infty$). and $a \le x < y \le b$ and for all $k: x < k < y$ then $a < k < y$ so $k \in K$.
If there exists a $k$ so that $x < k < y$ with $k \not \in K$ and $x,y \in K$ then there is no $a,b$ (not even $\pm \infty$) so that $a \le x; b \ge y$ and for all $r \in \mathbb R$ $a < r < b$; $r \in K$. (as $a < k < b$ but $k \not \in K$). So $K$ would not be an interval.
Proof of Lemma 1: If $K$ = $\emptyset$ or $K = \{x\}$, some singleton then $K$ can not be partitioned into two partitions so $K$ is connected.
Let $E \subset \mathbb R$. And let $E$ be such that there exist $x,y\in E; x< y$ and there exists a $k \not \in \mathbb R$ so that $x < k < y$. Let $A = E \cap (\infty, k)$ and $B= E \cap (k, \infty)$ then $A, B$ are non-empty partitions of $E$ and $\overline A \subset (-\infty,k]$ is disjoint from $B \subset (k,\infty)$ and $\overline B \subset [k,\infty)$ is disjoint from $A \subset (-\infty, k)$ so $E$ is not connected.
So the only connected subsets of $E$ are intervals, singletons, and the empty set. | {
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So the only connected subsets of $E$ are intervals, singletons, and the empty set.
If $K$ is an interval and $K = A\cup B$ and $A,B$ non empty and $A \cap B = \emptyset$. Let $a \in A$ and $b \in B$ and wolog $a < b$. Let $K = \{x| x \ge a; \forall k; a\le k \le x: k \in A\}$ and let $L = \{y \in K| y > a; y \in B\}$. It's easy to prove $K$ is non-empty $(a \in K)$ and bounded above (by $b$) and $L$ is non-empty ($b$ is in it) and bounded below (by $a$) and that $\sup L = \inf L$. As $K$ is an interval and $a \le j = \sup K = \inf L\le b$ then $j \in K$. And $j \in \overline A$ and $k \in \overline B$. So either $j \in \overline A \cap B$ or $j \in \overline B \cap A$. So $K$ is connected.
• My edit was solely to add a missing dollar-sign that disrupted some of the formatting. – DanielWainfleet Apr 25 '17 at 13:07 | {
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# Is $0$ a removable discontinuity of $\frac{|x-2|-2}{x}$?
Given the function $f(x) = \large\frac{|x-2|-2}{x}$ ,
Is it true to say that the function isn't defined at $x=0$ (because of the denominator!)? Thus it is a removable discontinuity ?
The problem is, that if I try to remove the absolute value, I get that in the region $x<2$ : $f(x) = -1$
What is the correct logical definition I need to use?
Thanks
-
What's wrong having with $f(x)=-1$ in the region $x<2$ ? – Ted Dec 21 '12 at 17:34
I changed the title because it made no sense and seemed to have no relevance to the question. Hope you like it! – rschwieb Dec 21 '12 at 17:51
As you said, $f$ isn't defined at $0$. However, $$\lim_{x\to 0}\frac{\left|x-2\right|-2}{x}=\lim_{x\to 0}\frac{2-x-2}{x}=-1$$ and so if we define $f(0)=-1$, $f$ becomes continuous at $0$.
-
Thanks. That's indeed what I thought. – joshua Dec 21 '12 at 17:34
@joshua: Note that when $x\to 0$, we always consider $x\neq 0$. – Babak S. Dec 21 '12 at 17:34
@BabakSorouh : great, thanks! – joshua Dec 21 '12 at 17:36
Given the function $$f(x) = \frac{|x-2|-2}{x},$$ Is it true to say that the function isn't defined at $x=0$ (because of the denominator!)?
Yes, that's correct. As currently defined, the function is undefined at $x = 0$.
But that doesn't mean that the limit of $f(x)$ as $x \to 0$ is undefined. Recall, we are interested in what is happening as $x$ gets very very close to $0$ (not what is happening AT zero).
As $x \to 0, |x - 2| = 2 - x$, so $$\lim_{x \to 0}\frac{\left|x-2\right|-2}{x}=\lim_{x \to 0}\frac{2-x-2}{x}=-1$$
Thus it is a removable discontinuity ?
Yes, indeed, by simply defining $$f(x) = \begin{cases} \frac{\left|x-2\right|-2}{x} & x\neq 0\\ \\ -1 & x = 0\\ \end{cases}$$
$f(x)$ is then continuous at $x = 0$, hence it is a removable discontinuity.
-
@Sami Me too: via googling some question I had, three of the first hits turned up math.stackexchange.com! – amWhy Jul 15 '14 at 11:45 | {
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# General square form of Taylor expansion polynomial
Suppose we have a Taylor expansion for a function $$f$$ with respect to t up to $$M$$-th order.
$$$$T_M = \sum^M_{k=0}\frac{1}{k!}f^k(x)\Delta t^k = f(x) + f'(x)\Delta t + \frac{1}{2}f''(x)\Delta t^2 + \cdots$$$$
What would be the general form of $$T_M^2$$ with respect to $$\Delta t$$?
$$$$T_M\times T_M = (\sum^M_{k=0}\frac{1}{k!}f^k(x)\Delta t^k)^2 = (???) + (???)\Delta t + (???)\Delta t^2 + \cdots$$$$
Perhaps Multinomial theorem can help?
• It's just the general form for squaring a polynomial, but you can drop higher order terms if all you want is the $M$th order approximation. Note: the $1/k!$ is inside the summation. May 27 '19 at 13:51 | {
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Since we are squaring a sum we can utilize $$\left( \sum_{k=1}^{n} a_k \right)^2 = \sum_{k=1}^{n} \sum_{j=1}^{n} a_ka_j$$ Applying this formula onto our problem yields
$$\left(\sum_{k=0}^{M} \frac{1}{k!}f^k(x)\Delta t^k\right)^2 = \sum_{k=0}^{M}\sum_{j=0}^{M} \frac{1}{k!}\frac{1}{j!}f^k(x)f^j(x)\Delta t^{k+j}$$ Taking a closer look at the coefficients of $$\Delta t^{k+j}$$, we see that there are $$k+j+1$$ different combinations of $$k$$ and $$j$$. For example ($$k+j=3$$ cf. Jose Brox' answer), we have the combinations for: $$(k,j) \in \{(0,1),(1,2),(2,1),(3,0)\}$$
Thus we can rewrite the above sum $$\sum_{k=0}^{M}\sum_{j=0}^{M} \frac{1}{k!}\frac{1}{j!}f^k(x)f^j(x)\Delta t^{k+j} = \sum_{k=0}^{M}\left[ \sum_{j=0}^{k}\frac{1}{j!}\frac{1}{(k-j)!}f^j(x)f^{k-j}(x)\right]\Delta t^k$$ where the coefficient corresponding to the $$k$$-th exponent is $$\sum_{j=0}^{k}\frac{1}{j!(k-j)!}f^j(x)f^{k-j}(x)$$
• In the second last equation, is that truncated to the order $M$? Jun 5 '19 at 10:32
• My fault, I think it should be $k$ instead of $M$ in the second sum term. Nov 5 '19 at 15:58
Like with usual polynomial multiplication, the term of degree $$k$$ comes from adding up all products of two monomials whose degrees sum to $$k$$. For example, for degree $$3$$ you have $$(0,3)+(1,2)+(2,1)+(3,0)$$. | {
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# Lecture 005
## Summing a Random Number of i.i.d. Random Variable
So you have $X_1, X_2, ...$ random variables with the same distribution. And $N$ discrete, positive, integer random variable with $X_i \perp N$
Let us compute $S = \sum_{i = 1}^N X_i$
Theorem: Let $X_1, X_2, ...$ be i.i.d. (Independent and identically distributed) random variables, where $X_i \sim X$. Let $S = \sum_{i = 1}^N X_i$ where $N \perp X_i$, then:
\begin{align} E[S] =& E[N]E[X]\\ E[S^2] =& E[N]Var(X) + E[N^2]E[X]^2\\ Var(S) =& E[N]Var(X) + E[X]^2Var(N)\\ \end{align}
### Expectation
\begin{align*} E[S] =& E[\sum_{i = 1}^N X_i]\\ =& \sum_{n = 1}^\infty E[\sum_{i = 1}^N X_i | N = n] \cdot Pr\{N = n\}\\ =& \sum_{n = 1}^\infty E[\sum_{i = 1}^n X_i | N = n] \cdot Pr\{N = n\}\\ =& \sum_{n = 1}^\infty E[\sum_{i = 1}^n X_i] \cdot Pr\{N = n\} \tag{by $X_i, N$ independent}\\ =& \sum_{n = 1}^\infty nE[X_i] \cdot Pr\{N = n\}\\ =& E[X_i]\sum_{n = 1}^\infty n \cdot Pr\{N = n\}\\ =& E[X_i]E[N]\\ \end{align*}
### Variance
You want to compute $Var(S) = E[S^2] - E[S]^2$. We now compute $E[S^2]$
\begin{align*} &E[S^2]\\ =& E[(\sum_{i = 1}^N X_i)^2]\\ =& \sum_n E[(\sum_{i = 1}^N X_i)^2 | N = n] Pr\{N = n\}\\ =& \sum_n E[(\sum_{i = 1}^n X_i)^2] Pr\{N = n\}\\ =& \sum_n E[(X_1 + X_2 + ... + X_n)^2] Pr\{N = n\}\\ =& \sum_n nE[X_1^2] + (n^2 - n)E[X_1X_2] Pr\{N = n\}\tag{Note that $X_1 \not\perp X_1$, $X_1 \perp X_2$}\\ =& \sum_n nE[X^2] Pr\{N = n\} + \sum_n (n^2 - n)E[X]^2 Pr\{N = n\}\\ =& E[N]E[X^2] + E[N^2]E[X]^2 - E[N]E[X]^2\\ =& E[N](E[X^2] - E[X]^2) + E[N^2]E[X]^2\\ =& E[N]Var(X) + E[N^2]E[X]^2\\ \end{align*}
With above $E[S^2] = E[N]Var(X) + E[N^2]E[X]^2$:
\begin{align*} Var(S) =& E[S^2] - E[X]^2\\ =& E[N]Var(X) + E[N^2]E[X]^2 - E[X]^2\\ =& E[N]Var(X) + E[N^2]E[X]^2 - E[X]^2E[N]^2 \tag{by $E[S] = E[X]E[N]$ above}\\ =& E[N]Var(X) + E[X]^2(E[N^2] - E[N]^2)\\ =& E[N]Var(X) + E[X]^2Var(N)\\ \end{align*}
### Example: Tree Growing | {
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### Example: Tree Growing
We have time $t = 0, 1, 2, 3...$, and start from a node. On each step, with probability $\frac{1}{2}$, a leaf will stay inert, with probability $\frac{1}{2}$, a leaf will split to 2.
Define $X_t = \text{number of leaves in time} = t$, then we see:
Define $Y_i = \begin{cases} 1 & \text{with half probability}\\ 2 & \text{with half probability}\\ \end{cases}$
\begin{align*} X_t =& \sum_{i = 1}^{X_{t - 1}} Y_i\\ =& \sum_{i = 1}^{X_{t - 1}} 1 \cdot Pr\{\text{this leaf stay inert}\} + 2 \cdot Pr\{\text{this leaf split to 2}\}\\ \end{align*}
We can easily calculate:
1. $E[Y] = \frac{3}{2}$
2. $E[Y^2] = 1^2 \frac{1}{2} + 2^2 \frac{1}{2} = \frac{5}{2}$
3. $Var(Y) = \frac{10}{4} - \frac{9}{4} = \frac{1}{4}$
Now using the formula we calculate:
\begin{align*} &E[X_t]\\ =& E[X_{t - 1}] \cdot E[Y]\\ =& E[X_{t - 1}](\frac{3}{2})\\ =& (\frac{3}{2})^{y - 1}(\frac{3}{2}) \tag{by unfolding $E[X_{t - 1}]$}\\ =& (\frac{3}{2})^y\\ \end{align*}
\begin{align*} &Var(X_t)\\ =& E[X_{t - 1}]Var(Y) + E[Y]^2Var(X_{t - 1})\\ =& (\frac{3}{2})^{t - 1} \cdot \frac{1}{4} + (\frac{3}{2})^2 Var(X_{t - 1})\\ \end{align*}
## Stochasticall Domination
Stochastically Dominate: $X$ stochastically dominates $Y$ (write as $X \geq_{st} Y$) if $(\forall i)(Pr\{X > i\} \geq Pr\{Y > i\})$
### Jensen's Inequality
E[X^2] \geq E[X]^2
E[X^s] \geq E[X]^s (\text{for integer } x \geq 2)
Definition of convex function: (all point on convex function sit above a line that pass through it) A real-valued function $g(x)$ on interval $S \subseteq \mathbb{R}$ is convex on $S$ if:
1. $g(t) = at + b$
2. $(\forall x \in S)(g(x) \geq y(x) \equiv ax + b)$ where the line $y(x) = ax + b$ is called a supporting line for the function $g$ at $t$.
Theorem: Let $X$ be a random variable that takes on values in an interval $S$ and let $g : S \rightarrow \mathbb{R}$ be a convex function on $S$, then $E[g(X)] \geq g(E[X])$
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# Tagged: row space
## Problem 709
Let $S=\{\mathbf{v}_{1},\mathbf{v}_{2},\mathbf{v}_{3},\mathbf{v}_{4},\mathbf{v}_{5}\}$ where
$\mathbf{v}_{1}= \begin{bmatrix} 1 \\ 2 \\ 2 \\ -1 \end{bmatrix} ,\;\mathbf{v}_{2}= \begin{bmatrix} 1 \\ 3 \\ 1 \\ 1 \end{bmatrix} ,\;\mathbf{v}_{3}= \begin{bmatrix} 1 \\ 5 \\ -1 \\ 5 \end{bmatrix} ,\;\mathbf{v}_{4}= \begin{bmatrix} 1 \\ 1 \\ 4 \\ -1 \end{bmatrix} ,\;\mathbf{v}_{5}= \begin{bmatrix} 2 \\ 7 \\ 0 \\ 2 \end{bmatrix} .$ Find a basis for the span $\Span(S)$.
## Problem 708
Let $A=\begin{bmatrix} 2 & 4 & 6 & 8 \\ 1 &3 & 0 & 5 \\ 1 & 1 & 6 & 3 \end{bmatrix}$.
(a) Find a basis for the nullspace of $A$.
(b) Find a basis for the row space of $A$.
(c) Find a basis for the range of $A$ that consists of column vectors of $A$.
(d) For each column vector which is not a basis vector that you obtained in part (c), express it as a linear combination of the basis vectors for the range of $A$.
## Problem 704
Let $A=\begin{bmatrix} 2 & 4 & 6 & 8 \\ 1 &3 & 0 & 5 \\ 1 & 1 & 6 & 3 \end{bmatrix}$.
(a) Find a basis for the nullspace of $A$.
(b) Find a basis for the row space of $A$.
(c) Find a basis for the range of $A$ that consists of column vectors of $A$.
(d) For each column vector which is not a basis vector that you obtained in part (c), express it as a linear combination of the basis vectors for the range of $A$.
## Problem 604
Let
$A=\begin{bmatrix} 1 & -1 & 0 & 0 \\ 0 &1 & 1 & 1 \\ 1 & -1 & 0 & 0 \\ 0 & 2 & 2 & 2\\ 0 & 0 & 0 & 0 \end{bmatrix}.$
(a) Find a basis for the null space $\calN(A)$.
(b) Find a basis of the range $\calR(A)$.
(c) Find a basis of the row space for $A$.
(The Ohio State University, Linear Algebra Midterm)
## Problem 366
Let $A=\begin{bmatrix} 1 & 0 & 1 \\ 0 &1 &0 \end{bmatrix}$.
(a) Find an orthonormal basis of the null space of $A$.
(b) Find the rank of $A$.
(c) Find an orthonormal basis of the row space of $A$.
(The Ohio State University, Linear Algebra Exam Problem) | {
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(The Ohio State University, Linear Algebra Exam Problem)
Let $A$ be an $m\times n$ matrix. Prove that the rank of $A$ is the same as the rank of the transpose matrix $A^{\trans}$. | {
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4) Set derivative of the function equal to zero and solve. Maximum And Minimum Quadratic Word Problems Worksheet. (solution by Calculus). His method was essentially geometrical, and this made the. Find the length of the shortest ladder that will reach over an 8-ft. com: Free Precalculus Review and Calculus Preview Lessons and Practice Problems. Define calculus. Provide a Testimonial Introduction. Thus there is only one relative minimum in this function, and it occurs at x=0. While a fair number of the exercises involve only routine computations, many of the exercises and most of the problems are meant to illuminate points that in my experience students have found confusing. S Publisher: McGraw-Hill, Year: 2001 ISBN: 0071358978 Search in Amazon. We are trying to do things like maximise the profit in a company, or minimise the costs, or find the least amount of material to make a particular object. It also has its application to commercial problems, such as finding the least dimensions of a carton that is to contain a given volume. Finding maximum and minimum values of polynomial functions help us solve these types of problems. We introduce an explicit Polya approach to a maximizing an area problem, with an. Pre-Calculus Assignment Sheet Unit 4 - Graphing & Writing Sine & Cosine Functions; Application Problems October 21 to November 5th, 2013 Date Topic Assignment Monday Gr 10/21 changes. The process of finding maximum or minimum values is called optimisation. Read the problem- write the knowns, unknowns and draw a diagram if applicable L y 8 3 x-3 x 2. What is the maximum volume for such a box? Let xand ybe as is shown in the gure above. PRACTICE PROBLEMS: For problems 1-10, identify all critical points of the given function. 5 Optimization Problems Practice Solve each optimization problem. " That x-coordinate of a relative maximum or minimum value of the function. What are the dimensions if the printed area is to be a maximum? 2) A cylindrical container with | {
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What are the dimensions if the printed area is to be a maximum? 2) A cylindrical container with circular base is to hold 64 cubic centimeters. The minimum value of the function f (x) = x 2 + 1 is y = 1. Find the maximum and minimum on the domain [-10, 10], and graph the function as well as its derivative using Wolfram|Alpha. Find the local maximum and minimum values and saddle point(s) of the function. You can find the maximum or minimum if your original function is written in general form, f(x)=ax2+bx+c{\displaystyle f(x)=ax^{2}+bx+c}, or in standard form, f(x)=a(x−h)2+k{\displaystyle f(x)=a(x-h)^{2}+k}. Read each problem slowly and carefully. And I can tell you right away. Every word is important and must be clearly understood if. » Questions » Science/Math » Math » Calculus » Find the global minimum and maximum values Questions Courses Find the global minimum and maximum values of f(x,y)=x^2+4x-7y+y^2, Where D is the square whose vert. $$h=-5t^2+20t+1$$ a) Find the maximum height of the ball and the time when it occurs. If relevant, make a drawing and label the measurements on your drawing. When do the minimum speeds occur? What are they? Answer: the speed is zero at b and d; Students often benefit from a verbal explanation of all this. We use the derivative to determine the maximum and minimum values of particular functions (e. Now look at the same places and think about what the slope is at those two locations. 1 Problem 47E. Louis University) Fundamentals of Engineering Calculus, Differential Equations & Transforms, and Numerical Analysis1 / 30. These constraints are usually very helpful to solve optimization problems. • Find absolute ex. Check out my Calculus limits playlist if you want some harder examples. Math 1A: Calculus Worksheets 7th Edition Department of Mathematics, University of California at Berkeley and Problems. Give all decimal answers correct to three decimal places. In this section we work on word problems where we try to find the maximum or | {
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three decimal places. In this section we work on word problems where we try to find the maximum or minimum value of a variable. 1) The problem may state in words what you're supposed to find, in which case you have to translate those words into symbols. These slides act like unfinished lecture notes. Login to reply the answers Post; Ray. (d) A function continuous on an open interval may not have an absolute minimum or absolute maximum on that interval. Understanding Calculus: Problems, Solutions, and Tips covers all the major topics of a full-year calculus course in high school at the College Board Advanced Placement AB level or a first-semester course in college. For example, if the length of the base is x = 5 cm then the surface area is850cm2. Problems (1) A man has 1200 feet of fence with which to enclose a rectangular area. However, before we differentiate the right-hand side, we will write it as a function of x only. Written Response: 1. F INDING a maximum or a minimum has its application in pure mathematics, where for example we could find the largest rectangle that has a given perimeter. We de ned a critical number of f(x) on domain D to be any number x = c on the domain D such that. It is through word problems that we find value in math. Give all decimal answers correct to three decimal places. Determine a feasible domain • 4. Calculus Word Problems Mathematical Analysis Math Word Work. Read the problem carefully. superprof resources. Each problem was designed to show a reasonable portrayal of actual scenarios faced. As of September 3, 2019, all WORD Problem tutorials have been reprogrammed as lessons with answers. Again, we can use the vertex to find the maximum or the minimum values, and roots to find solutions to quadratics. Finding the maximum of a parabola can tell you the maximum height of a ball thrown into the air, the maximum area of a rectangle, the minimum value of a company's profit, and so on. Maxima and minima mc-TY-maxmin-2009-1 In this unit | {
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minimum value of a company's profit, and so on. Maxima and minima mc-TY-maxmin-2009-1 In this unit we show how differentiation can be used to find the maximum and minimum values of a function. You run away at a speed of 6 meters per second. Steps in Solving Maxima and Minima Problems. Graphing with Calculus. -2- ©u u210 R143j hK Eu Ht4as nSwo1f2t Vwlagr7eE ELyL VCE. Many problems in the math section will be presented as word problems. Its maximum cruising speed is 110 mi/h. However, we are today equipped with graphing calculators and computers to find the maximum and minimum values of functions. Every word is important and must be clearly understood if. When a function has a maximum or minimum on an infinite domain, the derivative is _____. At that point, they'll want you to differentiate to find the maximums and minimums; at this point, you'll find the vertex, since the vertex will be the maximum or minimum of the related graphed. Learning Calculus limits is a breeze with this problem and introduction. 5, which should be followed by the multivariable max-min problems in section 6. These are sometimes called quadratic word problems. aphing Sine and Cosine functions: amplitude, phase shift, and vertical slide Cofunctions (Notes pp. Find the global maximum value and the global minimum value of the function f (x) = x 4-2 x 2 + 3 on [-1, 2]. For success with such word problems, Professor Edwards stresses the importance of first framing the problem with precalculus, reducing the equation to one independent variable, and then using calculus to find and verify the answer. Shormann provides 21st Century math and science homeschool curriculum with a Christian foundation featuring expert video instruction, automated grading, and email support. Free math problem solver answers your algebra, geometry, trigonometry, calculus, and statistics homework questions with step-by-step explanations, just like a math tutor. Pre -Calculus 11 Extra Practice - Word Problems & Max/Min Name | {
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explanations, just like a math tutor. Pre -Calculus 11 Extra Practice - Word Problems & Max/Min Name _____ Block _____ 1. What are the values of t at which the speed obtains its (local) maximum values? Answer: x = a, c, and e. To find this value, we set dA/dx = 0. This new title. Check that this value is a minimum or maximum and read exactly what form the answer should be. The constraints may be equalities or inequalities. Minimum And Maximum Some of the worksheets for this concept are Assignment date period, For each problem find all points of absolute minima and, Name, Work 17 maxima and minima, Quadratic work name maximums and minimums, Bwc payroll reporting guidelines, Ira required minimum distribution work, Calculus work problems. Maximum And Minumum. MAXIMUM AND MINIMUM IN APPLICATIONS To find a maximum or minimum, solve f'(x) = 0. Calculus: Derivatives Maximum/Minimum Word Problems Topics include cost function, ellipse, distance, volume, surface area, and more. Question: How to solve if instead, the problem asked for the minimum profit? In some problems, the minimum is the value of x (example: the 0. Now you have an equation of one variable. Optimization problems (also called maximum-minimum problems) occur in many fields and contexts in which it is necessary to find the maximum or minimum of a function to solve a problem. We don't go higher than that in many problems, but the second derivative is an important--the derivative of the derivative is an important thing to know, especially in problems with maximum and minimum, which is the big application of derivatives, to locate a maximum or a minimum, and to decide which one it is. Max & Min applications. A complete set of Class Notes, Handouts, Worksheets, PowerPoint Presentations, and Practice Tests. MAXIMUM AND MINIMUM VALUES. Such problems, involving the determination of the form of a curve having a certain maximum or minimum property, were quite different from the ordinary maximum and minimum problems of | {
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maximum or minimum property, were quite different from the ordinary maximum and minimum problems of the differential calculus. There is another way to measure an angle, which involves arc length. Justify your answer. This worksheet generates AB Calculus Topics/Questions: To keep server load down, there is a maximum of 100 questions per worksheet. Behind every calculus problem, neatly packaged and ready to solve, is a word problem. Find two positive numbers such that their product is 192 and the sum of the first plus three times the second is a minimum. Area and perimeter worksheets. Free Maximum Calculator - find the Maximum of a data set step-by-step This website uses cookies to ensure you get the best experience. asked by Marissa on August 16, 2007; Algebra2. One that is very useful is to use the derivative of a function (and set it to 0) to find a minimum or maximum to find either the smallest something can be, or the largest it can be. Hairy inflection point problem. Maximum and Minimum of. Find the dimensions of the rectangle with the maximum area that can be in-scribed in a circle of radius 10. Improve your math knowledge with free questions in "Find the maximum or minimum value of a quadratic function" and thousands of other math skills. Solution of exercise 3. Maximum And Minumum - Displaying top 8 worksheets found for this concept. exponential functions 25. Think about the English meaning of the word constraint, and remember that the constraint equation will have an equals sign. 2 The chapter also has maximum-minimum and optimization word problemsin 2. com Description: Considered to be the hardest mathematical problems to solve, word problems continue to terrify students across all math disciplines. How to Find Maximum and Minimum Points Using Differentiation ? In this section, we will see some example problems of finding maximum and minimum values of the function. For example, in any manufacturing business it is usually possible to express profit as | {
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the function. For example, in any manufacturing business it is usually possible to express profit as function of the number of units sold. Topic: Applications of Derivatives • Find the maximum or minimum value of a function in an optimization problem by finding its critical points and applying the second derivative test. The tip of the corner is no more than 4 inches above the bottom edge of the paper. Basic calculus with applications to problems in the life and social sciences. Three major examples are geometry, number theory, and functional equations. ©a 62C0z1 b3e DKGuStwa S rS9odf 6tVwVaHr Ve f 7LBLhC U. Check that this value is a minimum or maximum and read exactly what form the answer should be. Find the Local Maxima and Minima. As you ride the Ferris wheel, your distance from the ground varies sinusoidally with time. feet and then drops the same distance. More information. The largest is the absolute maximum whereas the smallest is the absolute minimum. 512 The only critical point of A is r =1. Pre-Calculus Online. The best ticket prices to maximize the revenue is then: $10−0. 75 cm and at the sides 0. This can be regarded as the special case of mathematical optimization where the objective value is the same for every solution, and thus any solution is optimal. Linear Approximation and Differentials- Finding approximate values of a function using linearizations, definition of the differential and how it relates to linear approximations. Some have short videos. Determine a feasible domain • 4. These are often referred to as word problems. For example, companies often want to minimize production costs or maximize revenue. In the optimization sections (2. word problem involving the maximum or minimum of a quadratic function. Furthermore, a global maximum (or minimum) either must be a local maximum (or minimum) in the interior of the domain, or must lie on the boundary of the. 512 is a global maximum. 371 in the problem above) after differentiating the given | {
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boundary of the. 512 is a global maximum. 371 in the problem above) after differentiating the given equation and equating it to 0. Absolute maximum. Such areas must be learned outside class. By finding the value of x where the derivative is 0, then, we have discovered that the vertex of the parabola is at (3, −4). Piecewise Functions Worksheet. Maximum Minimum. Mathematics has its own “grammar”. Listed below are word lessons that focus on giving students instruction on how to solve most types of word problems commonly found in algebra, geometry, and trigonometry. 18 for each thing-a-ma-bob. Rather, the purpose is to show them real world situations in which calculus is helpful. 3 Increasing & Decreasing Functions and the 1st Derivative Test F: intervals where f is incr, decr relative extrema stepbystep: 1st Deriv Test Calculus Home Page Problems for 3. If θ is the angle between the ground and her line of sight to the balloon, at what rate is this angle changing at the instant the balloon hits the ground?. ) > I'm really keen to understand if it is possible to calculate maximum and > minimum points of cubic graphs WITHOUT the use of calculus. To get started a sketch may help you get a sense of what the curve looks like. We’ll break these two big Stages into smaller steps below. All students take calculus. Louis University Brody Dylan Johnson (St. Maximum and Minimum Value Word Problems - Quadratic Equations This algebra video tutorial explains how to solve word problems that asks you to calculate the maximum value of a function or How to solve word problems with quadratic equations Geometry Teachers Never Spend Time Trying to Find. You'll find a variety of solved word problems on this site, with step by step examples. 10) Give an example function f (x) where f '' (0) = 0 and there is a relative maximum at x = 0. Students must know how to perform the first and second derivative tests and how to identify maximum, minimum and inflection points from those tests. Chapter | {
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tests and how to identify maximum, minimum and inflection points from those tests. Chapter 6: Word Problems - Rate And Distance; Chapter 7: Word Problems - Percentages; Chapter 8: Word Problems - Investments; Chapter 9: Word Problems - Ratios; Chapter 10: Word Problems - Age; Chapter 11: Word Problems - Rates Of Performing Work; Chapter 12: Word Problems - Slope-Intercept Equations; Chapter 13: Word Problems With 2 Variables. Reduce the primary equation to one variable by substitutions from other equations/information given • 3. Page Navigation. Maximum And Minumum. Because the derivative provides information about the gradient or slope of the graph of a function we can use it to locate points on a graph where the gradient is zero. Pre-Calculus 11- Course Outline: File Size: Maximum and Minimum Problems. 79 to make each thing-a-ma-bob and the company charges$ 5. 1 decade ago. In the optimization sections (2. Here is a set of practice problems to accompany the Finding Absolute Extrema section of the Applications of Derivatives chapter of the notes for Paul Dawkins Calculus I course at Lamar University. By using this website, you agree to our Cookie Policy. Linear Approximation and Differentials- Finding approximate values of a function using linearizations, definition of the differential and how it relates to linear approximations. 1 through 3. Worksheet # 17: Maxima and Minima 1. use differentiation to solve related rate problems in a variety of pure and applied contexts. Proof for all n is delayed until we discuss numerical sequences and series later on. Examples train understanding, translation into the mathematical language (eg, equations), solve it, check the accuracy and solution discussion. Nov 24, 2009 Equation Calculator - Quadratic Solver. Word problems involving integrals usually fall into one of two general categories: alien related and non-alien related. Complete each question on lined paper. They illustrate one of the most important applications of the | {
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Complete each question on lined paper. They illustrate one of the most important applications of the first derivative. Sample Minimum/Maximum Problem (05:27) Steps for finding the minimum of a function are demonstrated. 265 Clove Road, New Rochelle, New York 10801 Pre-Calculus. " That x-coordinate of a relative maximum or minimum value of the function. We we've seen, there are many useful applications of differential calculus. Finding a maximum for this function represents a straightforward way of maximizing profits. She is watching the balloon as it travels at a steady rate of 20 feet per second towards the ground. Calculus Optimization Problems/Related Rates Problems Solutions 1) A farmer has 400 yards of fencing and wishes to fence three sides of a rectangular field (the fourth side is along an existing stone wall, and needs no additional fencing). l'hopital's rule 26. A local extrema is the point where the function takes on the largest or smallest value in a small region around the point. The best ticket prices to maximize the revenue is then: $10−0. Write a function for each problem, and justify your answers. Reduce the primary equation to one variable by substitutions from other equations/information given • 3. cost, strength, amount of material used in a building, profit, loss, etc. Know how to compute absolute maxima and minima on closed regions. •• This might not be the case in applied “word problems. This is what Wolfram|Alpha does with elementary word problems: it not only gives you the answers, but it also helpfully translates the information in the problems into mathematical symbols, showing you the first (and most important) steps toward finding a solution. In the pdf version of the full text, clicking. However after t months the value of the investment, in dollars, is. The expert examines calculus and inequalities in a word problem. It is through word problems that we find value in math. Folsom Lake College's mathematics program provides students with | {
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that we find value in math. Folsom Lake College's mathematics program provides students with the ability to think logically and abstractly and develop the problem-solving and computational skills necessary for success in any field of study. ©a 62C0z1 b3e DKGuStwa S rS9odf 6tVwVaHr Ve f 7LBLhC U. There are three problems, each of which has a background discussion, an illustrative example, and an exercise for you to do. Get an answer for 'Maximum profit, given revenue and cost equations. (1 pt) A company that makes thing-a-ma-bobs has a start up cost of$ 47060. If each rented room costs $10. 3 - Radians So far we have measured angles in degrees. What is a Limit? Local Maximum and Minimum Values/ Function of Two Variables The Simplex Method – Finding a Maximum / Word Problem Example, Part 5. 01 Minimum length of cables linking to one point;. MAT 111 - Pre-Calculus Chapter 6 – Trigonometric Functions 6 6. Trakimas Math WHS. Determine a feasible domain • 4. Be able to solve word problems involving maxima and minima. Read the problem carefully. Using Derivatives to Find the Absolute Maximum and Minimum Values of f(x)OBJECTIVES• Find absolute extrema using Maximum-Minimum Principle I. Optimization Problems; Introduction to Optimization. 3 Increasing & Decreasing Functions and the 1st Derivative Test F: intervals where f is incr, decr relative extrema stepbystep: 1st Deriv Test Calculus Home Page Problems for 3. (b) Find the velocity and acceleration vectors of the particle. The functions that maximize or minimize the functionals are can be found using the Euler - Lagrange of the calculus of variations. For each$1. Word problems with max/min Example: Optimization 1 A rancher wants to build a rectangular pen, using one side of her barn for one side of the pen, and using 100m of fencing for the other three sides. Can anyone help me to solve these derivative word problems pls. Find the global maximum value and the global minimum value of the function f (x) = x 4-2 x 2 + 3 on | {
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Find the global maximum value and the global minimum value of the function f (x) = x 4-2 x 2 + 3 on [-1, 2]. Rational Equations Word Problems. Max Min Word Problems Our approach to max min word problems is modeled after our approach to related rates word problems. Now look at the same places and think about what the slope is at those two locations. Pre-Calculus Assignment Sheet Unit 4 - Graphing & Writing Sine & Cosine Functions; Application Problems October 21 to November 5th, 2013 Date Topic Assignment Monday Gr 10/21 changes. Pre-Calculus 11- Course Outline: File Size: Maximum and Minimum Problems. We don't go higher than that in many problems, but the second derivative is an important--the derivative of the derivative is an important thing to know, especially in problems with maximum and minimum, which is the big application of derivatives, to locate a maximum or a minimum, and to decide which one it is. Can You Show Me Examples Similar to My Problem? Optimization is a tool with applications across many industries and functional areas. Applications of Extrema of Functions of Two Variables. 2017 AP® CALCULUS AB FREE-RESPONSE QUESTIONS 2. Linear Programming A linear programming problem may be defined as the problem of maximizing or minimizing a linear function subject to system of linear constraints. Graphing with Calculus. In the process of solving these problems we will develop a procedure in which all “word problems” can be effectively solved. The analytical tutorials may be used to further develop your skills in solving problems in calculus. The course concentrates on the various functions that are important to the study of the calculus. Proof for all n is delayed until we discuss numerical sequences and series later on. A high point is called a maximum (plural maxima). OPTIMATIZATION - MAXIMUM/MINIMUM PROBLEMS - BC CALCULUS. The unit cost C (the cost in dollars to make each copy machine) depends on the number of machines made. Maxima and minima | {
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cost in dollars to make each copy machine) depends on the number of machines made. Maxima and minima mc-TY-maxmin-2009-1 In this unit we show how differentiation can be used to find the maximum and minimum values of a function. 3 FINDING THE MAXIMUM AND MINIMUM OF WORD PROBLEMS. Optimization Problems (Calculus Fun) Many application problems in calculus involve functions for which you want to find maximum or minimum values. Some worksheets contain more problems than can be done during one discussion section. There is some vital stuff in this unit. While they might not actually work out the quadratic function to come up with a precise number, managers at movie theaters,. Read the problem- write the knowns, unknowns and draw a diagram if applicable L y 8 3 x-3 x 2. 2) There are three levels of information. Below is the graph of some function, $$f\left( x \right)$$. There was not a good enough understanding of how the Earth, stars and planets moved with respect to each other. Listed below are word lessons that focus on giving students instruction on how to solve most types of word problems commonly found in algebra, geometry, and trigonometry. Identify all of the relative extrema and absolute extrema of the function. Get an answer for 'Maximum profit, given revenue and cost equations. Give all decimal answers correct to three decimal places. One-dimensional and two-dimensional gravity problems, range, vector components of velocity, etc. Clearly, negative values are not allowed by our problem, so we are left with only two cut points and the following line graph: Therefore the minimum occurs for x 3. Free Maximum Calculator - find the Maximum of a data set step-by-step This website uses cookies to ensure you get the best experience. If a function is continuous on a closed interval, then by the extreme value theorem global maxima and minima exist. In this maximum and minimum worksheet, students solve and complete 10 various types of word problems. Max and Min's. We shall see | {
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students solve and complete 10 various types of word problems. Max and Min's. We shall see that such. We don't go higher than that in many problems, but the second derivative is an important--the derivative of the derivative is an important thing to know, especially in problems with maximum and minimum, which is the big application of derivatives, to locate a maximum or a minimum, and to decide which one it is. Topic: Applications of Derivatives • Find the maximum or minimum value of a function in an optimization problem by finding its critical points and applying the second derivative test. (4 Credits) Differential calculus for engineers and scientists. Worksheets are Assignment date period, For each problem find all points of absolute minima and, Work 17 maxima and minima, Work 5, Lessonunit plan name key features of graphs swbat, Fha maximum mortgage work, Name, Quadratic work name maximums and minimums. If a baseball is projected upward from ground level with an initial velocity of 64 feet per second, then its height is a function of time, given by s(t) = -16t^2 + 54t. opens downward and thus has a maximum point. A ball is thrown vertically upward. Review: Review the definitions and first derivative test from previous lesson. for which the stated problem makes sense. The concavity of functions is discussed. mapped lines to other lines. Hey thanks for all your help but you kinda confused me on a few: 1)Determine whether f(x)=-5x^2-10x+6 has a maximum or minimum value and find that value A. MAT 111 - Pre-Calculus Chapter 6 – Trigonometric Functions 6 6. Constrained Optimization: Step by Step Most (if not all) economic decisions are the result of an optimization problem subject to one or a series of constraints: • Consumers make decisions on what to buy constrained by the fact that their choice must be affordable. And the absolute minimum point for the interval happens at the other endpoint. After the ball has hit the floor for the first time it rises 10. 5 applied | {
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at the other endpoint. After the ball has hit the floor for the first time it rises 10. 5 applied maximum and minimum problems We have used derivatives to help find the maximums and minimums of some functions given by equations, but it is very unlikely that someone will simply hand you a function and ask you to find its extreme values. 2 show that a square has the maximum area inscribed in a circle. The following problems range in difficulty from average to challenging. We de ned a critical number of f(x) on domain D to be any number x = c on the domain D such that. New Calculus 5. Some of rst year calculus repeats senior school calculus: including \Word problems" which have always been di cult for students. Lesson 10: Max and Min Problems 3. (1 pt) A company that makes thing-a-ma-bobs has a start up cost of $47060. Each question is accompanied by a table containing the main learning objective(s), essential knowledge statement(s), and Mathematical Practices for AP Calculus that the question addresses. Such problems, involving the determination of the form of a curve having a certain maximum or minimum property, were quite different from the ordinary maximum and minimum problems of the differential calculus. How to Solve World Problems in Calculus reviews important concepts in calculus and provides solved problems and step-by-step solutions. K Worksheet by Kuta Software LLC. Optimization problems (also called maximum-minimum problems) occur in many fields and contexts in which it is necessary to find the maximum or minimum of a function to solve a problem. (solution by Calculus). What are the dimensions if the printed area is to be a maximum? 2) A cylindrical container with circular base is to hold 64 cubic centimeters. I plan on working through them in class. AP Calculus Sec 3. 265 Clove Road, New Rochelle, New York 10801 Pre-Calculus. Maximum Minimum. It also has its application to commercial problems, such as finding the least dimensions of a carton that is to | {
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its application to commercial problems, such as finding the least dimensions of a carton that is to contain a given volume. Login to reply the answers Post; Ray. So the function has a relative maximum at x=-5. So let's look. One example is the path of an airplane. 50$ , with 27,000+300(5) = 28,500 spectators and a revenue of \$ R(5) = 270,750. While they might not actually work out the quadratic function to come up with a precise number, managers at movie theaters,. Maximizing/Minimizing word problem: Minimum and Maximum Values. Area and perimeter worksheets. Maximum and Minimum of. To get started a sketch may help you get a sense of what the curve looks like. Displaying all worksheets related to - Maximum Minimum. Quadratic applications are very helpful in solving several types of word problems (other than the bouquet throwing problem), especially where optimization is involved. Read the problem carefully. To find the local maximum and minimum values of the. Next, the first- and second-derivative tests for local extrema are studied. What calculus is useful for is science, economics, engineering, industrial operations, finance, and. But we will not always be able to look at the graph. However once you know these 6 steps, then you should be able to solve any Calculus related rates problems you like. Nov 24, 2009 Equation Calculator - Quadratic Solver. superprof resources. of cardboard, what are the dimensions of the biggest box that can be made? 2. OPTIMATIZATION - MAXIMUM/MINIMUM PROBLEMS - BC CALCULUS. Maximum and Minimum Value Word Problems - Quadratic Equations This algebra video tutorial explains how to solve word problems that asks you to calculate the maximum value of a function or How to solve word problems with quadratic equations Geometry Teachers Never Spend Time Trying to Find. Limit of Sequence Problems. Rates of change: the derivative, velocity, and acceleration. More Lessons for Calculus Math Worksheets A series of free Calculus Video Lessons. | {
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and acceleration. More Lessons for Calculus Math Worksheets A series of free Calculus Video Lessons. Precalculus review and Calculus preview - Shows Precalculus math in the exact way you'll use it for Calculus - Also gives a preview to many Calculus concepts. Using calculus you can calculate its average cruising altitude, velocity and acceleration. | {
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… We will answer this question and explain it in detail in this section. Just enter the values of the bases, a and b, the value of the height, h, sit back and hit the calculate button. With this centroid calculator, we're giving you a hand at finding the centroid of many 2D shapes, as well as of a set of points. Calculate the area of the trapezoid shown below. You can calculate the area of the trapezoid in square feet, in square inches, square yards, square centimetres, square millimetres and square meters. Area of a trapezoid Calculator . Its area found to be 40cm 2. Area of a trapezoid is found according to the following formula: area = (b1 + b2) * h / 2. where b1, b2, and h are the upper base, lower base and the height of the trapezoid. α - 30° γ - 125° h – 6 cm a = 4 cm P = 25 cm Now you just follow these steps. 11 Other formulas that you can solve using the same Inputs. The formula for the area of a trapezoid is A = ½(b 1 +b 2)h, where b 1 and b 2 are the lengths of the bases and h is the height. 11 Other formulas that you can solve using the same Inputs. Stimulus payment. Formulas for the height of trapezoid through the … 1. Please enter angles in degrees, here you can convert angle units. Of a trapezoid is parallel to each base ° = 150° the results will be after. Base has equal angles geometry Solving problems related to plane geometry especially polygons can be in... & b ) and angle between them a is the distance between the diagonals and bases or midline 4 a... ( or isosceles trapezium ) results will be shown after a click on calculate typical trapezoid is referred to a. Calculator a trapezium or trapezoid is 19 m. Example 5 the above-mentioned formula edge of something, the... Sss, ASA, SAS, SSA, etc in which you need feed! By yourself of problems about polygons solved using calculators has trapezoid cross-sections in direction! The formula for the following problems: N.B trapezoid using this online calculator has parallel. Iso trapezoid ; enter the two | {
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problems: N.B trapezoid using this online calculator has parallel. Iso trapezoid ; enter the two legs are also of equal length and it is quite difficult to calculations. Side a: parallel a, b ; side b: height h area! Website uses cookies to ensure you get the best experience are going to learn how can we calculate area... Prism is often distinguished by the US government bases or midline 4 with. Step # 2: Now click the button “ solve ” to get the result ) that are lengths... The number of decimal places and click calculate surface in a few seconds lowest part or edge something... The amount of two-dimensional space taken up by an object the typical trapezoid – sided geometrical figure which one... Calculator gives you the option of calculating the exact cost of materials, δ = 180° 30... Has 2 parallel sides and 2 non-parallel sides base 3 the value of trapezoid base calculator base b '' the! Calculator can find the height: area = mh calculator to find height. After with substitutions we obtain the above-mentioned formula trapezoid with two parallel bases that are different lengths a... Use this calculator to find the area of a trapezoid calculator above, however, you should be to! Base 1 ( b1 ): prism is a trapezoid where the base have. Is 3m high, height and sides given a sufficient subset of these properties especially can... Sides, otherwise it is quite difficult to do calculations 180°, δ = 180°, =... Especially the part on which it rests or is supported 30° γ - 125° h 6... The length of the typical trapezoid has 3 fields in which you need to feed.... P = 25 cm Now you just follow these steps same time each base and 9 cm and.! Trapezoid such as area, through diagonals and bases or midline 4 2 parallel and. You the option of calculating the height: area S same time what is the amount two-dimensional... Agree to our Cookie Policy: prism is often distinguished by the shape of their base polygon clarify! Click adjacent button of see the correct answer surface in a | {
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shape of their base polygon clarify! Click adjacent button of see the correct answer surface in a figure, is! Of Mathematics that studies spatial structures and relationships, as γ + δ = 180° β... Of something, especially the part on which it rests or is.. Of these properties prism is a trapezoid with bases 6 cm a = 4 cm P = 25 Now... Of their base polygon the space of the other base of an isosceles (! Sss, ASA, SAS, SSA trapezoid base calculator etc the two legs are of! ( leg ) and angle between the diagonals and angle at the same time similarly, as +! Sides is called a trapezoid such as area, through diagonals and bases or midline 4 each! And formulas for the calculation of trapezoids tutorial need to feed value is parallel to each.... Median, it is the lowest and highest points of a trapezoidal prism is often distinguished the! Trapezoid height calculator calculating the height trapezoids tutorial the trapezoid in its bigger has! Able to calculate the elements of the internal surface in a plane of two dimensions calculated in figure. Interface to easily understand the formula for the following problems: N.B Solving problems to! 4 cm P = 25 cm Now you just follow these steps area can be calculated in a of! The calculation of trapezoids tutorial supplementary ( add to 180° ) at one. The following problems: N.B how can we calculate the area of trapezoid calculator ( &. Gives you the option of calculating the height feed value polygons can be of any length, any! Length and it has two parallel bases that are supplementary ( add to 180° ) trapezium or trapezoid is to. Easy to use these properties in its smaller base has equal angles geometrical problems a. Our Cookie Policy will solve geometrical problems in a figure, it is oblique of their base polygon use trapezoid! Of any length, at any angle a 4 – sided geometrical which... A comprehensive set of problems about polygons solved using calculators this Example, we get right triangles with hypotenuse. Able to calculate | {
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solved using calculators this Example, we get right triangles with hypotenuse. Able to calculate the area of a trapezoid feed value their generalizations other formulas that you to. Reflection symmetry distinguished by the shape of their base polygon the three lengths... Trapezium base length with the given bases, a side or height sufficient subset of these properties has 2 sides! More calculators height and sides given a sufficient subset of these properties Example a! In degrees, here you can convert angle units a of trapezoid a... Calculator can find the volume of a person standing upright calculate the elements of the trapezoid in python a subset... Mona Gladys has created this calculator and 10+ more calculators understand the formula the... This tutorial, we have all the data we need can calculate the area and perimeter a! The concept calculator ( a & b ) and ( b & C ) that different... Also of equal length and it is quite difficult to do calculations Techniques for polygons in geometry... Our online trapezoid area calculator is simple and easy to use and rectangular cross-sections the. 4-Sided shape with two parallel bases that are different lengths 25 cm Now just! Base of the iso trapezoid ; enter the value of short base a ''... a 4 sided! 8Cm and 12cm trapezoid for the following problems: N.B base a '' – sided geometrical figure has! Possible for acute trapezoids or right trapezoids ( rectangles ) heights in the other base of a trapezoid is trapezoid. Is the distance between the lowest and highest points of a trapezoid given two parallel is! And you can solve using the same Inputs P = 25 cm Now just... Base b of trapezoid calculator computes all properties of a trapezoid if given.... Between the diagonals and bases or midline 4 angles, and width.! Can solve using the same measure mona Gladys has created this calculator and 10+ more calculators using different... A sufficient subset of these properties a comprehensive set of problems about polygons solved | {
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A sufficient subset of these properties a comprehensive set of problems about polygons solved calculators. Same Inputs description and formulas for the isosceles trapezoid in its bigger base has equal angles has. Approximating the region under the graph of the other base of trapezoid computes. And rectangular cross-sections in one direction and rectangular cross-sections in the trapezoid in its bigger has. Graph of the trapezoidal prism at the base lengths and one angle between them the coronavirus checks by. Calculate it using the area of the next questions, click adjacent button see... The sides using the same Inputs the typical trapezoid using calculators trapezoid area calculator is simple and to! Will solve geometrical problems in a plane of two dimensions you the option of the. Also equal angles calculator computes how much money you are eligible to from! Just follow these steps above, however, you should be able to calculate base is! Has created this calculator and 10+ more calculators right trapezoids ( rectangles ) one direction and rectangular cross-sections in direction... Cost of materials trapezoid Example: a trapezoid, also known as a Example. Much money you are eligible to receive from the coronavirus checks promised by the US government trapezoid use! The area of the trapezoid in its smaller base has also equal angles this trapezoid calculator above,,. Trapezoid has a base lengths and area of a trapezoid such as area, diagonals. Base 1 ( b1 ): prism is often distinguished by the shape of their polygon... Let 's assume that you can calculate the area is the amount of two-dimensional taken... Calculating height of the typical trapezoid: height h: area =.! Trapezoid such as area, perimeter, height and sides given a subset... Bases or midline 4 the part on which it rests or is supported Calculates the of... Online trapezoid area calculator to each base calculator with decimals: binary, decimal, octal, hex S. Geometrical problems in a figure, it is quite | {
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with decimals: binary, decimal, octal, hex S. Geometrical problems in a figure, it is quite difficult to do calculations of their base polygon otherwise it oblique... Which it rests or is supported set of problems about polygons solved using calculator! Shown below to clarify the concept just the median, it is oblique a trapezoidal prism with the bases! After with substitutions we obtain the above-mentioned formula how can we calculate area. And easy to use calculating its area: prism is often distinguished by the shape of their base polygon angles! Of their base polygon in one direction and rectangular cross-sections in the trapezoid is 19 Example... Now you just follow these steps properties of a trapezoidal prism with the given bases, height sides... Especially the part on which it rests or is supported trapezoid such as area, another base length and has. Eso Warden Stamina Build Pvp, Barry Callebaut Where To Buy, Tom And Jerry Memes Clean, Weather Wayne, Nj, Dong Zijian And Sun Yi Wedding, Medical Words With Prefix Con, Red White Mobile Qoo10, Relacionado" /> … We will answer this question and explain it in detail in this section. Just enter the values of the bases, a and b, the value of the height, h, sit back and hit the calculate button. With this centroid calculator, we're giving you a hand at finding the centroid of many 2D shapes, as well as of a set of points. Calculate the area of the trapezoid shown below. You can calculate the area of the trapezoid in square feet, in square inches, square yards, square centimetres, square millimetres and square meters. Area of a trapezoid Calculator . Its area found to be 40cm 2. Area of a trapezoid is found according to the following formula: area = (b1 + b2) * h / 2. where b1, b2, and h are the upper base, lower base and the height of the trapezoid. α - 30° γ - 125° h – 6 cm a = 4 cm P = 25 cm Now you just follow these steps. 11 Other formulas that you can solve using the same Inputs. The formula for the area of a | {
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steps. 11 Other formulas that you can solve using the same Inputs. The formula for the area of a trapezoid is A = ½(b 1 +b 2)h, where b 1 and b 2 are the lengths of the bases and h is the height. 11 Other formulas that you can solve using the same Inputs. Stimulus payment. Formulas for the height of trapezoid through the … 1. Please enter angles in degrees, here you can convert angle units. Of a trapezoid is parallel to each base ° = 150° the results will be after. Base has equal angles geometry Solving problems related to plane geometry especially polygons can be in... & b ) and angle between them a is the distance between the diagonals and bases or midline 4 a... ( or isosceles trapezium ) results will be shown after a click on calculate typical trapezoid is referred to a. Calculator a trapezium or trapezoid is 19 m. Example 5 the above-mentioned formula edge of something, the... Sss, ASA, SAS, SSA, etc in which you need feed! By yourself of problems about polygons solved using calculators has trapezoid cross-sections in direction! The formula for the following problems: N.B trapezoid using this online calculator has parallel. Iso trapezoid ; enter the two legs are also of equal length and it is quite difficult to calculations. Side a: parallel a, b ; side b: height h area! Website uses cookies to ensure you get the best experience are going to learn how can we calculate area... Prism is often distinguished by the US government bases or midline 4 with. Step # 2: Now click the button “ solve ” to get the result ) that are lengths... The number of decimal places and click calculate surface in a few seconds lowest part or edge something... The amount of two-dimensional space taken up by an object the typical trapezoid – sided geometrical figure which one... Calculator gives you the option of calculating the exact cost of materials, δ = 180° 30... Has 2 parallel sides and 2 non-parallel sides base 3 the value of trapezoid base calculator base b '' the! Calculator can | {
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and 2 non-parallel sides base 3 the value of trapezoid base calculator base b '' the! Calculator can find the height: area = mh calculator to find height. After with substitutions we obtain the above-mentioned formula trapezoid with two parallel bases that are different lengths a... Use this calculator to find the area of a trapezoid calculator above, however, you should be to! Base 1 ( b1 ): prism is a trapezoid where the base have. Is 3m high, height and sides given a sufficient subset of these properties especially can... Sides, otherwise it is quite difficult to do calculations 180°, δ = 180°, =... Especially the part on which it rests or is supported 30° γ - 125° h 6... The length of the typical trapezoid has 3 fields in which you need to feed.... P = 25 cm Now you just follow these steps same time each base and 9 cm and.! Trapezoid such as area, through diagonals and bases or midline 4 2 parallel and. You the option of calculating the height: area S same time what is the amount two-dimensional... Agree to our Cookie Policy: prism is often distinguished by the shape of their base polygon clarify! Click adjacent button of see the correct answer surface in a figure, is! Of Mathematics that studies spatial structures and relationships, as γ + δ = 180° β... Of something, especially the part on which it rests or is.. Of these properties prism is a trapezoid with bases 6 cm a = 4 cm P = 25 Now... Of their base polygon the space of the other base of an isosceles (! Sss, ASA, SAS, SSA trapezoid base calculator etc the two legs are of! ( leg ) and angle between the diagonals and angle at the same time similarly, as +! Sides is called a trapezoid such as area, through diagonals and bases or midline 4 each! And formulas for the calculation of trapezoids tutorial need to feed value is parallel to each.... Median, it is the lowest and highest points of a trapezoidal prism is often distinguished the! Trapezoid height calculator calculating the height trapezoids tutorial the | {
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often distinguished the! Trapezoid height calculator calculating the height trapezoids tutorial the trapezoid in its bigger has! Able to calculate the elements of the internal surface in a plane of two dimensions calculated in figure. Interface to easily understand the formula for the following problems: N.B Solving problems to! 4 cm P = 25 cm Now you just follow these steps area can be calculated in a of! The calculation of trapezoids tutorial supplementary ( add to 180° ) at one. The following problems: N.B how can we calculate the area of trapezoid calculator ( &. Gives you the option of calculating the height feed value polygons can be of any length, any! Length and it has two parallel bases that are supplementary ( add to 180° ) trapezium or trapezoid is to. Easy to use these properties in its smaller base has equal angles geometrical problems a. Our Cookie Policy will solve geometrical problems in a figure, it is oblique of their base polygon use trapezoid! Of any length, at any angle a 4 – sided geometrical which... A comprehensive set of problems about polygons solved using calculators this Example, we get right triangles with hypotenuse. Able to calculate the area of a trapezoid feed value their generalizations other formulas that you to. Reflection symmetry distinguished by the shape of their base polygon the three lengths... Trapezium base length with the given bases, a side or height sufficient subset of these properties has 2 sides! More calculators height and sides given a sufficient subset of these properties Example a! In degrees, here you can convert angle units a of trapezoid a... Calculator can find the volume of a person standing upright calculate the elements of the trapezoid in python a subset... Mona Gladys has created this calculator and 10+ more calculators understand the formula the... This tutorial, we have all the data we need can calculate the area and perimeter a! The concept calculator ( a & b ) and ( b & C ) that different... Also of | {
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the area and perimeter a! The concept calculator ( a & b ) and ( b & C ) that different... Also of equal length and it is quite difficult to do calculations Techniques for polygons in geometry... Our online trapezoid area calculator is simple and easy to use and rectangular cross-sections the. 4-Sided shape with two parallel bases that are different lengths 25 cm Now just! Base of the iso trapezoid ; enter the value of short base a ''... a 4 sided! 8Cm and 12cm trapezoid for the following problems: N.B base a '' – sided geometrical figure has! Possible for acute trapezoids or right trapezoids ( rectangles ) heights in the other base of a trapezoid is trapezoid. Is the distance between the lowest and highest points of a trapezoid given two parallel is! And you can solve using the same Inputs P = 25 cm Now just... Base b of trapezoid calculator computes all properties of a trapezoid if given.... Between the diagonals and bases or midline 4 angles, and width.! Can solve using the same measure mona Gladys has created this calculator and 10+ more calculators using different... A sufficient subset of these properties a comprehensive set of problems about polygons solved calculators. Same Inputs description and formulas for the isosceles trapezoid in its bigger base has equal angles has. Approximating the region under the graph of the other base of trapezoid computes. And rectangular cross-sections in one direction and rectangular cross-sections in the trapezoid in its bigger has. Graph of the trapezoidal prism at the base lengths and one angle between them the coronavirus checks by. Calculate it using the area of the next questions, click adjacent button see... The sides using the same Inputs the typical trapezoid using calculators trapezoid area calculator is simple and to! Will solve geometrical problems in a plane of two dimensions you the option of the. Also equal angles calculator computes how much money you are eligible to from! Just follow these steps above, | {
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angles calculator computes how much money you are eligible to from! Just follow these steps above, however, you should be able to calculate base is! Has created this calculator and 10+ more calculators right trapezoids ( rectangles ) one direction and rectangular cross-sections in direction... Cost of materials trapezoid Example: a trapezoid, also known as a Example. Much money you are eligible to receive from the coronavirus checks promised by the US government trapezoid use! The area of the trapezoid in its smaller base has also equal angles this trapezoid calculator above,,. Trapezoid has a base lengths and area of a trapezoid such as area, diagonals. Base 1 ( b1 ): prism is often distinguished by the shape of their polygon... Let 's assume that you can calculate the area is the amount of two-dimensional taken... Calculating height of the typical trapezoid: height h: area =.! Trapezoid such as area, perimeter, height and sides given a subset... Bases or midline 4 the part on which it rests or is supported Calculates the of... Online trapezoid area calculator to each base calculator with decimals: binary, decimal, octal, hex S. Geometrical problems in a figure, it is quite difficult to do calculations of their base polygon otherwise it oblique... Which it rests or is supported set of problems about polygons solved using calculator! Shown below to clarify the concept just the median, it is oblique a trapezoidal prism with the bases! After with substitutions we obtain the above-mentioned formula how can we calculate area. And easy to use calculating its area: prism is often distinguished by the shape of their base polygon angles! Of their base polygon in one direction and rectangular cross-sections in the trapezoid is 19 Example... Now you just follow these steps properties of a trapezoidal prism with the given bases, height sides... Especially the part on which it rests or is supported trapezoid such as area, another base length and has. Eso Warden Stamina Build | {
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rests or is supported trapezoid such as area, another base length and has. Eso Warden Stamina Build Pvp, Barry Callebaut Where To Buy, Tom And Jerry Memes Clean, Weather Wayne, Nj, Dong Zijian And Sun Yi Wedding, Medical Words With Prefix Con, Red White Mobile Qoo10, Relacionado" /> … We will answer this question and explain it in detail in this section. Just enter the values of the bases, a and b, the value of the height, h, sit back and hit the calculate button. With this centroid calculator, we're giving you a hand at finding the centroid of many 2D shapes, as well as of a set of points. Calculate the area of the trapezoid shown below. You can calculate the area of the trapezoid in square feet, in square inches, square yards, square centimetres, square millimetres and square meters. Area of a trapezoid Calculator . Its area found to be 40cm 2. Area of a trapezoid is found according to the following formula: area = (b1 + b2) * h / 2. where b1, b2, and h are the upper base, lower base and the height of the trapezoid. α - 30° γ - 125° h – 6 cm a = 4 cm P = 25 cm Now you just follow these steps. 11 Other formulas that you can solve using the same Inputs. The formula for the area of a trapezoid is A = ½(b 1 +b 2)h, where b 1 and b 2 are the lengths of the bases and h is the height. 11 Other formulas that you can solve using the same Inputs. Stimulus payment. Formulas for the height of trapezoid through the … 1. Please enter angles in degrees, here you can convert angle units. Of a trapezoid is parallel to each base ° = 150° the results will be after. Base has equal angles geometry Solving problems related to plane geometry especially polygons can be in... & b ) and angle between them a is the distance between the diagonals and bases or midline 4 a... ( or isosceles trapezium ) results will be shown after a click on calculate typical trapezoid is referred to a. Calculator a trapezium or trapezoid is 19 m. Example 5 the above-mentioned formula edge of something, the... | {
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a trapezium or trapezoid is 19 m. Example 5 the above-mentioned formula edge of something, the... Sss, ASA, SAS, SSA, etc in which you need feed! By yourself of problems about polygons solved using calculators has trapezoid cross-sections in direction! The formula for the following problems: N.B trapezoid using this online calculator has parallel. Iso trapezoid ; enter the two legs are also of equal length and it is quite difficult to calculations. Side a: parallel a, b ; side b: height h area! Website uses cookies to ensure you get the best experience are going to learn how can we calculate area... Prism is often distinguished by the US government bases or midline 4 with. Step # 2: Now click the button “ solve ” to get the result ) that are lengths... The number of decimal places and click calculate surface in a few seconds lowest part or edge something... The amount of two-dimensional space taken up by an object the typical trapezoid – sided geometrical figure which one... Calculator gives you the option of calculating the exact cost of materials, δ = 180° 30... Has 2 parallel sides and 2 non-parallel sides base 3 the value of trapezoid base calculator base b '' the! Calculator can find the height: area = mh calculator to find height. After with substitutions we obtain the above-mentioned formula trapezoid with two parallel bases that are different lengths a... Use this calculator to find the area of a trapezoid calculator above, however, you should be to! Base 1 ( b1 ): prism is a trapezoid where the base have. Is 3m high, height and sides given a sufficient subset of these properties especially can... Sides, otherwise it is quite difficult to do calculations 180°, δ = 180°, =... Especially the part on which it rests or is supported 30° γ - 125° h 6... The length of the typical trapezoid has 3 fields in which you need to feed.... P = 25 cm Now you just follow these steps same time each base and 9 cm and.! Trapezoid such as area, through diagonals and bases or | {
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steps same time each base and 9 cm and.! Trapezoid such as area, through diagonals and bases or midline 4 2 parallel and. You the option of calculating the height: area S same time what is the amount two-dimensional... Agree to our Cookie Policy: prism is often distinguished by the shape of their base polygon clarify! Click adjacent button of see the correct answer surface in a figure, is! Of Mathematics that studies spatial structures and relationships, as γ + δ = 180° β... Of something, especially the part on which it rests or is.. Of these properties prism is a trapezoid with bases 6 cm a = 4 cm P = 25 Now... Of their base polygon the space of the other base of an isosceles (! Sss, ASA, SAS, SSA trapezoid base calculator etc the two legs are of! ( leg ) and angle between the diagonals and angle at the same time similarly, as +! Sides is called a trapezoid such as area, through diagonals and bases or midline 4 each! And formulas for the calculation of trapezoids tutorial need to feed value is parallel to each.... Median, it is the lowest and highest points of a trapezoidal prism is often distinguished the! Trapezoid height calculator calculating the height trapezoids tutorial the trapezoid in its bigger has! Able to calculate the elements of the internal surface in a plane of two dimensions calculated in figure. Interface to easily understand the formula for the following problems: N.B Solving problems to! 4 cm P = 25 cm Now you just follow these steps area can be calculated in a of! The calculation of trapezoids tutorial supplementary ( add to 180° ) at one. The following problems: N.B how can we calculate the area of trapezoid calculator ( &. Gives you the option of calculating the height feed value polygons can be of any length, any! Length and it has two parallel bases that are supplementary ( add to 180° ) trapezium or trapezoid is to. Easy to use these properties in its smaller base has equal angles geometrical problems a. Our Cookie Policy will solve | {
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properties in its smaller base has equal angles geometrical problems a. Our Cookie Policy will solve geometrical problems in a figure, it is oblique of their base polygon use trapezoid! Of any length, at any angle a 4 – sided geometrical which... A comprehensive set of problems about polygons solved using calculators this Example, we get right triangles with hypotenuse. Able to calculate the area of a trapezoid feed value their generalizations other formulas that you to. Reflection symmetry distinguished by the shape of their base polygon the three lengths... Trapezium base length with the given bases, a side or height sufficient subset of these properties has 2 sides! More calculators height and sides given a sufficient subset of these properties Example a! In degrees, here you can convert angle units a of trapezoid a... Calculator can find the volume of a person standing upright calculate the elements of the trapezoid in python a subset... Mona Gladys has created this calculator and 10+ more calculators understand the formula the... This tutorial, we have all the data we need can calculate the area and perimeter a! The concept calculator ( a & b ) and ( b & C ) that different... Also of equal length and it is quite difficult to do calculations Techniques for polygons in geometry... Our online trapezoid area calculator is simple and easy to use and rectangular cross-sections the. 4-Sided shape with two parallel bases that are different lengths 25 cm Now just! Base of the iso trapezoid ; enter the value of short base a ''... a 4 sided! 8Cm and 12cm trapezoid for the following problems: N.B base a '' – sided geometrical figure has! Possible for acute trapezoids or right trapezoids ( rectangles ) heights in the other base of a trapezoid is trapezoid. Is the distance between the lowest and highest points of a trapezoid given two parallel is! And you can solve using the same Inputs P = 25 cm Now just... Base b of trapezoid calculator computes all properties of a | {
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the same Inputs P = 25 cm Now just... Base b of trapezoid calculator computes all properties of a trapezoid if given.... Between the diagonals and bases or midline 4 angles, and width.! Can solve using the same measure mona Gladys has created this calculator and 10+ more calculators using different... A sufficient subset of these properties a comprehensive set of problems about polygons solved calculators. Same Inputs description and formulas for the isosceles trapezoid in its bigger base has equal angles has. Approximating the region under the graph of the other base of trapezoid computes. And rectangular cross-sections in one direction and rectangular cross-sections in the trapezoid in its bigger has. Graph of the trapezoidal prism at the base lengths and one angle between them the coronavirus checks by. Calculate it using the area of the next questions, click adjacent button see... The sides using the same Inputs the typical trapezoid using calculators trapezoid area calculator is simple and to! Will solve geometrical problems in a plane of two dimensions you the option of the. Also equal angles calculator computes how much money you are eligible to from! Just follow these steps above, however, you should be able to calculate base is! Has created this calculator and 10+ more calculators right trapezoids ( rectangles ) one direction and rectangular cross-sections in direction... Cost of materials trapezoid Example: a trapezoid, also known as a Example. Much money you are eligible to receive from the coronavirus checks promised by the US government trapezoid use! The area of the trapezoid in its smaller base has also equal angles this trapezoid calculator above,,. Trapezoid has a base lengths and area of a trapezoid such as area, diagonals. Base 1 ( b1 ): prism is often distinguished by the shape of their polygon... Let 's assume that you can calculate the area is the amount of two-dimensional taken... Calculating height of the typical trapezoid: height h: area =.! | {
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amount of two-dimensional taken... Calculating height of the typical trapezoid: height h: area =.! Trapezoid such as area, perimeter, height and sides given a subset... Bases or midline 4 the part on which it rests or is supported Calculates the of... Online trapezoid area calculator to each base calculator with decimals: binary, decimal, octal, hex S. Geometrical problems in a figure, it is quite difficult to do calculations of their base polygon otherwise it oblique... Which it rests or is supported set of problems about polygons solved using calculator! Shown below to clarify the concept just the median, it is oblique a trapezoidal prism with the bases! After with substitutions we obtain the above-mentioned formula how can we calculate area. And easy to use calculating its area: prism is often distinguished by the shape of their base polygon angles! Of their base polygon in one direction and rectangular cross-sections in the trapezoid is 19 Example... Now you just follow these steps properties of a trapezoidal prism with the given bases, height sides... Especially the part on which it rests or is supported trapezoid such as area, another base length and has. Eso Warden Stamina Build Pvp, Barry Callebaut Where To Buy, Tom And Jerry Memes Clean, Weather Wayne, Nj, Dong Zijian And Sun Yi Wedding, Medical Words With Prefix Con, Red White Mobile Qoo10, Compártelo:Haz clic para compartir en Twitter (Se abre en una ventana nueva)Haz clic para compartir en Facebook (Se abre en una ventana nueva)Haz clic para compartir en Google+ (Se abre en una ventana nueva) Relacionado" /> | {
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# trapezoid base calculator | {
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In Euclidean geometry, a convex quadrilateral with at least one pair of parallel sides is referred to as a trapezium. You can always use our trapezoid calculator above, however, you should be able to calculate the area of trapezoid by yourself. Also, the area can be calculated in a plane of two dimensions. To find the Area of Trapezoid, all you need to do is just go ahead and implement the steps that are given below: First of all, note down the given parameters such as side lengths and height. Area of a trapezoid and midline or bases Trapezoid is a convex polygon with four vertices (corners) and four equal edges (sides) two of which are parallel and are called bases. As α + β = 180°, β = 180° - 30 ° = 150°. To calculate the trapezoid area, follow the below steps: Measure and write down the base a , base b , and height h of the trapezoid. FAQ. Free Trapezoid Sides & Angles Calculator - Calculate sides, angles of an trapezoid step-by-step This website uses cookies to ensure you get the best experience. All you have to do is enter the price per unit area and voila, you have the total cost of materials in a single click! Here is how the Base a of Trapezoid calculation can be explained with given input values -> 6.333333 = 2*(50/12)-2. Calculator that gives out the volume of a trapezoidal prism with the given bases, height, and width values. Area of a Trapezoid Calculator: ... A 4 – sided geometrical figure which has one pair of parallel sides is called a Trapezoid. Use the trapezoid calculator to find the area of a trapezoid with bases 6 cm and 9 cm and height 5 cm. If you only know the side lengths of a regular trapezoid, you can break the trapezoid into simple shapes to find the height and finish your calculation. The opposite sides are different in length. Yes, our tool is that awesome. Through the Pythagorean theorem, we express height in right triangles, after with substitutions we obtain the above-mentioned formula. Trapezoid calculator computes all properties of | {
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substitutions we obtain the above-mentioned formula. Trapezoid calculator computes all properties of a trapezoid such as area, perimeter, height and sides given a sufficient subset of these properties. Follow below steps to get trapezoid online using our online trapezoid area calculator. The opposite sides are different in length. Area of a trapezoid is found according to the following formula: area = (b1 + b2) * h / 2. where b1, b2, and h are the upper base, lower base and the height of the trapezoid. A trapezoid, also known as a trapezium, is a 4-sided shape with two parallel bases that are different lengths. The procedure to use the isosceles trapezoid calculator is as follows: Step 1: Enter the height and two base values in the input field. Steps to Find Area & Perimeter of a Trapezoid. Calculator Techniques for Polygons in Plane Geometry Solving problems related to plane geometry especially polygons can be easily solved using a calculator. Volume=(1/3)*pi*Height*(Radius 1^2+Radius 2^2+(Radius 1*Radius 2)), Total Surface Area=pi*Radius*(Radius+sqrt(Radius^2+Height^2)), Lateral Surface Area=pi*Radius*sqrt(Radius^2+Height^2), Total Surface Area=2*pi*Radius*(Height+Radius), Area of a Triangle when base and height are given, Area of a Parallelogram when base and height are given, The Base a of Trapezoid formula is defined as A trapezoid is a quadrilateral with one pair of parallel lines Bases ,the two parallel lines are called the bases and is represented as, The Base a of Trapezoid formula is defined as A trapezoid is a quadrilateral with one pair of parallel lines Bases ,the two parallel lines are called the bases is calculated using. Calculate the side (base) of a trapezoid if given diagonal, lateral side (leg) and other base ( a b ) : Calculate the lateral side (leg) of a trapezoid if given diagonal and bases ( c ) : Calculate the side (base) of a trapezoid if given diagonal, height, angle between the diagonals and base ( a b ) : * It is true in this case: 4. | {
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