Q stringlengths 70 13.7k | A stringlengths 28 13.2k | meta dict |
|---|---|---|
Find The Equation This is the question:
Find the equations of the tangent lines to the curve $y = x − \frac 1x + 1$ that are parallel to the line $x − 2y = 3$.
There are two answers: 1) smaller y-intercept; 2) larger y-intercept
The work:
The slope of the line is (1/2).
$y' = ((x + 1) - (x - 1))/(x + 1)^2 = 2/(x + 1)^2... | Two lines are parallel if they have the same slope, and the slope of a tangent line to a curve is the value of the derivative of the curve's function at that point. Therefore, follow this strategy.
1) Find the slope of the given line.
2) Find the derivative of the function defining the curve.
3) Set those expressions e... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1149505",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 2,
"answer_id": 0
} |
$\sqrt{4 -2 \sqrt{3}} = a + b\sqrt{3}$, where numbers $a$ and $b$ are rational If $a$ and $b$ are rational numbers such that $\sqrt{4 -2 \sqrt{3}} = a + b\sqrt{3}$
Then what is the value of $a$? The answer is $-1$.
$$\sqrt{4 - 2\sqrt{3}} = a + b\sqrt{3}$$
$$4 - 2\sqrt{3} = 2^2 - 2\sqrt{3}$$ Let $u =2$ hence,
$$\sqrt{u... | We can start with $\sqrt{4-2\sqrt{3}} = a + b\sqrt{3}$ and square both sides. This gives us
$$ 4-2\sqrt{3} = a^2 + 2ab\sqrt{3} + 3b^2 $$
Because $a$ and $b$ are said to be rational, we know that the only term which will be a multiple of $\sqrt{3}$ is $2ab\sqrt{3}$ From this, we can split our system into two equations:
... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1150497",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "5",
"answer_count": 3,
"answer_id": 1
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Question in relation to completing the square In description of "completing the square" at http://www.purplemath.com/modules/sqrquad.htm the following is given :
I'm having difficulty understanding the third part of the transformation.
Where is
$ -\frac{1}{4}$ derived from $-\frac{1}{2}$ ?
Why is $ -\frac {1}{4}$ squ... | In general, to complete the square of something like $x^2 \pm Bx = \pm C$, you take half the coefficient of $x$, which is $B$, and then square it.
For the third step, they first take half of the $x$ coefficient, which is $-\frac{1}{2}$. Half of $-\frac{1}{2}$ is $-\frac{1}{4}$ since
$$ \frac{-\frac{1}{2}}{2} = \frac{-... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1153260",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 3,
"answer_id": 0
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What is wrong with this approach evaluating $\int \sec \theta \ d \theta$? When integrating a function like $\sin^m \theta \cdot \cos^n \theta$ where where $m,n$ are nonnegative integers and $n$ is odd, a common approach is to peel off one power of $\cos \theta$ and then rewrite the resulting even power of $\cos \theta... | In your fifth line,
$$\log |1+u| + \log |1-u|$$
should be
$$\log |1+u| - \log |1-u|\ .$$
This then gives
$$\eqalign{
\frac12\log\Bigl|\frac{1+u}{1-u}\Bigr|
&=\frac12\log\Bigl(\frac{1+\sin\theta}{1-\sin\theta}\Bigr)\cr
&=\frac12\log\Bigl(\frac{1+\sin\theta}{1-\sin\theta}
\frac{1+\sin\theta}{1+\sin\theta}\Bigr)... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1155366",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "4",
"answer_count": 1,
"answer_id": 0
} |
Let $a$, $b$ and $c$ be the three sides of a triangle. Show that $\frac{a}{b+c-a}+\frac{b}{c+a-b} + \frac{c}{a+b-c}\geqslant3$.
Let $a$, $b$ and $c$ be the three sides of a triangle.
Show that $$\frac{a}{b+c-a}+\frac{b}{c+a-b} + \frac{c}{a+b-c}\geqslant3\,.$$
A full expanding results in:
$$\sum_{cyc}a(a+b-c)(a+c-b)\g... | Since $a,b,c$ are sides of a triangle, we can set $a = x+y$, $b = x+z$, $c = y+z$.
Plugging that in gives
\begin{align}
\frac{a}{b+c-a}+\frac{b}{a+c-b}+\frac{c}{a+b-c} &= \frac{x+y}{2z} + \frac{x+z}{2y} + \frac{y+z}{2x} \\
&= \frac{xy(x+y) + xz(x+z) + yz(y+z)}{2xyz}\\
&= \frac{x^2y+ xy^2 + x^2z + xz^2 + y^2z + yz^2}{2x... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1155955",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "5",
"answer_count": 7,
"answer_id": 6
} |
Integral $\int \limits _0 ^\pi |\sin x + \cos x|\; dx$ $$\int \limits _0 ^\pi |\sin x + \cos x|\; dx$$
If I divide integral in two parts $\int\limits_0^{\frac{\pi}{2}}{(\sin x + \cos x)\,dx}$ and $\int\limits_{\frac{\pi}{2}}^\pi{(\sin x - \cos x)\,dx}$...I am getting $4$...Am I right?
| Though answers were already given, I'd like to suggest a different approach:
We see that the absolute value of $\cos(x) + \sin(x)$ is required.
Also, we are dealing with sinusoidal functions, which can be easily dealt with the square function, so:
$$
\begin{align}
\int\limits_0^\pi \lvert \cos(x) + \sin(x) \rvert &= \... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1156042",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 4,
"answer_id": 3
} |
Sum $\cot^2\frac{\pi}{2m+1}+\cot^2\frac{2\pi}{2m+1}+\cot^2\frac{3\pi}{2m+1}+\ldots+\cot^2\frac{m\pi}{2m+1}?$ How to find
$$\cot^2\frac{\pi}{2m+1}+\cot^2\frac{2\pi}{2m+1}+\cot^2\frac{3\pi}{2m+1}+\ldots+\cot^2\frac{m\pi}{2m+1}?$$
Of course, the number $m$ is assumed to be a positive integer.
| Here is a rather elementary and simple way to sum this series:
We know from De Moivre's Formula that
$$(\cos\theta+\iota\sin\theta)^n=\cos n\theta+\iota\sin n\theta$$
Expanding the LHS using Binomial Theorem,
$$\binom{n}{0}\cos^n\theta+\binom{n}{1}\cos^{n-1}\theta(\iota\sin\theta)+\cdots+\binom{n}{n}\iota^n\sin^n\theta... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1157958",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 1,
"answer_id": 0
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How to calculate derivative of $f(x) = \frac{1}{1-2\cos^2x}$? $$f(x) = \frac{1}{1-2\cos^2x}$$
The result of $f'(x)$ should be equals
$$f'(x) = \frac{-4\cos x\sin x}{(1-2\cos^2x)^2}$$
I'm trying to do it in this way but my result is wrong.
$$f'(x) = \frac {1'(1-2\cos x)-1(1-2\cos^2x)'}{(1-2\cos^2x)^2} =
\frac {1-2\cos... | Avoid the quotient formula for functions of the form $1/g(x)$. Rather consider
$$
f(x)=\frac{1}{1-2\cos^2x}=\frac{1}{g(x)}
$$
where $g(x)=1-2\cos^2x$. Since $\Bigl(\dfrac{1}{x}\Bigr)'=-\dfrac{1}{x^2}$, you have, by the chain rule,
$$
f'(x)=-\frac{1}{g(x)^2}g'(x)
$$
and
$$
g'(x)=4\sin x\cos x,
$$
so
$$
f'(x)=-\frac{4\si... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1158652",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 5,
"answer_id": 2
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The Sum of $\sum\limits_{n=1}^\infty \left(\sin\frac{10}{n} -\sin\frac{10}{n+1}\right)$ $$\sum\limits_{n=1}^\infty \left(\sin\frac{10}{n} - \sin\frac{10}{n+1}\right)$$
I see that as $n$ approaches $\infty$ that it'll be 0 thus convergent. However, I'm unclear of the manipulation that's implemented to get an actual resu... | Just telescope it: look at the few initial terms: $$\sin 10 - \sin 5 + \sin 5 - \sin(10/3) + \sin(10/3)- \sin(10/4)+\cdots.$$ Meaning: $$\sum_{k=1}^n \left(\sin \frac{10}{k}-\sin\frac{10}{k+1}\right) = \sin 10 - \sin\frac{10}{n+1}.$$This way: $$\begin{align}\sum_{n=1}^{+\infty}\left(\sin\frac{10}{n} - \sin\frac{10}{n+1... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1159411",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 1,
"answer_id": 0
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Find all complex numbers satisfying the equation Find all complex numbers $z$ satisfying the equation $$\left|z+\frac{1}{z}\right|=2.$$
| Clearly, $z\ne0$
$$\implies|z^2+1|=2|z|$$
Let $z=re^{iy}$ where $r(\ge0),y$ are real
$$2r^2=(r^2\cos2y+1)^2+(r^2\sin2y)^2=r^4+2r^2\cos2y+1$$
$$(r^2)^2-4r^2\sin^2y+1=0$$
$$\implies r^2=\frac{4\sin^2y\pm\sqrt{(4\sin^2y)^2-4}}2=2\sin^2y\pm\sqrt{4\sin^4y-1}$$
As $4\sin^4y-1<4\sin^4y, r^2=2\sin^2y+\sqrt{4\sin^4y-1}$
As $r^2... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1159933",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 3,
"answer_id": 2
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contour integral piecewise Evaluate $$\int \limits_\gamma \frac1{z-1}dz$$
along the path:
$$\gamma(t) = \begin{cases}(1+i)t, & 0\leq t\leq 1 \\\\ t+i(2-t), & 1\leq t \leq2\end{cases}$$
I know how to do simple questions of these but I am unsure about this one. This is what I tried. Let $f(z)$ represent the equation give... | Actually $\gamma'(t) = 1 + i$ for $0 \le t < 1$ and $\gamma'(t) = 1 - i$ for $1 < t \le 2$. So your contour integral becomes
$$\int_0^1 f(\gamma(t))\gamma'(t)\, dt + \int_1^2 f(\gamma(t))\gamma'(t)\, dt = \int_0^1 \frac{1 + i}{(1 + i)t - 1}\, dt + \int_1^2 \frac{1 - i}{t - 1 + (2-t)i}\, dt.$$
To evaluate the last two i... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1162126",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 1,
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Probability urn-envelope question Urn 1 contains 14 envelopes--6 with \$1 bills and 8 with \$5 bills. Urn 2 has 8 envelopes--3 with \$1 bills and 5 with \$5 bills. 3 bills are randomly transferred from urn 1 to urn 2. What is the probability of drawing a\$1 bill from urn 2?
Attempted answer:
Let $O_x$ be the event wher... | Hint:
(1,1,1), (1,1,5), (5,5,1), (5,5,5) are the possible bills transferred
Similar to your approach:
P(USD1/Urn2) $$=\frac{{6\choose3}}{{14\choose3}}.\frac{6}{11}+\frac{{6\choose2}.{8\choose1}}{{14\choose3}}.\frac{5}{11}+\frac{{6\choose1}.{8\choose2}}{{14\choose3}}.\frac{4}{11}+\frac{{8\choose3}}{{14\choose3}}.\frac{... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1163245",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 1,
"answer_id": 0
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Assume that the integer $r$ is a primitive root of the prime $p$, where $p \equiv 1 \pmod{8}$. Assume that the integer $r$ is a primitive root of the prime $p$, where $p \equiv 1 \pmod{8}$.
Show that the solutions of the quadratic congruence $x^2 \equiv 2\pmod{p}$ are given by
$x \equiv \pm (r^{7(p-1)/8}+r^{(p-1)/8})... | Let $b=r^{(p-1)/8}$. Then $a=r^{7(p-1)/8}+r^{(p-1)/8}=b^7+b$ and so $a^2=b^{14}+2b^8+b^2$.
We want to prove that $a^2=2$.
Clearly, $b^8=1$. It remains to prove that $b^{14}+b^2=0$.
Now, $b^{14}+b^2=b^{6}+b^2=b^2(b^4+1)$.
But $b^4+1=0$ because $0=b^8-1=(b^4+1)(b^4-1)$, and $b^4-1\ne0$.
Thus, $x=\pm a$ are two solutions ... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1165869",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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Series Convergence $\sum_{n=1}^{\infty} \frac{1}{n} \left(\frac{2n+2}{2n+4}\right)^n$ I have to show if this series converges or diverges. I tried using asymptotics, but it's not formally correct as they should work only when the arguments are extremely small. Any ideas?
$$
\sum_{n=1}^{\infty} \frac{1}{n} \left(\frac{2... | Recall that, as $x \to 0$, by the Taylor expansion, we have
$$
\begin{align}
e^x& =1+x+\mathcal{O}(x^2)\\
\ln (1+x)&=x-\frac {x^2}{2}+\mathcal{O}(x^3)
\end{align}
$$ giving, as $n \to \infty$,
$$
n\ln \left(1-\frac {2}{2n+4}\right)=n \left(-\frac {2}{2n+4}+\mathcal{O}(\frac {1}{n^2})\right)=-1+\mathcal{O}(\frac {1}{n})... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1166192",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "7",
"answer_count": 4,
"answer_id": 0
} |
Prove that $n^3+2$ is not divisible by $9$ for any integer $n$ How to prove that $n^3+2$ is not divisible by $9$?
| Consider the following $9$ cases:
*
*$n\equiv0\pmod9 \implies n^3\equiv 0\pmod9 \implies n^3+2\equiv 2\pmod9$
*$n\equiv1\pmod9 \implies n^3\equiv 1\pmod9 \implies n^3+2\equiv 3\pmod9$
*$n\equiv2\pmod9 \implies n^3\equiv 8\pmod9 \implies n^3+2\equiv10\equiv1\pmod9$
*$n\equiv3\... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1167968",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 5,
"answer_id": 1
} |
Proof of an Limit Using the formal definition of convergence, Prove that $\lim\limits_{n \to \infty} \frac{3n^2+5n}{4n^2 +2} = \frac{3}{4}$.
Workings:
If $n$ is large enough, $3n^2 + 5n$ behaves like $3n^2$
If $n$ is large enough $4n^2 + 2$ behaves like $4n^2$
More formally we can find $a,b$ such that $\frac{3n^2+5n}{... | It simple
$$ \lim_{x \to \infty} \dfrac{3 + \dfrac{5}{n}}{4 + \dfrac{2}{n^2}}$$
$$ = \dfrac{3}{4}$$
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/1169336",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 3,
"answer_id": 1
} |
Polynomial division challenge Let $x,y,n \in \mathbb{Z} \geq 3$, Find $A,B$ such that $$x^{n-1}+x^{n-2}y+x^{n-3}y^2+\cdots+x^2y^{n-3}+xy^{n-2}+y^{n-1}= A(x^2+xy+y^2)+B$$ What is the best method to approach this?
| As we have homogeneous polynomials, we first solve the problem for the de-homogeneised polynomials, setting $y=1$, then homogeneise the results.
So we have to find $A, B$ such that
\begin{align*}
& x^{n-1}+x^{n-2}+x^{n-3}+\cdots+x^2+x+1= A(x^2+x+1)+B \\
\iff (&x-1)(x^{n-1}+x^{n-2}+x^{n-3}+\cdots+x^2+x+1)=A(x-1)(x^2+... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1170331",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 3,
"answer_id": 2
} |
In an equilateral spherical triangle, show that $\sec A=1+\sec a$ Q. In an equilateral spherical triangle, show that $\sec A=1+\sec a$
So $A$ is the vertex or the angle of the triangle and $a$ is the side of the equilateral spherical triangle.
I started off the proof by using the law of cosines:
$$\cos(a)-\cos(a)\cos(a... | lambda, mu are lattitude and longitude. $R = 1$.
$O = (0,0,0)$
$A = (\cos \lambda_a \cos \mu_a, \cos \lambda_a \sin \mu_a, \sin \lambda_a)$
$B = (\cos \lambda_b \cos \mu_b, \cos \lambda_b \sin \mu_b, \sin \lambda_b)$
$C = (\cos \lambda_c \cos \mu_c, \cos \lambda_c \sin \mu_c, \sin \lambda_c)$
The planes $AOB$ and $AOC$... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1171401",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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Find two linearly independent solutions of the differential equation $(3x-1)^2 y''+(9x-3)y'-9y=0 \text{ for } x> \frac{1}{3}$ I want to find two linearly independent solutions of the differential equation
$$(3x-1)^2 y''+(9x-3)y'-9y=0 \text{ for } x> \frac{1}{3}$$
Previously I have seen that the following holds for the... | Let $t = x - \frac{1}{3}$, then
$$\frac{dy}{dt} = y'$$
$$ \frac{d^2y}{dt^2} = y''$$
The equation becomes
$$ 9t^2 \frac{d^2y}{dt^2} + 9t \frac{dy}{dt} - 9y = 0 $$
This is of course the Cauchy-Euler equation. To solve, let $y = t^m$, so on and so forth.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/1172614",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "5",
"answer_count": 2,
"answer_id": 0
} |
Quadratic formula question: Missing multiplying factor of A? I have a very simple problem which must have a simple answer and I was wondering if anyone can point out my error.
I have the following quadratic equation to factor:
$2x^2+5x+1$
Which is of the form:
$Ax^2+Bx+C$
All I want to now do is factor this into the fo... | We can obtain the correct factorization by completing the square.
\begin{align*}
2x^2 + 5x + 1 & = 2\left(x^2 + \frac{5}{2}x\right) + 1\\
& = 2\left(x^2 + \frac{5}{2}x + \frac{25}{16}\right) - \frac{25}{8} + 1\\
& = 2\left(x + \frac{5}{4}\right)^2 - \frac{17}{8}\\
& = 2\left[\l... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1175195",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 3,
"answer_id": 1
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What does orthogonality mean in function space? The functions $x$ and $x^2 - {1\over2}$ are orthogonal with respect to their inner product on the interval [0, 1]. However, when you graph the two functions, they do not look orthogonal at all. So what does it truly mean for two functions to be orthogonal?
| Functions can be added together, scaled by constants, and taken in linear combination--just like the traditional Euclidean vectors.
$$\vec{u} = a_1\vec{v}_1 + a_2\vec{v}_2 + \cdots a_n\vec{v}_n$$
$$g(x) = a_1f_1(x) + a_2f_2(x) + \cdots + a_nf_n(x) $$
Just as $\left( \begin{array}{c} 5\\ 2\\ \end{array} \right) =
... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1176941",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "17",
"answer_count": 5,
"answer_id": 4
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Intersection between two planes and a line? What is the coordinates of the point where the planes: $3x-2y+z-6=0$ and $x+y-2z-8=0$ and the line: $(x, y, z) = (1, 1, -1) + t(5, 1, -1)$ intersects with eachother?
I've tried letting the line where the two planes intersect eachother be equal to the given line, this results ... | Isolate a variable in the planar equations and set the resulting expressions equal to each other (because they intersect):
$z= 6+2y-3x$, and $z=\frac{x+y-8}{2}$
so $6+2y-3x=\frac{x+y-8}{2}$.
Solving for x yields:
$\frac{20+3y}{7}=x$
and setting $y$ as the parameter ($y=s$), and substituting back into the original equat... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1181397",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 3,
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Prove $3a^4-4a^3b+b^4\ge0$ . $(a,b\in\mathbb{R})$ $$3a^4-4a^3b+b^4\ge0\ \ (a,b\in\mathbb{R})$$
We must factorize $3a^4-4a^3b+b^4\ge0$ and get an expression with an even power like $(x+y)^2$ and say an expression with an even power can not have a negative value in $\mathbb{R}$.
But I don't know how to factorize it sinc... | $$3a^4-4a^3b+b^4$$
$$=3a^4-3a^3b-a^3b+b^4$$
$$=3a^3(a-b)-b(a^3-b^3)$$
$$=(a-b)\Big(3a^3-b(a^2+b^2+ab)\Big)$$
$$=(a-b)(3a^3-a^2b-b^3-ab^2)$$
The cubic is zero if $a=b$, so try taking $(a-b)$ out again:
$$(a-b)(3a^3-3a^2b+2a^2b-2b^3+b^3-ab^2)$$
$$=(a-b)\Big(3a^2(a-b)+2b(a^2-b^2)-b^2(a-b)\Big)$$
$$=(a-b)^2(3a^2+b^2+2ab)$$... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1182152",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 4,
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Find the coordinate of third point of equilateral triangle. I have two points A and B whose coordinates are $(3,4)$ and $(-2,3)$ The third point is C. We need to calculate its coordinates.
I think there will be two possible answers, as the point C could be on the either side of line joining A and B.
Now I put AB = AC =... | Call the position of point $C$ by the coords $(a, b)$. The equations for $C$ are then
$$
\sqrt{(a-3)^2 + (b - 4)^2} = \sqrt{26} \\
\sqrt{(a+2)^2 + (b - 3)^2} = \sqrt{26}
$$
Squaring both, we get
$$
(a-3)^2 + (b - 4)^2 = 26 \\
(a+2)^2 + (b - 3)^2 = 26
$$
$$
a^2 - 6a + 9 + b^2 - 8b + 16= 26 \\
a^2 + 4a + 4 + b^2 - 6b +... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1183349",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "5",
"answer_count": 4,
"answer_id": 0
} |
Find all integer solutions of equality Find all integer solutions of equation $$x^3+(x+1)^3+...+(x+7)^3=y^3$$ I've solved it by opening brackets and consideration of signs but I think there is simpler way of solving it .
| Let
$$P(x)=x^3+(x+1)^3+(x+2)^3+(x+3)^3+(x+4)^3+(x+5)^3+(x+6)^3+(x+7)^3$$
so
$$P(-x-7)=-P(x)$$
on the other hand we can
$$P(x)=8x^3+84x^2+420x+784$$
if $x\ge 0$,we have
$$(2x+7)^3=8x^3+84x^2+294x+343<P(x)<8x^3+120x^2+600x+1000=(2x+10)^3$$
so we have
$$(2x+7)<y<2x+10$$
so
$$P(x)=(2x+8)^3\Longrightarrow -12x^2+36x+272=0... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1183914",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 1,
"answer_id": 0
} |
how to show $\frac{1+\cos x+\sin x}{1+\cos x-\sin x} = \frac{1+\sin x}{\cos x}$ I know that $\frac{1+\cos x+\sin x}{1+\cos x-\sin x} = \frac{1+\sin x}{\cos x}$ is correct. But having some hard time proofing it using trig relations. Some of the relations I used are
$$
\sin x \cos x= \frac{1}{2} \sin(2 x)\\
\sin^2 x + \... | Hint:
$$= \frac{(1 + \cos x + \sin x)(1 - \cos x + \sin x)}{(1 + \cos x - \sin x)(1 - \cos x + \sin x)}$$
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/1185058",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 4,
"answer_id": 3
} |
How to explain the formula for the sum of a geometric series without calculus? How to explain to a middle-school student the notion of a geometric series without any calculus (i.e. limits)? For example I want to convince my student that
$$1 + \frac{1}{4} + \frac{1}{4^2} + \ldots + \frac{1}{4^n} = \frac{1 - (\frac{1}{4... | $$
\frac{1}{3}-\frac{1}{4}=\frac{1}{12}=\frac{1}{4}\cdot\frac{1}{3}
$$
so
$$
\frac{1}{3}=\frac{1}{4}+\frac{1}{4}\cdot\frac{1}{3}
$$
Now ask your 14 year old to plug in this expression for $1\over 3$ into itself, quite funny, bewildering and strange at first sight:
$$
\frac{1}{3}=\frac{1}{4}+\frac{1}{4}\cdot \left(\frac... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1185259",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "15",
"answer_count": 13,
"answer_id": 2
} |
Find all possible values of $ a^3 + b^3$ if $a^2+b^2=ab=4$. Find all possible values of $a^3 + b^3$ if $a^2+b^2=ab=4$.
From $a^3+b^3=(a+b)(a^2-ab+b^2)=(a+b)(4-4)=(a+b)0$. Then we know $a^3+b^3=0$.
If $a=b=0$, it is conflict with $a^2+b^2=ab=4$.
If $a\neq0$ and $b\neq0$, then $a$ and $b$ should be one positive and one ... | As has been observed, one approach to this problem is to solve the equations over the complex numbers, and then check directly that $a^3+b^3=0$ in all cases. However, it's both useful and more general to observe that the conclusion can be reached without actually solving the equations: since $a^2+b^2=a b$, we have
$(... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1186290",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "5",
"answer_count": 5,
"answer_id": 4
} |
Finding eigenvalues to classify the steady state of a system I have this system of differential equations which model chemical concentrations in a certain reactions:
$$\dot{x}=a-x-\frac{2xy}{1+x^2}\qquad \dot{y}=bx\left(1-\frac{y}{1+x^2}\right)$$
for $a,b >0$ and $x(t),y(t)\geq 0$,
and I want to find the steady state o... | i am going to set $a = 3$ see what happens. i will see if by scaling time if this can be done. i will make a change of variable $$x = 1 + u, y = 2 + v.$$ then
$$u' = 3 - (1+u) - \frac{2(1+u)(2 + v)}{1 + (1 + u)^2 } = \frac{(2-u)(2+2u+u^2)-2(2+2u+v+uv)}{2+2u+u^2} = \frac{(4-2u+4u+\cdots) -(4+4u+2v+\cdots)}{2+2u+u^2} = -... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1186690",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 1,
"answer_id": 0
} |
Is equality $\sin(\frac{3\pi}{2}-x) = -\cos(x)$ always true? $\sin(\frac{3\pi}{2}-x) = -\cos(x)$
Is above equality always true even if $x \gt \frac{\pi}{2}$?
| why not $$\sin\left(\frac{3\pi} 2 - x\right) = \sin \left(\frac{3\pi} 2\right)\cos x -\cos\left(\frac{3\pi} 2\right) \sin x = -\cos x?$$
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/1189228",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 3,
"answer_id": 2
} |
Find the derivative using the chain rule and the quotient rule
$$f(x) = \left(\frac{x}{x+1}\right)^4$$ Find $f'(x)$.
Here is my work:
$$f'(x) = \frac{4x^3\left(x+1\right)^4-4\left(x+1\right)^3x^4}{\left(x+1\right)^8}$$
$$f'(x) = \frac{4x^3\left(x+1\right)^4-4x^4\left(x+1\right)^3}{\left(x+1\right)^8}$$
I know the fin... | I usually try to avoid the quotient rule if possible.
$$ f(x) = \left(\frac{x}{x+1}\right)^4 = \left(\frac{x+1-1}{x+1}\right)^4 = \left(\frac{x+1}{x+1} + \frac{-1}{x+1}\right)^4 = \left(1 - \frac{1}{x+1}\right)^4 $$
Applying the chain rule yields
$$ f'(x) = 4 \cdot \left(1-\frac{1}{x+1}\right)^3 \cdot \frac{1}{(x+1)^2}... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1189494",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "4",
"answer_count": 6,
"answer_id": 3
} |
For $x, y \in \Bbb R$ such that $(x+y+1)^2+5x+7y+10+y^2=0$. Show that $-5 \le x+y \le -2.$ I have a problem:
For $x, y \in \Bbb R$ such that $(x+y+1)^2+5x+7y+10+y^2=0$. Show that
$$-5 \le x+y \le -2.$$
I have tried:
I write $(x+y+1)^2+5x+7y+10+y^2=(x+y)^2+7(x+y)+(y+1)^2+10=0.$
Now I'm stuck :(
Any help will be a... | Set $x+y=c\iff y=c-x$
We have $(c+1)^2+5c+2y+10+y^2=0\iff y^2+2y+c^2+7c+11=0$
As $y$ is real, we need the discriminant $(2)^2-4\cdot1\cdot(c^2+7c+11)\ge0$
$\iff(c+2)(c+5)\le0$
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/1189778",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 2,
"answer_id": 0
} |
Integrate $\int_0^1 \frac{\ln(1+x^a)}{1+x}\, dx$ I have recently met with this integral:
$$\int_0^1 \frac{\ln(1+x^a)}{1+x}\, dx$$
I want to evaluate it in a closed form, if possible.
1st functional equation:
$\displaystyle f(a)=\ln^2 2-f\left ( \frac{1}{a} \right )$ since:
$$\begin{aligned}
f(a)=\int_{0}^{1}\frac{\ln (... | Another series approach might offer a hint:
$$\int_0^1 x^pdp=\frac{1}{p+1}$$
$$\int_0^1 x^{ak} x^ldp=\frac{1}{ak+l+1}$$
$$\int_0^1 \frac{\ln(1+x^a)}{1+x}dx=\sum_{k=1}^{\infty} \sum_{l=0}^{\infty} \frac{(-1)^{k+l+1}}{k(ak+l+1)}=\sum_{k=1}^{\infty} \frac{(-1)^{k+1}}{k} \Phi(-1,1,ak+1)$$
Here $\Phi$ is Lerch Trancendent... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1195297",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "17",
"answer_count": 1,
"answer_id": 0
} |
prove increasing/decreasing sequence Are these two statements true? If so, how does one prove them?
1) For each integer k (positive or negative), the sequence
$a_n = (1 + k/n) ^ n$ (1)
is increasing (at least after a certain number n).
2) For each integer k (positive or negative) the sequence
$a_n = (1 + k/n) ^{n+1... | Consider
\begin{align*}
\log(a_n) &= n\log\left(1 + \frac{k}{n}\right) \\
&= n\left(\frac{k}{n} - \frac{(\frac{k}{n})^2}{2} \mp \ldots \right)\\
& = \left(k - \frac{k^2}{2n} + \frac{k^3}{3n^2} \mp \ldots \right)
\end{align*}
where the series converges absolutely for $n > k$.
Then
\begin{align*}
\log(a_n) - \log(a_{n... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1198087",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 1,
"answer_id": 0
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Show that $\binom{2n}{n}$ is an even number, for positive integers $n$. I would appreciate if somebody could help me with the following problem
Show by a combinatorial proof that
$$\dbinom{2n}{n}$$
is an even number, where $n$ is a positive integer.
I tried to solve this problem but I can't.
| You should be familiar with the recursive definition of the binomial coefficients, that $\binom{n}{r} = \binom{n-1}{r-1}+\binom{n-1}{r}$
\begin{matrix}1\\
1 & 1\\
1 & 2 & 1\\
1 & 3 & 3 & 1\\
1 & 4 & 6 & 4 & 1\\
\vdots & & \vdots & & \vdots&\ddots\\
& \cdots& \binom{n-1}{r-1} & \binom{n-1}{r} & \cdots\\
& \cdots& & \b... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1199206",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 4,
"answer_id": 0
} |
Conditional Extremum, need help finding the extreme points in calculation. Find the conditonal extremums of the following $$u=xyz$$ if $$(1) x^2+y^2+z^2=1,x+y+z=0.$$
First i made the Lagrange function $\phi= xyz+ \lambda(x^2+y^2+z^2-1) + \mu (x+y+z) $, now making the derivatives in respect to x, y, z equal to zero i ge... | $x+y+z=0$ means $x,y,z$ have different signs,but for three ones ,there are two ones have same sign. WLOG, let $xy>0 \implies xy \le \dfrac{(x+y)^2}{4}$
we have $x^2+y^2+(x+y)^2=1 \ge \dfrac{(x+y)^2}{2}+(x+y)^2 \iff (x+y)^2 \le \dfrac{2}{3}$
when $x=y=\pm \dfrac{\sqrt{6}}{6} ,(x+y)^2$ get max and $xy$ get max also.
$u... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1200306",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 2,
"answer_id": 1
} |
Show limit of function $\frac{xy(x+y)}{x^2-xy+y^2}$ for $(x,y)\to(0,0)$ Show $$\lim_{(x,y)\to (0,0)} xy \frac{(x+y)}{x^2-xy+y^2}=0$$
If I approach from $y=\pm x$, I get $0$. Is that sufficient?
| From $\bigl(|x|-|y|\bigr)^2\geq0$ we get $2|xy|\leq x^2+y^2$ and then
$$|x|+|y|\leq\sqrt{2(x^2+y^2)},\qquad x^2-xy+y^2\geq{1\over2}(x^2+y^2)\ .$$
This implies
$$\bigl|f(x,y)\bigr|=\left|{xy(x+y)\over x^2-xy+y^2}\right|\leq{{1\over2}(x^2+y^2)\sqrt{2(x^2+y^2)}\over{1\over2}(x^2+y^2)}=\sqrt{2(x^2+y^2)}\ .$$
Therefore, giv... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1202183",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 3,
"answer_id": 0
} |
$f(x) = (\cos x - \sin x) (17 \cos x -7 \sin x) $ $f(x) = (\cos x - \sin x) (17 \cos x -7 \sin x)$
Determine the greatest and least values of $\frac{39}{f(x)+14}$ and state a value of x at which greatest values occurs.
Do I just use a graphing calculator for this? Is there a way I could do this without a graphing calc... | you can do this without calculus. here is a way. i will use $t$ for $x.$ we have
$$\begin{align}y &= 14 + (\cos t - \sin t)(17 \cos t - 7 \sin t) \\&= 14 + 17 \cos^2 t-24 \sin t \cos t+7 \sin ^2 t\\
&=14 + \frac{17}{2}(1 + \cos 2t)-12 \sin 2t+\frac 72(1-\cos 2t)\\
&=26+5\cos 2t-12\sin 2t\\
&=26 + 13\cos(2t+\phi), \t... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1202320",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 3,
"answer_id": 0
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Probability Question (Colored Socks)
In a drawer Sandy has 5 pairs of socks, each pair a different color. On Monday Sandy selects two individual socks at random from the 10 socks in the drawer. On Tuesday Sandy selects 2 of the remaining 8 socks at random and on Wednesday two of the remaining 6 socks at random. The pr... | There are three cases to consider:
*
*Four socks, each of a different color, are selected on Monday and Tuesday, then a pair is selected on Wednesday from the one pair and four single socks that remain. The probability of this occurring is
$$1 \cdot \frac{8}{9} \cdot \frac{6}{8} \cdot \frac{4}{7} \cdot \frac{\bino... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1202658",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 2,
"answer_id": 0
} |
Using Laplace Transforms to Evaluate Integrals
I'm trying to solve $$\int_0^{\infty} \, \frac{e^{-2t}\cos(3t)-e^{-4t}\cos(2t)}{t}dt.$$
I'm sure that this involves Laplace transforms, I'm just not sure how.
I would start by separating and making two integrals, where the second would solve in the same procedure as the... | *
*Method 1
By using the integral
\begin{align}
\int_{0}^{\infty} e^{-u t} \, du = \frac{1}{t}
\end{align}
the integral
\begin{align}
I = \int_0^{\infty} \, \frac{e^{-2t}\cos(3t)-e^{-4t}\cos(2t)}{t}dt
\end{align}
becomes
\begin{align}
I &= \int_0^{\infty} \int_{0}^{\infty} (e^{-2t}\cos(3t)-e^{-4t}\cos(2t) ) ds dt \\... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1203286",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 1,
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Sum $\pmod{1000}$
Let $$N= \sum_{k=1}^{1000}k(\lceil \log_{\sqrt{2}}k\rceil-\lfloor \log_{\sqrt{2}}k \rfloor).$$ Find $N \pmod{1000}$.
Let $\lceil x \rceil$ be represented by $(x)$ and $\lfloor x \rfloor$ be represented by $[x]$.
Consider $0 < x < 1$ then:
$$(x) - [x] = 1 - 0 = 1$$
Consider $x=0$ then:
$$(x) - [x] ... | You've correctly identified two statements:
$\lceil x \rceil - \lfloor x \rfloor = \begin{cases} 0 & \text{ if } x \in \mathbb{Z} \\ 1 & \text{ if } x \notin \mathbb{Z} \end{cases}$
For integers $k$, $\log_{\sqrt{2}} k \in \mathbb{Z} \iff k = 2^r$ for some integer $r$.
Combining these, we get
$\lceil \log_{\sqrt{2}} k... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1204494",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 2,
"answer_id": 1
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Limit of $\sqrt{25x^{2}+5x}-5x$ as $x\to\infty$ $\hspace{1cm} \displaystyle\lim_{x\to\infty} \left(\sqrt{25x^{2}+5x}-5x\right) $
The correct answer seems to be $\frac12$, whereas I get $0$.
Here's how I do this problem:
$$ \sqrt{25x^{2}+5x}-5x \cdot \frac{\sqrt{25x^{2}+5x}+5x}{\sqrt{25x^{2}+5x}+5x} = \frac{25x^2+5x - 2... | \begin{align}
\lim_{x\to\infty} \left(\sqrt{25x^{2}+5x}-5x\right)
&= \lim_{x\to\infty} x\left(\sqrt{25+5/x}-5\right) \\
&= \lim_{x\to\infty} \frac{\sqrt{25+5/x}-5}{1/x}
\end{align}
now we can apply L'Hospital's rule:
\begin{align}
= \lim_{x\to\infty}
\left.\frac{-5}{2x^2\sqrt{25+5/x} }
\right/
\frac{-1}{x^2}
&=
\lim_{... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1205626",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 4,
"answer_id": 2
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Showing $1+2+\cdots+n=\frac{n(n+1)}{2}$ by induction (stuck on inductive step) This is from this website:
Use mathematical induction to prove that
$$1 + 2 + 3 +\cdots+ n = \frac{n (n + 1)}{2}$$
for all positive integers $n$.
Solution to Problem 1: Let the statement $P(n)$ be
$$1 + 2 + 3 + \cdots + n = \frac{n... | induction is basically saying that if it is true for this step, it is true for the next step. so assuming $1+2+3...+k=k(k+1)/2$, ie it is true for step k, we have to show that it must be true for step k+1, the next step. the final line shows how, by going through some algebra, adding all the numbers up to k+1 equals pu... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1206645",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 3,
"answer_id": 2
} |
Divergence of $\prod_{n=2}^\infty(1+(-1)^n/\sqrt n)$. Looking looking for a verification of my proof that the above product diverges.
$$\begin{align}
\prod_{n=2}^\infty\left(1+\frac{(-1)^n}{\sqrt n}\right) & =\prod_{n=1}^\infty\left(1+\frac1{\sqrt {2n}}\right)\left(1-\frac1{\sqrt{2n+1}}\right)\\
& =\prod_{n=1}^\infty\l... | Your calculation is wrong, but the conclusion is correct.
$$ \left( 1 + \dfrac{1}{\sqrt{2n}}\right) \left( 1 - \dfrac{1}{\sqrt{2n+1}}\right) = 1 - \dfrac{1}{2n} + O(n^{-3/2})$$
The infinite product diverges, but to $0$, not $+\infty$.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/1206733",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "4",
"answer_count": 2,
"answer_id": 0
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If $A,B>0$ and $A+B = \frac{\pi}{3},$ Then find Maximum value of $\tan A\cdot \tan B$. If $A,B>0$ and $\displaystyle A+B = \frac{\pi}{3},$ Then find Maximum value of $\tan A\cdot \tan B$.
$\bf{My\; Try::}$ Given $$\displaystyle A+ B = \frac{\pi}{3}$$ and $A,B>0$.
So we can say $$\displaystyle 0< A,B<\frac{\pi}{3}$$. No... | By using $1-\tan A\tan B=\frac{\tan A+\tan B}{\tan(A+B)}$, it follows that for $A>0,B>0,A+B=\pi/3$,
$$\tan A\tan B=\frac{\tan A(\sqrt{3}-\tan A)}{1+\sqrt{3}\tan A}.$$
Hence
$$\begin{align}\max_{A>0,B>0,A+B=\pi/3} \tan A\tan B&=\max_{0<A<\pi/3} \frac{\tan A(\sqrt{3}-\tan A)}{1+\sqrt{3}\tan A}=\max_{1<s<4} \frac{(s-1)(4-... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1207197",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "5",
"answer_count": 3,
"answer_id": 1
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Find the product, if we know 5 of them We have 4 positive,but not necessarily integers $a, b, c,$ and $d$. So we have 6 options how multiply two of them. And we know 5 of 6 products $2, 3, 4, 5$ and $6$. Find the last product.
| One way is to find out which two pairs of the $5$ known numbers give the same product, and then use this product to divide the remaining known number. In your example, by inspection, $2 \times 6$ = $3 \times 4$ = $12$, then $12 / 5 = 2.4$, so the last product is $2.4$.
The reason is: We assume the one pair of product i... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1207464",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "7",
"answer_count": 1,
"answer_id": 0
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Limit as $x$ tend to zero of: $x/[\ln (x^2+2x+4) - \ln(x+4)]$ Without making use of LHôpital's Rule solve:
$$\lim_{x\to 0} {x\over \ln (x^2+2x+4) - \ln(x+4)}$$
$ x^2+2x+4=0$ has no real roots which seems to be the gist of the issue.
I have attempted several variable changes but none seemed to work.
| Rewrite it as
$$\lim\limits_{x\rightarrow0}\frac{x}{\ln \left( \frac{x^2+2x+4}{x+4}\right)}$$
Apply L'Hopital's rule:
$$\lim\limits_{x\rightarrow0}\frac{1}{\frac{x^2+8x+4}{(x+4)(x^2+2x+4)}}$$
Then simply evaluate the limit:
$$\frac{1}{\frac{0^2+8\times0+4}{(0+4)(0^2+2\times0+4)}} = \frac{1}{\frac{4}{16}} = 4$$
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/1213655",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 5,
"answer_id": 3
} |
Generating function for sequence $a_n = \lceil \sqrt{n} \rceil $ In one of books for discrete mathematics i came across sum to calculate $$\sum_{k=0}^n \lceil \sqrt{n} \rceil$$ which was fairly easy, but this sum intrigued me what is generating function for $$\sum_{n=0}^\infty \lceil \sqrt{n} \rceil x^n $$ so from that... | As far as I can tell, the answer can only be written in terms of the Jacobi theta functions (or the like). Specifically, we have
$$
\begin{align}
\sum_{n=0}^\infty \lceil \sqrt{n} \rceil x^n
& = x + 2x^2 + 2x^3 + 2x^4 + 3x^5 + 3x^6 + 3x^7 + 3x^8 + 3x^9 + \cdots \\
& = (x + x^2 + x^3 + \cdots) + (x^2 + x^3 + x^... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1216282",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 1,
"answer_id": 0
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Prove that equation $x^6+x^5-x^4-x^3+x^2+x-1=0$ has two real roots Prove that equation
$$x^6+x^5-x^4-x^3+x^2+x-1=0$$ has two real roots
and $$x^6-x^5+x^4+x^3-x^2-x+1=0$$
has two real roots
I think that:
$$x^{4k+2}+x^{4k+1}-x^{4k}-x^{4k-1}+x^{4k-2}+x^{4k-3}-..+x^2+x-1=0$$
and
$$x^{4k+2}-x^{4k+1}+x^{4k}+x^{4k-1}-x^{... | since $ f(\pm \infty) = \infty, f(0) = -1$ shows that $f$ has one positive root and one negative root. so $f$ has at least two real roots.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/1217316",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 5,
"answer_id": 2
} |
How to differentiate $y=\sqrt{\frac{1+x}{1-x}}$? I'm trying to solve this problem but I think I'm missing something.
Here's what I've done so far:
$$g(x) = \frac{1+x}{1-x}$$
$$u = 1+x$$
$$u' = 1$$
$$v = 1-x$$
$$v' = -1$$
$$g'(x) = \frac{(1-x) -(-1)(1+x)}{(1-x)^2}$$
$$g'(x) = \frac{1-x+1+x}{(1-x)^2}$$
$$g'(x) = \frac{2... | Write:
$$ y^2 = \frac{ 1 + x}{1 -x } \iff y^2 = 1 + \frac{2x}{1-x}$$
so taking derivative with repect to $x$ gives
$$ 2y y' = \frac{2(1-x) + 2x}{(1-x)^2} =\frac{2}{(1-x)^2} $$
so
$$ y' = \sqrt{ \frac{1-x}{1+x} } (1-x)^2$$
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/1223727",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "8",
"answer_count": 6,
"answer_id": 3
} |
Proof of sum results I was going through some of my notes when I found both these sums with their results
$$
x^0+x^1+x^2+x^3+... = \frac{1}{1-x}, |x|<1
$$
$$
0+1+2x+3x^2+4x^3+... = \frac{1}{(1-x)^2}
$$
I tried but I was unable to prove or confirm that these results are actually correct, could anyone please help me conf... | Hints:
$$(x^0+x^1+x^2+x^3+...x^n)(1-x)=x^0-x^1+x^1-x^2+x^2-x^3+x^3\cdots+x^{n}-x^{n+1}\\
=1-x^{n+1}$$
and
$$(x^0+2x^1+3x^2+4x^3+...(n+1)x^n)(1-2x+x^2)\\
=x^0-2x^1+x^2+2x^1-4x^2+2x^3+3x^2-6x^3+3x^4\cdots\\+(n+1)x^n-2(n+1)x^{n+1}+(n+1)x^{n+2}\\
=1-(n+2)x^{n+1}+(n+1)x^{n+2}.$$
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/1223811",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 6,
"answer_id": 2
} |
If $\frac{a}{b}=\frac{x}{y}$, is $\frac{x-a}{y-b}=\frac{x}{y}$? Does this hold? $b,y \neq 0$, $b \neq y$.
| \begin{align}
\frac{a}{b}&=\frac{x}{y}\\
\frac{x}{a}&=\frac{y}{b}\\
\frac{x}{a}-1&=\frac{y}{b}-1\\
\frac{x-a}{a}&=\frac{y-b}{b}\\
\frac{x-a}{y-b}&=\frac{a}{b}\\
\frac{x-a}{y-b}&=\frac{x}{y}
\end{align}
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/1223913",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 3,
"answer_id": 2
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To prove $\prod\limits_{n=1}^\infty\cos\frac{x}{2^n}=\frac{\sin x}{x},x\neq0$ Prove $$\prod_{n=1}^\infty\cos\frac{x}{2^n}=\frac{\sin x}{x},x\neq0$$
This equation may be famous, but I have no idea how to start. I suppose it is related to another eqution:
(Euler)And how can I prove the $follwing$ eqution?
$$\sin x=x(1-\... | By double angle formula we have $$\sin\left(x\right)=2\sin\left(\frac{x}{2}\right)\cos\left(\frac{x}{2}\right)=4\sin\left(\frac{x}{4}\right)\cos\left(\frac{x}{4}\right)\cos\left(\frac{x}{2}\right)=\dots=2^{n}\sin\left(\frac{x}{2^{n}}\right)\prod_{k\leq n}\cos\left(\frac{x}{2^{k}}\right)$$ now remains to note that $$\li... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1226868",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 3,
"answer_id": 2
} |
Solve $\frac{|x|}{|x-1|}+|x|=\frac{x^2}{|x-1|}$ Solve $\frac{|x|}{|x-1|}+|x|=\frac{x^2}{|x-1|}$.What will be the easiest techique to solve this sum ?
Just wanted to share a special type of equation and the fastest way to solve it.I am not asking for an answer and i have solved it in my answer given below.Thank You for ... | To me the easiest and most systematic way to solve it is to explicitly write out the absolute value which means breaking the equation into regions:
$$
\frac{|x|}{|x - 1|} + |x| = \frac{x^2}{|x - 1|} \rightarrow \begin{cases}
\left(\frac{1}{x - 1} + 1\right)x = \frac{x^2}{x - 1} & 1 < x < \infty \\
\left(\frac{1}{1 - x... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1229647",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 3,
"answer_id": 0
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Demand $z=x+y$ and $x^2/4 + y^2/5 + z^2/25 = 1$. What is the maximum value of $f(x,y,z) = x^2+y^2+z^2$?
Demand $z=x+y$ and $x^2/4 + y^2/5 + z^2/25 = 1$. What is the maximum value of $f(x,y,z) = x^2+y^2+z^2$?
I've been attempting this with Lagrange multipliers in a few different ways. However, the resulting system of ... | Beginning with the equation for the Lagrange Multipliers, it is a matter of (tedious) algebraic manipulation with a goal to systematically eliminate parameters. We have $x+y-z=0$ and $\frac{x^2}{4}+\frac{y^2}{5}+\frac{z^2}{25}=1$, $f(x,y,z)=x^2+y^2+z^2$
$$\begin{pmatrix} 2x \\ 2y \\ 2z \end{pmatrix} = \lambda \begin{... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1230637",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 3,
"answer_id": 0
} |
Calculate $\int_0^{2\pi} \frac{\sin(t) + 4}{\cos(t) + \frac{5}{3}}dt$ I have to calculate $\int_0^{2\pi} \frac{\sin t + 4}{\cos t + \frac{5}{3}}dt$ using complex analysis.
I was thinking of setting $z(t) = re^{it} $ but I'm not sure what $r$ to pick or can I just pick any and is this even useful? Do I have to worry ab... | HINT: split the integral into two summands:
$$\int_0^{2\pi} \frac{\sin t + 4}{\cos t + \frac{5}{3}} dt = \int_0^{2\pi} \frac{\sin t}{\cos t + \frac{5}{3}} dt + \int_0^{2\pi} \frac{dt}{\cos t + \frac{5}{3}} =$$
$$=\left. -\log \left( \cos t + \frac{5}{3} \right) \right|_0^{2\pi} + 4\int_{|z|=1} \frac{1}{\frac{z+z^{-1}}{... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1231171",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 1,
"answer_id": 0
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How to find value of $x$ in this formula I have this formula:
$$1-\frac 1x=y$$
How do I invert this so that, if I have value of $y$, I want to find value of $x$.
I know, but I am pretty dense in math :(
I dont even know what category to put this under! Any help is appreciated!
| \begin{align}
1 - \frac{1}{x} &= y \\
1 &= \frac{1}{x} + y \\
1 - y &= \frac{1}{x} \\
x &= \frac{1}{1-y}
\end{align}
If $y = 1$, the above solution is undefined because there is no $x$ such that $1 - \frac{1}{x} = y$ because we would have
\begin{align}
1 - \frac{1}{x} &= 1 \\
\frac{1}{x} &= 0 \\
\end{align}
which is un... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1233042",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 2,
"answer_id": 0
} |
How can you derive $\sin(x) = \sin(x+2\pi)$ from the Taylor series for $\sin(x)$? \begin{eqnarray*}
\sin(x) & = & x - \frac{x^3}{3!} + \frac{x^5}{5!} - \ldots\\
\sin(x+2π) & = & x + 2\pi - \frac{(x+2π)^3}{3!} + \frac{(x+2π)^5}{5!} - \ldots \\
\end{eqnarray*}
Those two series must be equal, but how can you show that by ... | Note that $\sin$ is the unique solution of $y'' = -y$ subject to $y(0) = 0, y'(0) = 1$. Note that $x \mapsto \sin (x+2 \pi)$ also satisfies the same
differential equation, so the question boils down to showing that
$\sin (2 \pi) = 0, \cos(2 \pi) = 1$ (where $\cos = \sin'$).
Consider $\eta(x) = \sin^2 x + \cos^2 x$, we ... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1233961",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "24",
"answer_count": 5,
"answer_id": 1
} |
How prove this geometry inequality $R_1^4+R_2^4+R_3^4+R_4^4+R_5^4\geq {4\over 5\sin^2 108^\circ}S^2$ Zhautykov Olympiad 2015 problem 6 This links discusses the olympiad problem which none of students could solve , meaning it is very hard.
Question:
The area of a convex pentagon $ABCDE$ is $S$, and the circumradii of ... | Dan Schwarz (one of the major problem proposers for EGMO, RMM, Balkan...) has posted a solution at here. I'll briefly sketch it here.
First, one takes the midpoints $M_1$, ..., $M_5$ of $A_1A_2$, ..., $A_5A_1$. Then at each angle $A_i$, one takes the circumcircle of triangle $M_{i-1}A_iM_{i+1}$ (which has radius $\frac... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1234591",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "38",
"answer_count": 1,
"answer_id": 0
} |
Euclidean distance and dot product I've been reading that the Euclidean distance between two points, and the dot product of the two points, are related. Specifically, the Euclidean distance is equal to the square root of the dot product. But this doesn't work for me in practice.
For example, let's say the points are $(... | In $\mathbb{R}^{2}$, the distance between $\displaystyle X = \begin{pmatrix} x_{1} \\ x_{2} \end{pmatrix}$ and $\displaystyle Y = \begin{pmatrix} y_{1} \\ y_{2} \end{pmatrix}$ is defined as :
$$ d(X,Y) = \Vert Y - X \Vert = \sqrt{\left\langle Y-X,Y-X \right\rangle}. $$
with $\left\langle Y-X,Y-X \right\rangle = (y_{1}-... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1236465",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "28",
"answer_count": 4,
"answer_id": 2
} |
Solving integral $\int \frac{3x-1}{\left(x^2+16\right)^3}$ I need to solve this one integral.
$$\int \frac{3x-1}{\left(x^2+16\right)^3}$$
You need to use the method of undetermined coefficients. That's what I get:
$$(3x-1) = (Ax + B)(x^{2}+16)^{2} + (Cx + D)(x^{2}+16) + (Ex + F)$$
$$1: 256B + 16D + F = -1$$
$$x: 256... | As your integrand is already decomposed in partial fractions, start with the next step. The derivative of $x^2 + 16$ is $2x$, hence we have
$$ \int \frac{3x-1}{(x^2+ 16)^3}\, dx = \frac 32 \int\frac{2x}{(x^2 +16)^3}\,dx -\int\frac{1}{(x^2+16)^3}\, dx $$
In the first term, let $u = x^2+16$, giving
$$ \int \frac{2x}{(x^2... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1237289",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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"answer_id": 0
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Prove the set $\mathcal{O}_d := \left\{\frac{a + b\sqrt{d}}{2}:a,b \in \mathbb{Z}, a \equiv b \mod 2 \right\}$ is a ring.
Prove that if $d$ is a non-square integer with $d \equiv 1 \mod 4$ then the set $$\mathcal{O}_d := \left\{\frac{a + b\sqrt{d}}{2}:a,b \in \mathbb{Z}, a \equiv b \mod 2 \right\}$$ is a ring, and in ... | $aa'+bb'd\equiv aa'+bb'\equiv aa'+aa'\equiv 0\bmod 2$, so $u=\dfrac{aa'+bb'd}{2}\in\mathbb Z$.
$ab'+ba'\equiv aa'+aa'\equiv 0\bmod 2$, so $v=\dfrac{ab'+ba'}{2}\in\mathbb Z$.
Now note that $x\circ y$ writes $\dfrac{u+v\sqrt d}{2}$ with $u,v\in\mathbb Z$.
In the following $a-b=2k$, $a'-b'=2k'$ and $d=1+4l$.
We have $u-v=... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1237953",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 3,
"answer_id": 2
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$a^2b + abc + a^2c + ac^2 + b^2a + b^2c + abc +bc^2$ factorisation
I came across this from a university mathematics resource page but they do not provide answer to this.
What I did was this:
$(a^2+b^2+c^2)(a+b+c) - (a^3 + b^3 + c^3) + 2abc$
But I don't think this is the correct solution.
How should I spot how to facto... | $$a^2b+abc+a^2c+ac^2+b^2a+b^2c+abc+bc^2$$
$$=ab(a+b+c)+ac(a+b+c)+b^2c+bc^2$$
$$=(a+b+c)(ab+ac)+bc(b+c)$$
$$=a(a+b+c)(b+c)+bc(b+c)$$
$$=(b+c)[a(a+b+c)+bc]$$
$$=(b+c)[a^2+ab+ab+bc]$$
$$=(b+c)[a(a+b)+c(a+b)]$$
$$=(b+c)(a+b)(a+c)$$
This is the simplest method I could think of. Hope it helps.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/1238712",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "4",
"answer_count": 4,
"answer_id": 0
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Proving $6^n - 1$ is always divisible by $5$ by induction I'm trying to prove the following, but can't seem to understand it. Can somebody help?
Prove $6^n - 1$ is always divisible by $5$ for $n \geq 1$.
What I've done:
Base Case:
$n = 1$: $6^1 - 1 = 5$, which is divisible by $5$ so TRUE.
Assume true for $n = k$, whe... | This is the inductive step written out:
$$
6 \cdot 6^k - 1 = 5 \cdot Q |+1; \cdot \frac{1}{6};-1 \Leftrightarrow 6^k - 1 = \frac{5\cdot Q-5}{6}\underset{P}{\rightarrow}5\cdot P = \frac{5\cdot Q - 5 }{6} | \cdot \frac{1}{5}; \cdot 6\Leftrightarrow Q=6\cdot P + 1
$$
$$
6^k - 1 = \frac{5\cdot Q-5}{6} \overset{Q}{\rightarr... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1239106",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "4",
"answer_count": 6,
"answer_id": 4
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Logic behind finding a $ (2 \times 2) $-matrix $ A $ such that $ A^{2} = - \mathsf{I} $. I know the following matrix "$A$" results in the negative identity matrix when you take $A*A$ (same for $B*B$, where $B=-A$):
$$A=\begin{pmatrix}0 & -1\cr 1 & 0\end{pmatrix}$$
However, I am not certain how one would go about findin... | If you assume that $A$ is invertible, then you can write:
$$
AA = -\mathbb{I}_2 \rightarrow A^{-1}AA = -A^{-1}\mathbb{I}_2 \rightarrow A = -A^{-1}\mathbb{I}_2
$$
But we know the inverse for a $2\times2$ matrix--and it's going to create a set of four linear equations for four variables:
$$
\begin{pmatrix}
a& b \\
c & d
... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1241232",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 4,
"answer_id": 2
} |
How to find the number of values for $x$ and $y$? I have come across numerous questions where I am asked for example, if $x$ and $y$ are non-negative integers and $3x + 4y = 96$, how many pairs of $(x,y)$ are there?
Usually, I just use trial and error and look at the multiples of $3$ and $4$. However, I was wondering w... | To solve your particular problem, you can see that what you need to find is all values of $x$ for which $96 - 3x$ is a positive multiple of $4$.
That means you want $96-3x$ to be divisible by $4$, and since $96$ is divisible by $4$, that means that $3x$ must be as well. Now, since $3$ and $4$ are coprime, $3x$ is divis... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1248213",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 2,
"answer_id": 1
} |
How to Find the Function of a Given Power Series? (Please see edit below; I originally asked how to find a power series expansion of a given function, but I now wanted to know how to do the reverse case.)
Can someone please explain how to find the power series expansion of the function in the title? Also, how would you... | $f_{n+3}=f_{n+2}+f_{n}; n\geq 1$ then $\sum^{\infty}_{n=1}f_{n+3}x^n = \sum^{\infty}_{n=1}f_{n+2}x^n+\sum^{\infty}_{n=1}f_{n}x^n$ with $f(x)=\sum^{\infty}_{n=1}f_{n}x^n$ then $\frac{f(x)-f_1x-f_2x^2-f_3x^3}{x^3}= \frac{f(x)-f_1x-f_2x^2}{x^2}+f(x)$ then
$$\frac{f(x)-x-x^2-x^3}{x^3}= \frac{f(x)-x-x^2}{x^2}+f(x)$$ then
$$... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1249107",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "4",
"answer_count": 6,
"answer_id": 4
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Inconsistent Matrices I'm teaching myself Linear Algebra and am not sure how to approach this problem:
Let A be a 4×4 matrix, and let b and c be two vectors in R4. We are told that the system Ax = b is inconsistent. What can you say about the number of solutions of the system Ax = c?
Bretscher, Otto (2013-02-21). Lin... | Perhaps things will go faster with a simpler example. Consider the inconsistent equations
$$
\begin{align}
x + y &= 1 \\
x + y &= 0
\end{align}
$$
The linear system is
$$
\begin{align}
%
\mathbf{A} x &= b\\
%
\left[
\begin{array}{cc}
1 & 1 \\
1 & 1 \\
\end{array}
\right]
\left[
\begin{array}{c}
x \\
y \\
\end{ar... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1250334",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 2,
"answer_id": 0
} |
How find $\max _{z: \ |z|=1} \ f \left( z \right)$ for $f \left( z \right) = |z^3 - z +2|$ Let $f : C \mapsto R $, $f \left( z \right) = |z^3 - z +2|$. How find $\max _{z: \ |z|=1} \ f \left( z \right)$ ?
| $\bf{My\; Solution::}$ Let $z=x+iy\;,$ Then $|x+iy| = 1\Rightarrow x^2+y^2 =1 $
Now We have To Maximize $$f(z) = \left|z^3-z+2\right| = \left|(x+iy)^3-(x+iy)+2\right|$$
We Get
$$f(x,y) = \left|x^3-iy^3+3ix^2y-3xy^2-x-iy+2\right|$$
$$f(x,y)=\left|x(x^2-3y^2)+iy(3x^2-y^2)-x-iy+2\right|\;,$$ Using $x^2+y^2 = 1$
$$f(x,y) =... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1252505",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 2,
"answer_id": 0
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Area of an equilateral triangle
Prove that if triangle $\triangle RST$ is equilateral, then the area of $\triangle RST$ is $\sqrt{\frac34}$ times the square of the length of a side.
My thoughts:
Let $s$ be the length of $RT$. Then $\frac s2$ is half the length of $\overline{RT}$. Construct the altitude from the $S$ t... | Let the lenght of the side be $s.$
In an equilateral triangle, the lengths of the sides are equal.
So, $RS=ST=TR=s.$
All angles are $\frac{π}{3}$ $radians.$
Area of a triangle as we know, is $$\frac{(RS)(ST)\sin(S)}{2}.$$
$$=\frac{s^2\sin(\frac{π}{3})}{2}$$
The area of the $ΔRST$ is thus, $$\frac{s^2\sqrt3}{4}$$
$Quo... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1252840",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "4",
"answer_count": 2,
"answer_id": 0
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Integration with Limits Find $$\displaystyle \lim_{n \to \infty} \int^{1}_{0}(x^{n}+(1-x)^{n})^{\frac{1}{n}}dx$$
Now, the solution was hinted like this:
using the property $$f(x)=f(2a-x)$$ the limit becomes half
$$2 \displaystyle \int^{\frac{1}{2}}_{0} (x^{n}+(1-x)^{n})^{\frac{1}{n}} dx$$.....$$(1)$$
plz see carefully.... | $$y=(x^{n}+(1-x)^{n})^{\frac{1}{n}}=(1-x)\left(1+\left(\frac{x}{1-x}\right)^{n}\right)^{\frac{1}{n}}$$
Now, in $0\leq x \leq \frac{1}{2}$
$$0\leq \frac{x}{1-x} \leq 1$$
$$0\leq \left(\frac{x}{1-x}\right)^{n} \leq 1$$
$$1\leq \left(1+\left(\frac{x}{1-x}\right)^{n}\right)^{\frac{1}{n}} \leq 2^{\frac{1}{n}}$$
$$(1-x)\leq ... | {
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How can $n^5+4$ be a perfect square? How can one find all $n \in \mathbb{N}$ such that $n^5+4$ is a perfect square?
I see that $n^5=(x+2)(x-2)$ here im suck can someone help ?
| Partial solution:
$d=\gcd(x+2,x-2)$ is $1$, $2$ or $4$.
If $d=1$ then $x+2$ and $x-2$ are perfect fifth powers. That is clearly impossible.
If $d=2$ then $x+2=2a$ and $x-2=2^4b=16b$ where $a$ is odd and $b$ is an integer (or vice versa). Either way, we have that $x=a+8b$, which is odd; a contradiction.
If $d=4$ then $x... | {
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Prove that $f=x^4-4x^2+16\in\mathbb{Q}[x]$ is irreducible
Prove that $f=x^4-4x^2+16\in\mathbb{Q}[x]$ is irreducible.
I am trying to prove it with Eisenstein's criterion but without success: for p=2, it divides -4 and the constant coefficient 16, don't divide the leading coeficient 1, but its square 4 divides the cons... | Below is an explicit proof. Note that $x^4-4x^2+16 = (x^2-2)^2 + 12$, which clearly has no real root. Hence, the only possible way to reduce $x^4-4x^2+16$ over $\mathbb{Q}$ is $(x^2+ax+b)(x^2+cx+d)$. However, the roots of $x^4-4x^2+16$ are $x = \pm \sqrt{2 \pm i\sqrt{12}}$, which are all complex numbers. Since complex ... | {
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"source": "stackexchange",
"question_score": "17",
"answer_count": 9,
"answer_id": 7
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Area enclosed by an equipotential curve for an electric dipole on the plane I am currently teaching Physics in an Italian junior high school. Today, while talking about the electric dipole generated by two equal charges in the plane, I was wondering about the following problem:
Assume that two equal charges are placed... | At the first step, I will introduce a proper curve linear coordinates for this problem. This will help to construct the integral for area. We can write the equation of these equi-potential curves as
$$\frac{1}{r_1}+\frac{1}{r_2}=C \tag{1}$$
where $C$ is some real constant and $r_1$ and $r_2$ are defined as
$$r_1=\sqrt{... | {
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"source": "stackexchange",
"question_score": "20",
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The residue of $9^{56}\pmod{100}$ How can I complete the following problem using modular arithmetic?
Find the last two digits of $9^{56}$.
I get to the point where I have $729^{18} \times 9^2 \pmod{100}$. What should I do from here?
| $\bmod 4\!:\ 9^{56}\equiv 1^{56}\equiv 1\,\Rightarrow\, 9^{56}=4m+1$.
$$\bmod 25\!:\ 9^{56}\stackrel{(1)}\equiv 9^{56\pmod{\phi(25)}}\equiv 9^{56\pmod {20}}\equiv 9^{-4}\equiv \left(\frac{1}{81}\right)^2$$
$$\equiv \left(\frac{1+5\cdot 25}{6+3\cdot 25}\right)^2\equiv \left(\frac{126}{6}\right)^2\equiv 21^2\equiv ... | {
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What is the connection between the discriminant of a quadratic and the distance formula? The $x$-coordinate of the center of a parabola $ax^2 + bx + c$ is $$-\frac{b}{2a}$$
If we look at the quadratic formula
$$\frac{-b \pm \sqrt{b^2 - 4ac}}{2a}$$
we can see that it specifies two points at a certain offset from the cen... | We have $$f(x)=ax^2+bx+c=a(x+\frac{b}{2a})^2+\frac{4ac-b^2}{4a}$$ with $b^2-4ac\geq 0$ and $a\neq 0$. The vertex of the parabola is $$V\left(-\frac{b}{2a},\frac{4ac-b^2}{4a}\right)$$ and the root(s) are at $$R_1\left(-\frac{b}{2a}+\sqrt{\frac{b^2-4ac}{4a^2}},0\right)~\text{and}~R_2\left(-\frac{b}{2a}-\sqrt{\frac{b^2-4a... | {
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"question_score": "19",
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Help me understand this solution $$2x^4yy'+y^4=4x^6$$
The way my teacher did it is:
First, he made a substitution:
$y=z^m$
$y'=mz^{m-1}z'$
$$2x^4 z^m mz^{m-1} z'+z^{4m}=4x^6$$
$$z'=\frac{4x^6-z^{4m}}{2mx^4z^{2m-1}}=f\left(\frac{z}{x}\right)$$
Then he wrote the part I don't understand (how he got $m$):
$2m-1=2$
$2m+1=4$... | here ia nother way to do this problem.
a change of variable $y = u^k$ in $$2x^4 yy'+y^4 = 4x^6 \to 2kx^4u^{2k-1}u'+u^{4k} = 4x^6$$ now we will choose $k$ so that $2k-1 = 4k \to k = -1/2.$ and we have $$-x^4u'+u= 4x^6 \to e^{x^3/3}u' - \frac{e^{x^3/3}}{x^4}u = -4x^6e^{x^3/3} $$ on integration, we get $$u =-4e^{-x^3/3}... | {
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Finding the inverse of a number under a certain modulus How does one get the inverse of 7 modulo 11?
I know the answer is supposed to be 8, but have no idea how to reach or calculate that figure.
Likewise, I have the same problem finding the inverse of 3 modulo 13, which is 9.
| Here's an illustration of finding the multiplicative inverse of $37 \bmod 100$ using the extended Euclidean algorithm. (I used bigger numbers for this example so that the relationships are a little clearer).
On each line, $n=100s+37t$. We start the table with two lines giving $n=100$ and $n=37$ in the obvious way. Then... | {
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"question_score": "2",
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For $n>2, n\in\mathbb{Z}$, why is this true: $\left\lfloor 1/\left(\frac{1}{n^2}+\frac{1}{(n+1)^2}+\cdots+\frac{1}{(n+n)^2}\right)\right\rfloor=2n-3$ Let $n>2$ be a positive integer, prove that
$$\left\lfloor \dfrac{1}{\dfrac{1}{n^2}+\dfrac{1}{(n+1)^2}+\cdots+\dfrac{1}{(n+n)^2}}\right\rfloor=2n-3?$$
before I use hand C... | Asymptotically,
$$ \sum_{i=n}^{2n} \dfrac{1}{i^2} = \dfrac{1}{2n} + \dfrac{5}{8n^2} + O(1/n^3)$$
so
$$ \dfrac{1}{\displaystyle\sum_{i=n}^{2n} \dfrac{1}{i^2}} = 2n - \dfrac{5}{2} + O(1/n) $$
Thus your equation will be true for sufficiently large $n$. With sufficiently good explicit bounds on the $O(1/n^3)$ term, you sh... | {
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"source": "stackexchange",
"question_score": "3",
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How to solve this nonstandard system of equations? How to solve this system of equations
$$\begin{cases}
2x^2+y^2=1,\\
x^2 + y \sqrt{1-x^2}=1+(1-y)\sqrt{x}.
\end{cases}$$
I see $(0,1)$ is a root.
| Solution.
First way
From the first equation, we have
$$\begin{cases}
2x^2\leqslant 1,\\
y^2 \leqslant 1
\end{cases}
\Leftrightarrow
\begin{cases}
-\dfrac{1}{\sqrt{2}} \leqslant x \leqslant \dfrac{1}{\sqrt{2}},\\
- 1 \leqslant y \leqslant 1.
\end{cases}$$
Then, the conditions of $x$ and $y$ are
$$\begin{cases}
0 \leqs... | {
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"source": "stackexchange",
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For which values of $a,b,c$ is the matrix $A$ invertible? $A=\begin{pmatrix}1&1&1\\a&b&c\\a^2&b^2&c^2\end{pmatrix}$
$$\Rightarrow\det(A)=\begin{vmatrix}b&c\\b^2&c^2\end{vmatrix}-\begin{vmatrix}a&c\\a^2&c^2\end{vmatrix}+\begin{vmatrix}a&b\\a^2&b^2\end{vmatrix}\\=ab^2-a^2b-ac^2+a^2c+bc^2-b^2c\\=a^2(c-b)+b^2(a-c)+c^2(b-a)... | The matrix is the special case of so-called Vandermonde Matrix (See Induction for Vandermonde Matrix).
det($A$)$=(a-b)(a-c)(b-c)$. So if $a\neq b,a\neq c, b\neq c, \space$ det$(A)\neq 0$.
| {
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"source": "stackexchange",
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If $\frac{a}{b}=\frac{b}{c}=\frac{c}{d}$, prove that $\frac{a}{d}=\sqrt{\frac{a^5+b^2c^2+a^3c^2}{b^4c+d^4+b^2cd^2}}$ What I've done so far;
$$\frac{a}{b}=\frac{b}{c}=\frac{c}{d}=k\\
a=bk, b=ck, c=dk\\
a=ck^2, b=dk^2\\
a=dk^3$$
I tried substituting above values in the right hand side of the equation to get $\frac{a}{d}$... | You did fine. Repalicng your values in the term inside the radical, you have
$$a^5+b^2c^2+a^3c^2=d^5 k^{15}+d^5 k^{11}+d^4 k^6$$ $$b^4c+d^4+b^2cd^2=d^5 k^9+d^5 k^5+d^4$$ thet is to say that the ratio is $k^6$, its square root is $k^3$ which is $\frac ad$.
| {
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$\lim\limits_{h\to0} \frac{(x+h)^3 -x^3}{h}$ Compute the following limit I can not seem to figure out the direction to go with this problem. I'm not quite sure how to break it up.
| Others have pointed out here that $(x+h)^3 = x^3+3x^2h+3xh^2+h^3$.
Here's different approach. Recall that $a^3-b^3=(a-b)(a^2+ab+b^2)$. Consequently
\begin{align}
(x+h)^3-x^3 & = \Big((x+h)-x\Big)\Big( (x+h)^2 +(x+h)x + x^2 \Big) \\[12pt]
& = h\Big( (x+h)^2 +(x+h)x + x^2 \Big).
\end{align}
Now we might be tempted to d... | {
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Prove that number of zeros at the right end of the integer $(5^{25}-1)!$ is $\frac{5^{25}-101}{4}.$
Prove that number of zeros at the right end of the integer $(5^{25}-1)!$ is $\frac{5^{25}-101}{4}.$
Attempt:
I want to use the following theorem:
The largest exponent of $e$ of a prime $p$ such that $p^e$ is a divisor... | Actually it would be:
$$\small e_1=\sum_{k=1}^{\infty} \left\lfloor\frac{5^{25}-1}{5^k}\right\rfloor=\sum_{k=1}^{\infty} \left\lfloor5^{25-k}-\frac1{5^k}\right\rfloor=\sum_{k=1}^{25} (5^{25-k}-1)\tag{$\because \left\lfloor-\frac1{5^k}\right\rfloor=-1$}$$
| {
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"question_score": "3",
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Diagonalize tri-diagonal symmetric matrix How to diagonalize the following matrix?
\begin{pmatrix}
2 & -1 & 0 & 0 & 0 & \cdots \\
-1 & 2 & -1 & 0 & 0 & \cdots \\
0 & -1 & 2 & -1 & 0 & \cdots \\
\cdots & \cdots & \cdots & \cdots & \cdots & \cdots \\
\cdots & 0 & 0 & -1 & 2 & -1 \\
\cdots & 0 & 0 & 0 & -1 & 2 \\
\en... | Let $\mathbf{x}$ be the eigenvector of $A$ with eigenvalue $\lambda$.
$$
A \mathbf{x} = \lambda \mathbf{x}
$$
Note that $\mathbf{x}$ is a solution to the following difference equation:
$$
-x_{k+1} + 2x_k - x_{k-1} = \lambda x_k,\quad k = 1, \dots, n\\
x_0 = 0,\quad x_{n+1} = 0.
$$
The characteristic equation for that l... | {
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"source": "stackexchange",
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How does $-\frac{1}{x-2} + \frac{1}{x-3}$ become $\frac{1}{2-x} - \frac{1}{3-x}$ I'm following a solution that is using a partial fraction decomposition, and I get stuck at the point where $-\frac{1}{x-2} + \frac{1}{x-3}$ becomes $\frac{1}{2-x} - \frac{1}{3-x}$
The equations are obviously equal, but some algebraic mani... | This problem all boils down to the following relationship $$-1 = \frac{-1}{1}=\frac{1}{-1}$$
Part one is easy if you just express the division as a multiplication
$$x=\frac{-1}{1}\implies -1=1\cdot x\implies -1=x$$
For part two,
$$x=\frac{1}{-1}\implies1=-1\cdot x$$
$$1+(-1)+x=-1\cdot x+(-1)+x$$
$$x+0=-1\cdot x + 1\cdo... | {
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"source": "stackexchange",
"question_score": "5",
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Determine values of k for a matrix to have a unique solution I have the following system and need to find for what values of $k$ does the system have
i) a unique solution
ii) no solution
iii) an infinite number of solutions
$(k^3+3k)x + (k+5)y + (k+3)z = k^5+(k+3)^2
ky + z = 3
(k^3+k^2-6k)z = k(k^2-9)$
Putting this i... | Coefficient matrix is given by
$A = \begin{pmatrix}
k^3+3k & k+5 & k+3\\
0 & k & 1 \\
0 & 0 & k^3+k^2-6k
\end{pmatrix}$
And, augmented matrix is given by
$[A:b] = \begin{pmatrix}
k^3+3k & k+5 & k+3 & k^5+(k+3)^2\\
0 & k & 1 & 3\\
0 & 0 & k^3+k^2-6k & k(k^2-9)
\end{pmatrix}$
(1) UNIQUE S... | {
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"source": "stackexchange",
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Primitive element theorem, simple extension
Let $X$, $Y$ be indeterminates over $F_2$, the finite field with 2 elements.
Let $L = F_2(X, Y )$ and $K = F_2(u, v)$, where $u = X + X^2$, $v = Y + Y^2$.
Explain why $L$ is a simple extension of $K$. Find an element $\gamma \in L$ such
that $L = K(\gamma)$. [Hint: First... | Note that sending $X$ to $X+1$ is a $K$-automorphism of $L$ since this is always an automorphism of $K$ since $X$ is transcendental over $F_2$, and it fixes $u$ since $X+1+(X+1)^2=X+1+X^2+1=X+X^2$. Similarly sending $Y$ to $Y+1$ is a $K$-automorphism. Note that these automorphisms necessarily generate the Galois group ... | {
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"timestamp": "2023-03-29T00:00:00",
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Fermats Little Theorem: How $(3^7)^{17}$ will leave the same remainder when divided by $17$ as $3^7$? As it is used in this explanation, how and why $(3^7)^{17}$ will leave the same remainder when divided by $17$ as $3^7$?
Thanks!
| The question referred to is
Find the remainder when $54^{124}$ is divided by $17$
Step 1: Reduce the base according to the modulus
We can see that $54 \equiv 3 \bmod 17$, so $54^{124} \equiv 3^{124} \bmod 17$.
Step 2: Reduce the exponent according to the order of the base
We know that, since $17$ is prime, $a^{17} \... | {
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Show that $f:= \frac{1 - 2x}{x^2 - 1}$ is monotonically increasing in the open interval $(-1, 1)$ Show that $f:= \frac{1 - 2x}{x^2 - 1}$ is monotonically increasing in the open interval $(-1, 1)$
To show a function is monotonically increasing, I started by saying that:
A function $f$ is monotonically increasing in an ... | Without derivatives:
Note that we can write for $|x|<1$
$$f(x)=\frac{1-2x}{x^2-1}= \frac{-1/2}{x-1}+ \frac{-3/2}{x+1}$$
If we take $y>x$, then we have
$$f(y)-f(x) =\frac{y-x}{2(x-1)(y-1)}+\frac{3(y-x)}{2(x+1)(y+1)}>0$$
for all $-1<x<y<1$
| {
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"source": "stackexchange",
"question_score": "3",
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$\lim_{x\rightarrow0}\frac{1-\left(\cos x\right)^{\ln\left(x+1\right)}}{x^{4}}$ Could you please check if I derive the limit correctly?
$$\lim_{x\rightarrow0}\frac{1-\left(\cos x\right)^{\ln\left(x+1\right)}}{x^{4}}=\lim_{x\rightarrow0}\frac{1-\left(O\left(1\right)\right)^{\left(O\left(1\right)\right)}}{x^{4}}=\lim_{x\... | $$1-(\cos{x})^{\log{(1+x)}} = 1-e^{\log{\cos{x}} \log{(1+x)}} $$
$$\begin{align}\log{\cos{x}} &= \log{\left ( 1-\frac{x^2}{2!} + \frac{x^4}{4!}+\cdots \right )}\\ &= \left (-\frac{x^2}{2!} + \frac{x^4}{4!}+\cdots \right ) - \frac12\left (-\frac{x^2}{2!} + \frac{x^4}{4!}+\cdots \right )^2+\cdots \\ &= -\frac{x^2}{2}-\fr... | {
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"timestamp": "2023-03-29T00:00:00",
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$ \lim_{x\to o} \frac{(1+x)^{\frac1x}-e+\frac{ex}{2}}{ex^2} $
$$ \lim_{x\to o} \frac{(1+x)^{\frac1x}-e+\frac{ex}{2}}{ex^2} $$
(can this be duplicate? I think not)
I tried it using many methods
$1.$ Solve this conventionally taking $1^\infty$ form in no luck
$2.$ Did this, expand $ {(1+x)^{\frac1x}}$ using binomial th... | let $y = \left(1 + \frac1n\right)^n.$ taking logarithm, we get
$$\begin{align} \ln y &= n\ln\left(1 + \frac 1n \right)\\
&= n \left(\frac 1n - \frac1{2n^2} +\frac1{3n^3}+\cdots \right) \\
& = 1 - \frac1{2n} + \frac{1}{3n^2}+ \cdots \end{align}$$
therefore $$\begin{align} y &= ee^{-\fra... | {
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"source": "stackexchange",
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How to find cotangent? Need to find a $3\cot(x+y)$ if $\tan(x)$ and $\tan(y)$ are the solutions of $x^2-3\sqrt{5}\,x +2 = 0$.
I tried to solve this and got $3\sqrt{5}\cdot1/2$, but the answer is $-\sqrt{5}/5$
| From quadratic equation given,
$$\tan{x}+\tan{y}=3\sqrt5$$
$$\tan{x}\tan{y}=2$$
So by the Compound Angle Formula,
$$\tan{(x+y)}=\frac{\tan{x}+\tan{y}}{1-\tan{x}\tan{y}}=\frac{3\sqrt5}{-1}=-3\sqrt5$$
So we have
$$3\cot{(x+y)}=3\cdot{\frac1{\tan{(x+y)}}}=3\cdot{\frac1{-3\sqrt5}}=\frac{-\sqrt5}{5}$$
| {
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"source": "stackexchange",
"question_score": "5",
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Proving simple trigonometric identity I need help with this one
$$
\frac{\sin^2 \alpha}{\sin\alpha - \cos\alpha} + \frac{\sin\alpha + \cos \alpha}{1- \mathrm{tan}^2\alpha} - \cos\alpha = \sin \alpha
$$
I tried moving sin a on the other side of the eqation
$$
\frac{\sin^2 \alpha}{\sin\alpha - \cos\alpha} + \frac{\sin\... | Hint: Put the fractions over the same denominator $\left(\sin\left(\alpha\right)-\cos\left(\alpha\right)\right)\left(1-\tan^{2}\left(\alpha\right)\right)$, then eliminate the denominator on the left side. Now you have just to expand the left and right side, and finally use the identity $\tan\left(\alpha\right)=\frac{\s... | {
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
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} |
Integration of complex functions with trig functions: $\int_0^{2 \pi} \frac{ d\theta}{5-\cos( \theta )}$ $\int_0^{2 \pi} \frac{ d\theta}{5-\cos( \theta )}$
How should I integrate this? Using the exponantial identities of trig? Any hints will be great...
Thank you!
| Here's an answer using Fourier series.
First fact: if $\beta\in(-1,1)$,
$$\sum_{n=0}^{+\infty}\beta^n\cos(n\theta)=\Re\left(\frac1{1-\beta\mathrm{e}^{i\theta}}\right)=\frac{1-\beta\cos\theta}{1+\beta^2-2\beta\cos\theta}=\frac12\frac{2-2\beta\cos\theta}{1+\beta^2-2\beta\cos\theta}=\frac12\frac{\bigl(1+\beta^2-2\beta\cos... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1286109",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 3,
"answer_id": 2
} |
Another combined limit I've tried to get rid of those logarithms, but still, no result has came.
$$\lim_{x\to 0 x \gt 0} \frac{\ln(x+ \sqrt{x^2+1})}{\ln{(\cos{x})}}$$
Please help
| HINT:
To get rid of the $ln$ in the denominator, remember that $\ln \cos x= \frac{1}{2} \ln(1-\sin^2(x))$.
As for the $ln$ in the nominator, you'll have to calculate the limit: $\lim_{x\to 0^+} {\ln(x+ \sqrt{x^2+1})^{1/x^2}}$ wich is easier (it's zero).
SOLUTION:
\begin{align}
\lim_{x\to 0^+} \frac{\ln(x+ \sqrt{x^2+1})... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1286651",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 6,
"answer_id": 0
} |
Find the integer x: $x \equiv 8^{38} \pmod {210}$ Find the integer x: $x \equiv 8^{38} \pmod {210}$
I broke the top into prime mods:
$$x \equiv 8^{38} \pmod 3$$
$$x \equiv 8^{38} \pmod {70}$$
But $x \equiv 8^{38} \pmod {70}$ can be broken up more:
$$x \equiv 8^{38} \pmod 7$$
$$x \equiv 8^{38} \pmod {10}$$
But $x \equiv... | The quick solution to the Chinese Remainder Theorem exercise is to find integers $w,x,y,z$ such that $105w + 70x+42y + 30z=1$. Then:
$$x\equiv 0\cdot 105w + 1\cdot 70x + 4\cdot 42y + 1\cdot 30z\pmod {210}$$
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/1286716",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "8",
"answer_count": 8,
"answer_id": 0
} |
show that $\frac{1}{F_{1}}+\frac{2}{F_{2}}+\cdots+\frac{n}{F_{n}}<13$ Let $F_{n}$ is Fibonacci number,ie.($F_{n}=F_{n-1}+F_{n-2},F_{1}=F_{2}=1$)
show that
$$\dfrac{1}{F_{1}}+\dfrac{2}{F_{2}}+\cdots+\dfrac{n}{F_{n}}<13$$
if we use
Closed-form expression
$$F_{n}=\dfrac{1}{\sqrt{5}}\left(\left(\dfrac{1+\sqrt{5}}{2}\right... | Let $\phi=\frac{1+\sqrt{5}}{2}$. Then $\phi^2=\phi+1$.
By induction, we have $F_n > \phi^{n-2}$ for all $n\ge 1$, and so
$$
\sum_{n=1}^{N} \frac{n}{F_n}
<
\sum_{n=1}^{\infty} \frac{n}{F_n}
< \sum_{n=1}^{\infty} \frac{n}{\phi^{n-2}}
= \sum_{n=1}^{\infty} \frac{n\phi^2}{\phi^{n}}
= \phi^2 \sum_{n=0}^{\infty} \frac{n}{\ph... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1287613",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "14",
"answer_count": 6,
"answer_id": 0
} |
Trigonometric Substitution in $\int _0^{\pi/2}{\frac{ x\cos x}{ 1+\sin^2 x} dx }$ Evaluate $$ \int _{ 0 }^{ \pi /2 }{ \frac { x\cos { (x) } }{ 1+\sin ^{ 2 }{ x } } \ \mathrm{d}x } $$
$$$$
The solution was suggested like this:$$$$
SOLUTION:
First of all its, quite obvious to have substitution $ \sin(x) \rightarrow x $
$... | You may obtain
$$
\int_0^{\pi/2} \frac{x\cos x}{1+\sin^2 x}\:dx=\frac12 \log^2 (1+\sqrt{2}) . \tag1
$$
Proof. First observe that
$$\begin{align}
\int_0^\pi \frac{x\cos x}{1+\sin^2 x}\:dx &= \int_0^{\pi/2} \frac{x\cos x}{1+\sin^2 x}\:dx + \int_{\pi/2}^{\pi} \frac{x\cos x}{1+\sin^2 x}\:dx \\
& = \int_0^{\pi/2} \frac{x... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1289387",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 3,
"answer_id": 2
} |
Matrix Problem of form Ax=B The matrix $A$ is given by
$$\left(\begin{array}{ccc}
1 & 2 & 3 & 4\\
3 & 8 & 11 & 8\\
1 & 3 & 4 & \lambda\\
\lambda & 5 & 7 & 6\end{array} \right)$$
Given that $\lambda$=$2$, $B$=$\left(\begin{array}{ccc}
2 \\
4 \\
\mu \\
3 \end{array} \right)$ and $X$=$\left(\begin{array}{ccc}
x \\
y \\
z ... | Gauss-Jordan elimination is one of standard methods for solving linear systems.
$
\left(\begin{array}{cccc|c}
1 & 2 & 3 & 4 & 2 \\
3 & 8 &11 & 8 & 4 \\
1 & 3 & 4 & 2 & \mu \\
2 & 5 & 7 & 6 & 3
\end{array}\right)\sim
\left(\begin{array}{cccc|c}
1 & 2 & 3 & 4 & 2 \\
2 & 5 & 7 & 6 & 3 \\
3 & 8 &11 & 8 & 4 \\... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1290492",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 3,
"answer_id": 2
} |
Subsets and Splits
Fractions in Questions and Answers
The query retrieves a sample of questions and answers containing the LaTeX fraction symbol, which provides basic filtering of mathematical content but limited analytical insight.