https://liberty.ioCurated star highlights with fresh energy.en-usSat, 29 Aug 2026 23:33:15 -0400https://liberty.io/calculating-a-minimum-surface-area-of-a-box.htmlhttps://liberty.io/calculating-a-minimum-surface-area-of-a-box.htmlI want to calculate the minimum surface area of a (closed) box for a given volume. So let’s say I have a given volume V (e.g. V=10m^3). And I need a box where all the surface area is as minimal as ...Sun, 30 Aug 2026 03:28:00 -0400https://liberty.io/how-does-tree-3-grow-to-get-so-big-laymen-explanation.htmlhttps://liberty.io/how-does-tree-3-grow-to-get-so-big-laymen-explanation.htmlI am not a mathematician but I am interested in big numbers. I find them to be really interesting, almost god-like. I am watching a series of videos from David Metzler on YouTube. I have a basic ...Sun, 30 Aug 2026 03:19:02 -0400https://liberty.io/trial-solutions-for-differential-equations.htmlhttps://liberty.io/trial-solutions-for-differential-equations.htmlI have a question from my Differential Equations & Linear Algebra class. When you're trying to find the general solution to an nth order linear non-homogeneous differential equation, you have t......Sun, 30 Aug 2026 03:02:57 -0400https://liberty.io/what-is-the-difference-between-10-choose-2-and-10-choose-1-times-9-cho.htmlhttps://liberty.io/what-is-the-difference-between-10-choose-2-and-10-choose-1-times-9-cho.htmlI'm not able to understand the difference between these two. Don't both just give the number of ways of selecting $2$ objects from a total of $10$? Maybe the difference is that ${10 \choose 2}$ gi......Sun, 30 Aug 2026 02:54:33 -0400https://liberty.io/how-do-i-compute-a-double-weighted-average-in-other-words-an-average-w.htmlhttps://liberty.io/how-do-i-compute-a-double-weighted-average-in-other-words-an-average-w.htmlThis might be a simple question... I was wondering, how does one take a double weighted average? (honestly, don't know if that is what it is called). For example, let's say that we are working wit......Sun, 30 Aug 2026 02:49:27 -0400https://liberty.io/expected-number-of-coin-flips-to-get-three-heads-or-three-tails.htmlhttps://liberty.io/expected-number-of-coin-flips-to-get-three-heads-or-three-tails.htmlWe both play a game where we flip a coin. You win if 3 heads appear, I win if 3 tails appear. What is the expected number of flips for the game to end. The heads/tails doesn't need to be consecutive. ...Sun, 30 Aug 2026 02:47:56 -0400https://liberty.io/how-are-the-known-digits-of-pi-guaranteed.htmlhttps://liberty.io/how-are-the-known-digits-of-pi-guaranteed.htmlWhen discussing with my son a few of the many methods to calculate the digits of $\pi$ (15 yo school level), I realized that the methods I know more or less (geometric approximation, Monte Carlo and ...Sun, 30 Aug 2026 02:47:43 -0400https://liberty.io/proving-a-statement-using-the-definition-of-a-limit.htmlhttps://liberty.io/proving-a-statement-using-the-definition-of-a-limit.htmlLet $f$ be a function that is defined on a deleted neighbourhood of $x_0 ∈ \Bbb R$, such that $f(x) \ne 0$ for every $x$ on that deleted neighbourhood. Suppose that $$\lim _{x\to x_0} f(x) = \lim_{......Sun, 30 Aug 2026 02:46:56 -0400https://liberty.io/what-is-the-measure-of-angle-f.htmlhttps://liberty.io/what-is-the-measure-of-angle-f.htmlThis is an isosceles triangle with base angles being the only part filled in. I have trouble understanding that if the base angles are not like terms, then how can I solve my equation (which is angle ...Sun, 30 Aug 2026 02:45:46 -0400https://liberty.io/some-tips-on-3d-trig.htmlhttps://liberty.io/some-tips-on-3d-trig.htmlI understand this isn't a MATHS question specifically, however, sometimes I have trouble identifying when to use 3D trig, and trouble with it in general. Can some of the experienced/advanced people ...Sun, 30 Aug 2026 02:29:14 -0400https://liberty.io/what-is-an-indexed-set.htmlhttps://liberty.io/what-is-an-indexed-set.htmlI stuck on something. I am trying to understand what is said in the image I linked above. "A function I from a set Λ onto a set a is said to index the set a by Λ. The set Λ is called the index an......Sun, 30 Aug 2026 02:21:43 -0400https://liberty.io/question-on-the-039-hat-check-039-problem.htmlhttps://liberty.io/question-on-the-039-hat-check-039-problem.htmlThe famous 'Hat Check Problem' goes like this, 'n' men enter the restaurant and put their hats at the reception. Each man gets a random hat back when going back after having dinner. The goal is to ......Sun, 30 Aug 2026 02:21:34 -0400https://liberty.io/notation-for-elements-of-a-vector.htmlhttps://liberty.io/notation-for-elements-of-a-vector.htmlIf I want $x_i$ to be an arbitrary element of a vector $\vec{x}$ can I use the following notation: $x_i \in \vec{x}=[x_1\;x_2\;\cdots\;x_n]^T\in R^n$ ? And if I later want to spesify the interval o......Sun, 30 Aug 2026 02:19:25 -0400https://liberty.io/find-the-number-of-real-solutions-of-the-equation-x-3-1-2-2x-1-frac-1-.htmlhttps://liberty.io/find-the-number-of-real-solutions-of-the-equation-x-3-1-2-2x-1-frac-1-.htmlQuestion: Find the number of real solutions of the following equation: $$x^3+1=2(2x-1)^{\frac{1}{3}}$$ My Approach: Since the question was under the topic "Inverse Functions", I just tri......Sun, 30 Aug 2026 02:17:19 -0400https://liberty.io/relationship-between-cartesian-velocity-and-polar-velocity.htmlhttps://liberty.io/relationship-between-cartesian-velocity-and-polar-velocity.htmlI am trying to convert an equation from Cartesian to polar coordinates. I know for a given $x$ and $y$ Cartesian, we can get polar $r=\sqrt{x^2+y^2}$ and $\theta = \arctan(y/x)$. However, for a g......Sun, 30 Aug 2026 02:17:01 -0400https://liberty.io/converge-in-probability-and-big-oh-pee-1.htmlhttps://liberty.io/converge-in-probability-and-big-oh-pee-1.htmlI have read an econometrics textbook on "converge in distribution" and found the following sentence. \begin{equation} \text{If} \hspace{0.25 cm} Z_{N} \xrightarrow{d} Z, Z_{N} = \large{O......Sun, 30 Aug 2026 02:10:41 -0400https://liberty.io/how-to-define-a-vector-without-a-basis.htmlhttps://liberty.io/how-to-define-a-vector-without-a-basis.htmlWe know that a vector is an element of a vector space that does not depend by the basis that is chosen to represent it. But, in a finite dimensional vector space, when we want define a specific vec......Sun, 30 Aug 2026 02:09:06 -0400https://liberty.io/are-abelian-subgroups-normal.htmlhttps://liberty.io/are-abelian-subgroups-normal.htmlLet $G$ be a group, and let $H < G$ be such that all the elements of $H$ commute with each other, i.e. $H$ is abelian. Then is $H$ necessarily normal in $G$, i.e. $H \unlhd G$?...Sun, 30 Aug 2026 01:50:55 -0400https://liberty.io/how-to-show-an-ancillary-statistic-is-a-first-order-ancillary-statisti.htmlhttps://liberty.io/how-to-show-an-ancillary-statistic-is-a-first-order-ancillary-statisti.htmlA statistic $A(X)$ is first order ancillary if $\mathbb{E}_{\theta}[A(X)]$ does not depend on $\theta$ where $X \sim P_{\theta}$. Show when distribution of $A(X)$ is independent of $\theta$, $\math......Sun, 30 Aug 2026 01:50:13 -0400https://liberty.io/formal-definition-of-the-differential-of-a-function.htmlhttps://liberty.io/formal-definition-of-the-differential-of-a-function.htmlThe formal definition of the differential of a differentiable function $f: x \mapsto y=f(x)$ is that it's a two-variable function, its name is $df$ and its value is $df(x,\Delta_X) = f'(x)\cdot\Del......Sun, 30 Aug 2026 01:50:10 -0400https://liberty.io/prove-that-every-year-has-at-least-one-friday-the-13th.htmlhttps://liberty.io/prove-that-every-year-has-at-least-one-friday-the-13th.htmlEveryone knows Friday the 13th is regarded as a day of bad luck. Why does every year have at least one of this bad day? Source: 2010 ISI B Math UGB...Sun, 30 Aug 2026 01:48:20 -0400https://liberty.io/chain-rule-with-triple-composition.htmlhttps://liberty.io/chain-rule-with-triple-composition.htmlWe are supposed to apply the chain rule on the following function $f$: $$ f(x) = \sqrt{x+\sqrt{2x+\sqrt{3x}}} $$ I assumed we could rewrite this as $$ f(x) = g(h(j(x))) $$ However, I was not su......Sun, 30 Aug 2026 01:42:43 -0400https://liberty.io/finding-a-equation-for-a-tangent-line-to-the-curve-y-e-x-which-also-go.htmlhttps://liberty.io/finding-a-equation-for-a-tangent-line-to-the-curve-y-e-x-which-also-go.htmlI have an assignment Find an equation for a tangent line to the curve $y=e^x$ which also goes through the origin. However, in my formula it is asserted that the slope between at a point "p" an......Sun, 30 Aug 2026 01:37:00 -0400https://liberty.io/how-do-i-find-the-cumulative-distribution-function-of-a-binomial-rando.htmlhttps://liberty.io/how-do-i-find-the-cumulative-distribution-function-of-a-binomial-rando.htmlHow would I find the cumulative distribution function of a binomial? I know I'd have to integrate it with its given parameters but how would someone go about doing that?...Sun, 30 Aug 2026 01:34:26 -0400https://liberty.io/understanding-the-proof-of-sqrt-2-is-irrational-by-contradiction.htmlhttps://liberty.io/understanding-the-proof-of-sqrt-2-is-irrational-by-contradiction.htmlI have some difficulties in understanding the proof of "$\sqrt{2}$is irrational" by contradiction. I am reading it in 10th class(in India) Mathematics book( available online, here ) This ......Sun, 30 Aug 2026 01:33:20 -0400https://liberty.io/when-square-is-inscribed-how-to-identify-the-similar-triangles.htmlhttps://liberty.io/when-square-is-inscribed-how-to-identify-the-similar-triangles.htmlIn the picture, $ \square PQRS $ is a square inscribed in a right triangle. How can we come to conclusion that $ \triangle ABC \sim \triangle PBQ $? My understanding is that $ \triangle ABC $ and $ \...Sun, 30 Aug 2026 01:29:00 -0400https://liberty.io/find-adjacent-only-knowing-angle-and-opposite.htmlhttps://liberty.io/find-adjacent-only-knowing-angle-and-opposite.htmlCan you find the length of the adjacent side of a right triangle only knowing the length of the opposite side and the angle? If so how do you calculate it?...Sun, 30 Aug 2026 01:09:19 -0400https://liberty.io/backpropagation-with-linear-algebra-how.htmlhttps://liberty.io/backpropagation-with-linear-algebra-how.htmlI have an intresset in control theory - optimal control. But unfortunately all control theories are for linear models. That's become a very difficult issue when to implement a linear controller for a ...Sun, 30 Aug 2026 01:08:17 -0400https://liberty.io/basic-open-set-question.htmlhttps://liberty.io/basic-open-set-question.htmlIn $\mathbb{R}^2$, if I have an open set, call it $U$, and a point $u_0\in U$. Is the set $U^{'} = U - \{ u_0 \}$ is open as well? Or perhaps it is non-open & non-closed? My intuition is that it ...Sun, 30 Aug 2026 01:06:35 -0400https://liberty.io/calculate-the-radius-of-the-cylinder-such-that-the-tsa-is-a-max.htmlhttps://liberty.io/calculate-the-radius-of-the-cylinder-such-that-the-tsa-is-a-max.htmlGiven: Volume 30m^2 Diameter 6x Height H Calculate the radius of the cylinder such that the Total Surface Area is a maximum? Can someone help me understand this question and point me in the correct ...Sun, 30 Aug 2026 01:02:21 -0400https://liberty.io/examples-of-bijective-map-from-mathbb-r-3-rightarrow-mathbb-r.htmlhttps://liberty.io/examples-of-bijective-map-from-mathbb-r-3-rightarrow-mathbb-r.htmlCould any one give an example of a bijective map from $\mathbb{R}^3\rightarrow \mathbb{R}$? Thank you....Sun, 30 Aug 2026 00:58:06 -0400https://liberty.io/proof-of-uniqueness-of-the-bounded-linear-transformation-extended-in-t.htmlhttps://liberty.io/proof-of-uniqueness-of-the-bounded-linear-transformation-extended-in-t.htmlB.L.T Theorem (from Reed/Simon): Suppose $T$ is a bounded linear transformation from a normed linear space $\langle V_1, \|\cdot\|\rangle$ to a complete normed linear space $\langle V_2, \|\cdot\|\...Sun, 30 Aug 2026 00:46:07 -0400https://liberty.io/can-this-boolean-expression-be-simplified-any-further.htmlhttps://liberty.io/can-this-boolean-expression-be-simplified-any-further.htmlI have simplified a Boolean expression to $$(\lnot a \land \lnot b \land \lnot c) \lor (a \land (b \lor c)).$$ Is there any way to simplify this further, e.g. using De Morgan's or anything?...Sun, 30 Aug 2026 00:43:07 -0400https://liberty.io/tangentplane-of-a-hyperboloid.htmlhttps://liberty.io/tangentplane-of-a-hyperboloid.htmlI have a hyperboloid $H$, which is $x^2+\frac{y^2}{4}-z^2=1$. I want to show that all points $P$ on $H$, in which the tangentplane to $H$ that goes through the point $(1,4,2)$, all are in a plane.......Sun, 30 Aug 2026 00:41:02 -0400https://liberty.io/what-is-the-general-equation-for-rotated-ellipsoid.htmlhttps://liberty.io/what-is-the-general-equation-for-rotated-ellipsoid.htmlI have general equation for ellipsoid not in center: $$ \frac{(x-x_0)^2}{a^2}+\frac{(y-y_0)^2}{b^2}+\frac{(z-z_0)^2}{c^2}=1.$$ What is the equation when it's rotated based on $\alpha$(over $x$ axi......Sun, 30 Aug 2026 00:39:07 -0400https://liberty.io/what-is-the-derivative-of-x-y-with-respect-to-y.htmlhttps://liberty.io/what-is-the-derivative-of-x-y-with-respect-to-y.htmlI'm looking for the derivative of $x^y$ with respect to $y$. I have done it by taking log of both sides, how do I do it if I try to write $\log e^{x^y} = x^y$?...Sun, 30 Aug 2026 00:36:59 -0400https://liberty.io/how-to-calculate-average-of-sine-squared.htmlhttps://liberty.io/how-to-calculate-average-of-sine-squared.htmlI have problem with calculating average of sine. In my book, there is claim, that $$\langle\sin^2\omega t\rangle=\frac12$$ Now I have a question how to get this result mathematically? I know you ......Sun, 30 Aug 2026 00:36:02 -0400https://liberty.io/does-every-mathematical-operation-has-an-inverse-operation.htmlhttps://liberty.io/does-every-mathematical-operation-has-an-inverse-operation.htmlFor example, we say that the addition and subtraction are inverse operations like that does each and every mathematical operation has an inverse operation?...Sun, 30 Aug 2026 00:30:19 -0400https://liberty.io/what-039-s-the-differece-between-cot-x-and-arctan-x-duplicate.htmlhttps://liberty.io/what-039-s-the-differece-between-cot-x-and-arctan-x-duplicate.htmlI know that $\displaystyle \cot(x)=\frac{1}{\tan(x)}$ and $\space \displaystyle \arctan(x)=\tan(x)^{-1}=\frac{1}{\tan(x)}$ What is the difference between these two function? Is $\cot(x)$ the reci......Sun, 30 Aug 2026 00:24:01 -0400https://liberty.io/solve-the-following-equation-sqrt-x-sqrt-4x-sqrt-16x-sqrt-64x-5-sqrt-x.htmlhttps://liberty.io/solve-the-following-equation-sqrt-x-sqrt-4x-sqrt-16x-sqrt-64x-5-sqrt-x.htmlA past examination paper had the following question that I found interesting. I tried having a go at it but haven't come around with any solutions. How would one go about tackling it? $$\sqrt {x +......Sun, 30 Aug 2026 00:23:52 -0400https://liberty.io/what-is-the-fastest-way-to-multiply-and-divide-polynomials.htmlhttps://liberty.io/what-is-the-fastest-way-to-multiply-and-divide-polynomials.htmlI want to get faster at multiplying and dividing polynomials. Is there a way I can do it faster? For multiplying I just do it normally. But for division I have to use the box method....Sun, 30 Aug 2026 00:19:53 -0400https://liberty.io/how-would-you-define-twice-as-many-when-0-is-concerned.htmlhttps://liberty.io/how-would-you-define-twice-as-many-when-0-is-concerned.htmlConsider the statement that "$\mathscr{L}$ is defined as a language where all strings contained in it contain twice as many b's as a's." Clearly a string containing a single a will yield ......Sun, 30 Aug 2026 00:12:16 -0400https://liberty.io/what-are-some-applications-of-linear-approximation-in-the-real-world.htmlhttps://liberty.io/what-are-some-applications-of-linear-approximation-in-the-real-world.htmlWhat are examples you can give to Calculus I high school students? Here is a link to show the level linear approximations will be taught. I've found some applications by a simple google search. M......Sun, 30 Aug 2026 00:11:15 -0400https://liberty.io/base-and-height-of-equilateral-triangle-that-are-closest-to-integers.htmlhttps://liberty.io/base-and-height-of-equilateral-triangle-that-are-closest-to-integers.htmlI know that formula for the height of an equilateral triangle is height = (1/2) * √3 * base, so the height and base can never both be integers. But how could I find values that are as close as poss......Sun, 30 Aug 2026 00:08:07 -0400https://liberty.io/solve-sin-4x-sin-x-closed.htmlhttps://liberty.io/solve-sin-4x-sin-x-closed.htmlSolve $\sin(4x) = \sin(x)$ for $0 < x < 180$ and solve $\tan(4x) = \tan(x)$ for $0 < x < 2 \pi$ I think you have to use the addition formulas however I keep on getting stuck on the wo......Sun, 30 Aug 2026 00:01:16 -0400https://liberty.io/velocity-definite-integral.htmlhttps://liberty.io/velocity-definite-integral.htmlI am not sure how to approach this problem. Problem Suppose a certain object moves in a straight line with the following velocity, where $v$ is in meters per second and $t$ is in seconds: $v(t) =......Sat, 29 Aug 2026 23:56:45 -0400https://liberty.io/help-me-understand-the-formal-logic-notation.htmlhttps://liberty.io/help-me-understand-the-formal-logic-notation.htmlI'm trying to describe the predicate/proposition n>1 in formal logic notation. Apparently the answer is ∃y. y=1 $\land$ n>y. If I try to read this, it reads: There exists a y. y is 1......Sat, 29 Aug 2026 23:47:53 -0400https://liberty.io/why-do-you-add-1-in-counting-test-questions.htmlhttps://liberty.io/why-do-you-add-1-in-counting-test-questions.htmlHere's an example question from the SAT question of the day: On the last day of a one-week sale, customers numbered 149 through 201 were waited on. How many customers were waited on that day? ...Sat, 29 Aug 2026 23:37:32 -0400https://liberty.io/how-to-calculate-the-volume-of-an-ellipsoid-with-triple-integral.htmlhttps://liberty.io/how-to-calculate-the-volume-of-an-ellipsoid-with-triple-integral.htmlI'm having some troubles since this morning on an exercise. I need to find the volume of the ellipsoid defined by $\frac{x^2}{a^2} + \frac{y^2}{a^2} + \frac{z^2}{a^2} \leq 1$. So at the beginning I ...Sat, 29 Aug 2026 23:26:51 -0400https://liberty.io/what-is-the-probability-of-picking-a-random-number-from-1-to-100-100-t.htmlhttps://liberty.io/what-is-the-probability-of-picking-a-random-number-from-1-to-100-100-t.htmlSay I pick a number from 1 to 100, and repeat this process 99 more times. What is the probability that no numbers will repeat? Is it correct to say the answer is 1/100! or is there more to it?...Sat, 29 Aug 2026 23:21:34 -0400https://liberty.io/check-my-proof-of-0-r-q-epsilon-real-rational-duplicate.htmlhttps://liberty.io/check-my-proof-of-0-r-q-epsilon-real-rational-duplicate.htmlI am working on this exercise: $$ \forall \varepsilon > 0, \ \exists q \in Q \text{ where } 0 < |r - q| < \varepsilon $$ To clarify, r is a real number, q is a rational number. This is ......Sat, 29 Aug 2026 23:19:24 -0400https://liberty.io/disproving-a-s-s-angle-side-side-congruence-condition.htmlhttps://liberty.io/disproving-a-s-s-angle-side-side-congruence-condition.htmlI know that $A-S-S$$(Angle-Side-Side)$ congruence does not exist.But I cannot disprove it. Every time I draw a figure,I get two congruent triangles. My Attempt- So,we draw two lines such that $A......Sat, 29 Aug 2026 23:18:20 -0400https://liberty.io/converting-between-two-frames-of-reference-given-two-points.htmlhttps://liberty.io/converting-between-two-frames-of-reference-given-two-points.htmlI have two points ($P_1$ & $P_2$) with their coordinates given in two different frames of reference ($A$ & $B$). Given these, what I'd like to do is derive the transformation to be able to ...Sat, 29 Aug 2026 23:17:24 -0400https://liberty.io/euclidean-norm-of-a-vector-with-respect-to-some-basis-other-than-the-s.htmlhttps://liberty.io/euclidean-norm-of-a-vector-with-respect-to-some-basis-other-than-the-s.htmlWe know that if $v=a_1 e_1+\cdots+a_n e_n$ where $(e_{i})$ is the standard basis on $\mathbb{R}^{n}$ that the Euclidean norm is given by $$\|v\|^2 = a_1^2 +\cdots+a_n^2.$$ Suppose that $v$ were wri......Sat, 29 Aug 2026 23:12:56 -0400https://liberty.io/what-039-s-the-difference-between-list-and-sequence-as-mathematical-co.htmlhttps://liberty.io/what-039-s-the-difference-between-list-and-sequence-as-mathematical-co.htmlWhat distinguishes between set, multiset and list is whether the order is important or not, and whether repetitions of elements is allowed: List: order is important and repetition is allowed Multi......Sat, 29 Aug 2026 23:12:39 -0400https://liberty.io/cosine-similarity-vs-angular-distance.htmlhttps://liberty.io/cosine-similarity-vs-angular-distance.htmlWhile checking Google's Universal sentence encoder paper, I found that they mention that using a similarity based on angular distance performs better than raw cosine similarity. More specifically......Sat, 29 Aug 2026 23:04:00 -0400https://liberty.io/property-of-subordinate-matrix-norm-ab-leq-a-b.htmlhttps://liberty.io/property-of-subordinate-matrix-norm-ab-leq-a-b.htmlI do not understand why the following property for Matrix subordinate norms holds: \begin{equation} \|AB\| \leq \|A\|\|B\| \end{equation} Please explain clearly as I know that it should be shown b......Sat, 29 Aug 2026 23:02:00 -0400https://liberty.io/when-to-use-congruent-vs-approximately.htmlhttps://liberty.io/when-to-use-congruent-vs-approximately.htmlWhat is the appropriate usage of the congruent ($\cong$) vs. the approximately ($\approx$) symbols?...Sat, 29 Aug 2026 23:01:15 -0400https://liberty.io/thales-properties-with-similar-triangles.htmlhttps://liberty.io/thales-properties-with-similar-triangles.htmlLet : $\Delta ABC$ is a triangle $0<\beta<1$ We put $M$ and $N$ two points such that $AM=\beta AB$ and $AN=\beta AC$ We want to prove that $(MN)$ parallel to $(BC)$ and $MN= \beta BC$. I m......Sat, 29 Aug 2026 22:48:30 -0400https://liberty.io/counterclockwise-rotation-matrix.htmlhttps://liberty.io/counterclockwise-rotation-matrix.htmlIf I take the basis $(\vec{e_x},\vec{e_y})$ and make a rotation counterclockwise of angle $\theta$, I end up with two new vectors $(\vec{u},\vec{v})$ such that : $\vec{u} = \cos\theta \vec{e_x} + ......Sat, 29 Aug 2026 22:43:53 -0400https://liberty.io/riemann-sums-finding-the-lower-sum.htmlhttps://liberty.io/riemann-sums-finding-the-lower-sum.htmljust doing some review and I am a bit confused on how I pick the upper vs lower sum of a Riemann sum. I get for the upper sum I choose the maximum value of $f$ on $[x_{k-1},x_k]$ and the lower sum ......Sat, 29 Aug 2026 22:36:54 -0400https://liberty.io/how-to-calculate-matrix-rotation.htmlhttps://liberty.io/how-to-calculate-matrix-rotation.htmlGiven the following rotation matrix $$\left[ \begin{matrix} -1/3 & 2/3 & -2/3 \\ 2/3 & -1/3 & -2/3 \\ -2/3 & -2/3 & -1/3 \\ \end{matrix} \right]$$ ......Sat, 29 Aug 2026 22:30:20 -0400https://liberty.io/how-to-compute-the-gradient-of-the-norm-for-linear-least-squares.htmlhttps://liberty.io/how-to-compute-the-gradient-of-the-norm-for-linear-least-squares.htmlI am reading about Solution of the Linear Least Squares Problem. Given the function $$f(\theta) = \frac{1}{2} \lVert y - \Phi \theta \rVert _{2}^{2}$$ To find the minimizer we need to compute the ...Sat, 29 Aug 2026 22:26:00 -0400https://liberty.io/prove-the-determinant-is-the-product-of-its-diagonal-entries.htmlhttps://liberty.io/prove-the-determinant-is-the-product-of-its-diagonal-entries.htmlProve that the determinant of an upper triangular matrix is the product of its diagonal entries. What I have so far: We will prove this by induction for an $n$ $\times$ $n$ matrix. For the case o......Sat, 29 Aug 2026 22:20:23 -0400https://liberty.io/show-that-unit-circle-is-compact.htmlhttps://liberty.io/show-that-unit-circle-is-compact.htmlQuick question. Say we are given the unit circle $\{ (x,y)\in \mathbb{R}^2: x^2+y^2=1 \}$. Is this set compact? How can I prove that this is closed? Bounded? Do I have to take the complement of th......Sat, 29 Aug 2026 22:16:39 -0400https://liberty.io/derive-formula-for-number-of-cables-in-full-mesh-network.htmlhttps://liberty.io/derive-formula-for-number-of-cables-in-full-mesh-network.htmlI am trying to determine how they derived number of cables needed in a full mesh network According to networking books it is $\dfrac{N * (N-1)}{2}$, where N is the number of nodes. I tried drawin......Sat, 29 Aug 2026 22:14:36 -0400https://liberty.io/what-on-earth-is-the-difference-between-calculus-and-analysis-duplicat.htmlhttps://liberty.io/what-on-earth-is-the-difference-between-calculus-and-analysis-duplicat.htmlI noticed that there isn't a word for Calculus in my native language, Dutch. So I just went to the English wikipedia entry on Calculus, and tried searching for the Dutch article, and as I suspected......Sat, 29 Aug 2026 22:09:55 -0400https://liberty.io/does-the-equation-x-3-9-0-have-any-solutions-in-mathbb-z-31-mathbb-z.htmlhttps://liberty.io/does-the-equation-x-3-9-0-have-any-solutions-in-mathbb-z-31-mathbb-z.htmlThis is my first time "solving an equation" in a group so I feel like that may be the source of my troubles. Here's what I have so far, although I'm not sure how to progress: Assume $c$ is a sol......Sat, 29 Aug 2026 22:01:24 -0400https://liberty.io/intersection-of-2-tangent-lines-on-a-circle.htmlhttps://liberty.io/intersection-of-2-tangent-lines-on-a-circle.htmlSay there are two lines that are tangent to a circle at points A and B, so that they intersect at an external point C, as shown below. Two congruent right triangles are formed, since the tangent l......Sat, 29 Aug 2026 21:59:31 -0400https://liberty.io/what-does-200-faster-mean-how-can-something-be-more-than-100-faster.htmlhttps://liberty.io/what-does-200-faster-mean-how-can-something-be-more-than-100-faster.htmlI'm a simple man living my every day life and have not much understanding of math or science. Today I read an article where someone claimed they can charge a battery $200\%$ faster. This got me th......Sat, 29 Aug 2026 21:58:54 -0400https://liberty.io/if-x-3-is-positive-then-which-one-of-the-following-must-be-positive.htmlhttps://liberty.io/if-x-3-is-positive-then-which-one-of-the-following-must-be-positive.htmlSee: Nova GRE Math Bible. Page-$235$. Problem#$09$ If $x+3$ is positive, then which one of the following must be positive? (A) $x-3$ (B) $(x-3)(x-4)$ (C) $(x-3)(x+3)$ (D) $(x-3)(x+4......Sat, 29 Aug 2026 21:56:26 -0400https://liberty.io/proving-that-there-exists-no-natural-number-between-2-consecutive-natu.htmlhttps://liberty.io/proving-that-there-exists-no-natural-number-between-2-consecutive-natu.htmlFrom an axiomatic description of $\Bbb R$ as given in (for example) Calculus by Apostol, which assumes the field axioms, order relations and the completeness axiom. The Definition of natural numbers ...Sat, 29 Aug 2026 21:48:41 -0400https://liberty.io/rearranging-formulas-involving-fractions.htmlhttps://liberty.io/rearranging-formulas-involving-fractions.htmlI'm currently studying some basic algebra (Not sure what the equivalent is for the US etc) in school currently. I'm having trouble with rearranging certain formulas. I attempted to try and solve ......Sat, 29 Aug 2026 21:31:24 -0400https://liberty.io/obtaining-the-frac-1-2-pi-factor-in-the-fourier-transform.htmlhttps://liberty.io/obtaining-the-frac-1-2-pi-factor-in-the-fourier-transform.htmlThis MathWorld page gives this definition of a Fourier transform: $$F(k) = \int_{-\infty}^{\infty} f(x) e^{-2\pi i k x}dx.$$ But, I wish to speak in terms of linear frequency $\nu$ and time $t$ ra......Sat, 29 Aug 2026 21:20:36 -0400https://liberty.io/centroid-of-a-region.htmlhttps://liberty.io/centroid-of-a-region.html$$y = x^3, x + y = 2, y = 0$$ I am suppose to find the centroid bounded by those curves. I have no idea how to do this, it isn't really explained well in my book and the places I have looked onlin......Sat, 29 Aug 2026 21:19:47 -0400https://liberty.io/proving-monotonic-sequence-theorem.htmlhttps://liberty.io/proving-monotonic-sequence-theorem.htmlA sequence $b_n$ is decreasing and bounded. Prove it it convergent. Proof: Since $b_n$ is bounded, $b_n > L$ where L is the greatest lower bound as per the completeness Axiom. Consider some ......Sat, 29 Aug 2026 21:06:08 -0400https://liberty.io/why-is-the-graph-of-x-y-2-and-y-sqrt-x-not-the-same.htmlhttps://liberty.io/why-is-the-graph-of-x-y-2-and-y-sqrt-x-not-the-same.htmlSo if you take $x=y^2$ and get the sqrt of both sides you get $y=\sqrt{x}$ so they are the same right? But when you graph them, $y=\sqrt{x}$ only shows the positive $y$ values because you can't sqr......Sat, 29 Aug 2026 21:05:44 -0400https://liberty.io/obscuring-squares-of-rubik-039-s-cube.htmlhttps://liberty.io/obscuring-squares-of-rubik-039-s-cube.htmlThis is a combinatorial question related to Rubik's cube $3\times3\times3$ (and, in the end, its generalizations $n\times n\times n$). I assume that the readers are familiar with this puzzle. Let's ...Sat, 29 Aug 2026 21:05:28 -0400https://liberty.io/writing-out-the-number-of-zeros-from-1-1-000-000.htmlhttps://liberty.io/writing-out-the-number-of-zeros-from-1-1-000-000.htmlIf you had to write out all the numbers between 1 and 1,000,000. How many zeros would you have to write in total. I was given that the answer is 488,895. Can any one show the clearest way possible......Sat, 29 Aug 2026 21:02:34 -0400https://liberty.io/in-set-theory-what-does-it-mean-for-a-variable-to-have-a-bar-symbol-ab.htmlhttps://liberty.io/in-set-theory-what-does-it-mean-for-a-variable-to-have-a-bar-symbol-ab.htmlPlease see the image below. What does the bar symbol mean in this context?...Sat, 29 Aug 2026 20:58:52 -0400https://liberty.io/how-to-simplify-abs-x-x.htmlhttps://liberty.io/how-to-simplify-abs-x-x.htmlI've been trying to find a way to simplify $\frac{|x|}{x}$ if $x$ is real and $\neq{0}$. The two possible outcomes to this are $\pm{1}$ but I believe there is one required answer. I've noticed that......Sat, 29 Aug 2026 20:57:22 -0400https://liberty.io/integrate-int-x-sin-2-x-dx.htmlhttps://liberty.io/integrate-int-x-sin-2-x-dx.htmlIntegrate $\int x\sin^2 (x) dx$ My attempt: $$=\int x\sin^2 (x) dx\\ =x^2\sin^2 (x) - \int 2\sin (x)\cos (x)x^2 dx\\ =x^2\sin^2 (x) - \int \sin (2x) x^2 dx.$$...Sat, 29 Aug 2026 20:54:02 -0400https://liberty.io/3-d-geometry-three-vertices-of-a-gm-abcd-is-3-1-2-1-2-4-1-1-2-find-the.htmlhttps://liberty.io/3-d-geometry-three-vertices-of-a-gm-abcd-is-3-1-2-1-2-4-1-1-2-find-the.htmlThe question is Three vertices of a parallelogram ABCD are A(3,-1,2), B(1,2,-4) and C(-1,1,2). Find the coordinate of the fourth vertex. To get the answer I tried the distance formula, equated ......Sat, 29 Aug 2026 20:45:25 -0400https://liberty.io/calculate-the-vector-normal-to-the-plane-by-given-points.htmlhttps://liberty.io/calculate-the-vector-normal-to-the-plane-by-given-points.htmlHow can one calculate the vector normal to the plane that is determined by given points? For example, given three points $P_1(5,0,0)$, $P_2(0,0,5)$ and $P_3(10,0,5)$, calculate the vector normal t......Sat, 29 Aug 2026 20:41:00 -0400https://liberty.io/prove-the-limit-is-sqrt-e.htmlhttps://liberty.io/prove-the-limit-is-sqrt-e.htmlHow do you show $$\lim\limits_{k \rightarrow \infty} \frac{\left(2+\frac{1}{k}\right)^k}{2^k}=\sqrt{e}$$ I know that $$\lim\limits_{k \to \infty} \left(1+\frac{1}{k}\right)^k=e$$ but I don't kno......Sat, 29 Aug 2026 20:39:06 -0400https://liberty.io/how-to-lower-upper-bound-n-using-1-x-leq-e-x.htmlhttps://liberty.io/how-to-lower-upper-bound-n-using-1-x-leq-e-x.htmlI need to prove for all positive integer $n$ $$ e\left(\frac{n}{e}\right)^n\leq n!\leq en\left(\frac{n}{e}\right)^n, $$ using the hint $1+x\leq e^x$ for all $x\in \mathbb{R}$. I did this: The h......Sat, 29 Aug 2026 20:27:06 -0400https://liberty.io/what-is-the-relationship-between-the-trigonometric-secant-and-the-geom.htmlhttps://liberty.io/what-is-the-relationship-between-the-trigonometric-secant-and-the-geom.htmlWhat is the difference between the geometric secant(the line that cuts two points of a curve) of a curve, and the trigonometric secant(=1/cosinex) ? If they are the same, can you explain how they a......Sat, 29 Aug 2026 20:22:10 -0400https://liberty.io/why-can-039-t-my-graphing-calculator-find-the-rref-of-the-transpose-of.htmlhttps://liberty.io/why-can-039-t-my-graphing-calculator-find-the-rref-of-the-transpose-of.htmlI know this is somewhat of an odd question, but I am having trouble with my TI-84 calculator and I don't know why. I'm trying to find the RREF of the transpose of a $4\times6$ matrix; for some reas......Sat, 29 Aug 2026 20:19:33 -0400https://liberty.io/one-to-one-correspondence-between-infinite-sets.htmlhttps://liberty.io/one-to-one-correspondence-between-infinite-sets.htmlWhat does it mean to have one-to-one correspondence between infinite sets. How would you solve them?...Sat, 29 Aug 2026 20:13:15 -0400https://liberty.io/area-enclosed-by-a-circle-and-leminscate.htmlhttps://liberty.io/area-enclosed-by-a-circle-and-leminscate.htmlFind the area enclosed by a circle $r=4\sin\theta$ and out of $r^2=8\cos 2\theta$ I have tried the following integral $\int_{\frac{\pi}{6}}^{\frac{\pi}{4}}\int_{\sqrt{8\cos2\theta}}^{4\sin\theta}d......Sat, 29 Aug 2026 20:09:18 -0400https://liberty.io/int-sin-6-x-cos-x-dx.htmlhttps://liberty.io/int-sin-6-x-cos-x-dx.html$$\int\sin^6(x)\cos(x)\,dx.$$ Apparently... $$\int\sin^6(x)\cos(x)\,dx= \int\sin^6(x)\,d\sin(x)x = \frac{1}{7}\sin^7(x)+c.$$ Can somebody explain the second equality? Shouldn't there be a $dx$ a......Sat, 29 Aug 2026 20:05:46 -0400https://liberty.io/rational-equations-what-are-excluded-values.htmlhttps://liberty.io/rational-equations-what-are-excluded-values.htmlI have an example rational equation in a textbook which I was able to solve: $$\dfrac{3}{x-6} = \dfrac{5}{x}$$ Least common denominator is : $x(x-6)$ so: $$\frac{x(x-6)3}{x-6} = \dfrac{x(x-6)5}{x}......Sat, 29 Aug 2026 20:00:37 -0400https://liberty.io/why-the-derivative-of-inverse-secant-has-an-absolute-value.htmlhttps://liberty.io/why-the-derivative-of-inverse-secant-has-an-absolute-value.html$y=\operatorname{arcsec}x$ can be defined in two ways. The first restricts the domain of $\sec y$ to $[0,\pi], y\neq\frac{\pi}{2}$. So the range of $y$ goes between $[0,\frac{\pi}{2})\cup(\frac{\pi......Sat, 29 Aug 2026 20:00:07 -0400https://liberty.io/difficult-conversion-from-polar-equation-to-rectangular-equation.htmlhttps://liberty.io/difficult-conversion-from-polar-equation-to-rectangular-equation.htmlHow do we convert this into rectangular equation? $r=5\theta$...Sat, 29 Aug 2026 19:58:21 -0400https://liberty.io/volume-of-a-tent-with-canvas-stretched-from-a-circular-base-to-a-verti.htmlhttps://liberty.io/volume-of-a-tent-with-canvas-stretched-from-a-circular-base-to-a-verti.htmlFrom George Simmons' Calculus With Analytical Geometry, page 229 question 7: A tent consists of canvas stretched from a circular base of radius $a$ to a vertical semicircular rod fastened to the ...Sat, 29 Aug 2026 19:34:17 -0400https://liberty.io/is-it-possible-to-change-the-pole-and-or-the-polar-axis-in-a-polar-coo.htmlhttps://liberty.io/is-it-possible-to-change-the-pole-and-or-the-polar-axis-in-a-polar-coo.htmlCiting Wikipedia's article on polar coordinates... In mathematics, the polar coordinate system is a two-dimensional coordinate system in which each point on a plane is determined by a distance f......Sat, 29 Aug 2026 19:31:42 -0400https://liberty.io/is-polynomial-in-general-the-same-as-polynomial-function.htmlhttps://liberty.io/is-polynomial-in-general-the-same-as-polynomial-function.htmlThe algebra text book says, a polynomial in one variable over $\mathbb{R}$ is given by, $$f(x)= a_n x^n + a_{n-1} x^{n-1} + \cdots + a_1x + a_0$$ Where $x$ is an unknown quantity which commutes ......Sat, 29 Aug 2026 19:27:23 -0400https://liberty.io/does-the-fact-that-a-17-i-2-imply-that-the-matrix-a-must-be-i-2.htmlhttps://liberty.io/does-the-fact-that-a-17-i-2-imply-that-the-matrix-a-must-be-i-2.htmlDoes the fact that $A^{17} = I_2$ imply that the matrix $A$ must be $I_2$? Since the question does not specify whether the entries of $A$ are allowed to be complex valued functions, I said that IF......Sat, 29 Aug 2026 19:23:25 -0400https://liberty.io/convert-vector-into-diagonal-matrix.htmlhttps://liberty.io/convert-vector-into-diagonal-matrix.htmlGiven a vector $[x_1,x_2,x_3, \dots, x_n]^T$, is it possible to obtain a diagonal matrix, $ \left[\begin{array}{c c c c c} x_1 & 0 & 0 & \dots & 0\\ 0 & x_2 & 0 & \dots......Sat, 29 Aug 2026 19:22:18 -0400https://liberty.io/c-alculating-flux-using-the-divergence-theorem-when-the-divergence-is-.htmlhttps://liberty.io/c-alculating-flux-using-the-divergence-theorem-when-the-divergence-is-.htmlI calculated the divergence of my vector field $\langle x^2 + y^2, y^2 + z^2, 1 − 2xz − 2yz\rangle$ to be $0$. The flux is meant to be over the unit hemisphere. If I do use the divergence theorem, ......Sat, 29 Aug 2026 19:10:45 -0400