https://pop.com.sgExplosive pop stories with broad entertainment appeal.en-usTue, 01 Sep 2026 05:28:08 -0400https://pop.com.sg/how-to-integrate-a-three-products.htmlhttps://pop.com.sg/how-to-integrate-a-three-products.htmlI tried to integrate $x e^x \sin x$, using integration by parts, and setting $\frac{\, \mathrm dv}{\, \mathrm dx} = e^x \sin x$. Even though I got really close, I kept getting it wrong. Can some......Tue, 01 Sep 2026 09:26:57 -0400https://pop.com.sg/odds-of-a-specific-dice-distribution-occuring-when-playing-catan-2d6.htmlhttps://pop.com.sg/odds-of-a-specific-dice-distribution-occuring-when-playing-catan-2d6.htmlSo this is messing with me, because I have no idea how I would even go about starting calculating this. I have played a game of Catan. For those that don't know, it's a board game that involves rol......Tue, 01 Sep 2026 09:22:45 -0400https://pop.com.sg/change-of-basis-matrix-exercise-find-the-basis-given-the-matrix.htmlhttps://pop.com.sg/change-of-basis-matrix-exercise-find-the-basis-given-the-matrix.htmlThe matrix of basis change from basis $B = \{b_1,b_2\}$ to the basis $C = \{(1,1),(0,2)\}$ is $$ M = \begin{bmatrix} 1 & 0 \\2&3\end{bmatrix}$$ Find basis $B$. Well, here's what I did: ...Tue, 01 Sep 2026 09:22:32 -0400https://pop.com.sg/how-to-find-indicial-equation.htmlhttps://pop.com.sg/how-to-find-indicial-equation.htmlHow can I find the indicial equation of $x(x-1)y''+3y'-2y=0$? I tried the method of Frobenius but I keep getting lost in the algebra. Is there any other way to get the indicial equation?...Tue, 01 Sep 2026 09:04:19 -0400https://pop.com.sg/how-to-recognise-a-problem-requiring-the-use-of-calculus.htmlhttps://pop.com.sg/how-to-recognise-a-problem-requiring-the-use-of-calculus.htmlI've taken a calculus class for a semester at university as a mandatory part of my curriculum. In this class, we were taught formal definitions of what are derivatives and integrals, how to calculate ...Tue, 01 Sep 2026 08:57:51 -0400https://pop.com.sg/is-the-hilbert-schmidt-norm-of-a-matrix-induced-from-a-vector-norm-dup.htmlhttps://pop.com.sg/is-the-hilbert-schmidt-norm-of-a-matrix-induced-from-a-vector-norm-dup.htmlA general way to get a matrix norm is inducing it from a vector norm. The question is, can the Hilbert-Schmidt norm obtained this way? As a norm general question, is every matrix norm obtained th......Tue, 01 Sep 2026 08:47:54 -0400https://pop.com.sg/what-textbook-is-being-used-in-these-lectures-linear-algebra.htmlhttps://pop.com.sg/what-textbook-is-being-used-in-these-lectures-linear-algebra.htmlI am learning Linear Algebra from these lectures by Prof. Adrian Banner (Princeton University) Does anyone know what textbook they are using? This is a link to the playlist on YouTube: https://www....Tue, 01 Sep 2026 08:39:20 -0400https://pop.com.sg/what-exactly-is-significand.htmlhttps://pop.com.sg/what-exactly-is-significand.htmlper wiki, The significand is part of a number in scientific notation or a floating-point number, consisting of its significant digits. in this example $$123.45 = 0.12345 × 10^3$$ which part is ......Tue, 01 Sep 2026 08:31:00 -0400https://pop.com.sg/is-it-propositional-function-or-simply-proposition.htmlhttps://pop.com.sg/is-it-propositional-function-or-simply-proposition.htmlI was going through the text "Discrete Mathematics and Its Application" by Kenneth H Rosen (5th Edition) where I came across the use of $P(n)$ in the mathematical induction chapter and felt ...Tue, 01 Sep 2026 08:30:59 -0400https://pop.com.sg/how-to-calculate-x-with-percent-of-x-y.htmlhttps://pop.com.sg/how-to-calculate-x-with-percent-of-x-y.htmlI am sorry if this is impossible to compute in one equatation. I am trying to compute the following: for example y = 574 x - ((16% of x) + y) = 0 the goal is to find out what is x so as to make it ...Tue, 01 Sep 2026 08:16:29 -0400https://pop.com.sg/why-aren-039-t-logarithms-defined-for-negative-x.htmlhttps://pop.com.sg/why-aren-039-t-logarithms-defined-for-negative-x.htmlGiven a logarithm is true, if and only if, $y = \log_b{x}$ and $b^y = x$ (and $x$ and $b$ are positive, and $b$ is not equal to $1$)[1], are true, why aren't logarithms defined for negative ...Tue, 01 Sep 2026 08:08:30 -0400https://pop.com.sg/what-is-the-double-bracket-notation-used-here.htmlhttps://pop.com.sg/what-is-the-double-bracket-notation-used-here.htmlIts kind of a bracket but I'm not sure what it means. I have two ideas about it: It means the number of times the expression in satisfied or it changes for $1$ or $0$ depending on the result every......Tue, 01 Sep 2026 07:56:57 -0400https://pop.com.sg/why-is-pi-2-so-close-to-10.htmlhttps://pop.com.sg/why-is-pi-2-so-close-to-10.htmlNoam Elkies explained why $\pi^2=9.8696...$ is so close to $10$ using an inequality on Euler's solution to Basel problem $$\frac{\pi^2}{6}=\sum_{k=0}^{\infty} \frac{1}{\left(k+1\right)^2}$$ to f......Tue, 01 Sep 2026 07:55:23 -0400https://pop.com.sg/how-can-i-find-the-maximum-velocity-if-i-039-ve-already-found-when-it-.htmlhttps://pop.com.sg/how-can-i-find-the-maximum-velocity-if-i-039-ve-already-found-when-it-.htmlI've been working on this problem for awhile and got the second half figured out, but I can't seem to get what the maximum actually is. Here's the question: A bullet is fired in the air verticall......Tue, 01 Sep 2026 07:53:22 -0400https://pop.com.sg/deriving-the-derivative-formula-for-arcsecant-correctly.htmlhttps://pop.com.sg/deriving-the-derivative-formula-for-arcsecant-correctly.htmlI have been trying to derive the derivative of the arcsecant function, but I can't quite get the right answer (the correct answer is the absolute value of what I get). I first get $\frac{d}{dy}\sec......Tue, 01 Sep 2026 07:50:44 -0400https://pop.com.sg/interpreting-divergence-of-velocity-field.htmlhttps://pop.com.sg/interpreting-divergence-of-velocity-field.htmlThe wikipedia article on divergence describes one interpretation of divergence: "The velocity of the air at each point defines a vector field. While air is heated in a region, it expands in all ...Tue, 01 Sep 2026 07:48:48 -0400https://pop.com.sg/martingale-and-submartingale.htmlhttps://pop.com.sg/martingale-and-submartingale.htmlWhen $X_n$ is a martingale we know $(X_n)^+$, i.e. the positive part of $X_n$, is also a submartingale. Is this true? It is easy to show that this is SUB-m.g. but I couldn't find a counter-example ......Tue, 01 Sep 2026 07:47:56 -0400https://pop.com.sg/can-the-fibonacci-sequence-be-written-as-an-explicit-rule.htmlhttps://pop.com.sg/can-the-fibonacci-sequence-be-written-as-an-explicit-rule.htmlWhen I learned sums and sequences in algebra II with trig I learned about recursive rules and explicit rules. A recursive rule written with the formula of: $$a_n = r * a_{n-1}$$ Or as: $$a_n = a_{......Tue, 01 Sep 2026 07:46:45 -0400https://pop.com.sg/proving-the-identity-sin3x-sin-x-2-sin2x-cos-x.htmlhttps://pop.com.sg/proving-the-identity-sin3x-sin-x-2-sin2x-cos-x.htmlI need some help proving this identity: $$\sin3x + \sin x = 2\sin2x\cos x$$ I don't know where to start. I thought about expanding $\sin 3x$ into $\sin (2x + x)$ but I don't think that does me ......Tue, 01 Sep 2026 07:45:02 -0400https://pop.com.sg/integration-of-frac-1-x-ln-x-by-parts.htmlhttps://pop.com.sg/integration-of-frac-1-x-ln-x-by-parts.htmlI tried to solve $\int \frac{dx}{x\ln(x)}$ by parts. I have done it as usual: $$\int \frac{dx}{x\ln(x)}=\begin{bmatrix} f(x)=\frac{1}{\ln x} & g'(x)=\frac{1}{x}\\ f'(x)=-\frac{1}{x\ln^2x} &amp......Tue, 01 Sep 2026 07:34:19 -0400https://pop.com.sg/expressing-the-trace-of-a-2-by-trace-of-a.htmlhttps://pop.com.sg/expressing-the-trace-of-a-2-by-trace-of-a.htmlLet $A$ be a a square matrix. Is it possible to express $\operatorname{trace}(A^2)$ by means of $\operatorname{trace}(A)$ ? or at least something close?...Tue, 01 Sep 2026 07:22:56 -0400https://pop.com.sg/what-is-mutually-disjoint-sets.htmlhttps://pop.com.sg/what-is-mutually-disjoint-sets.htmlWhat is mutually disjoint sets? I know it has something to do with subsets but I don't know for sure....Tue, 01 Sep 2026 07:00:58 -0400https://pop.com.sg/a-small-doubt-about-the-dominated-convergence-theorem.htmlhttps://pop.com.sg/a-small-doubt-about-the-dominated-convergence-theorem.htmlTheorem $\mathbf{A.2.11}$ (Dominated convergence). Let $f_n : X \to \mathbb R$ be a sequence of measurable functions and assume that there exists some integrable function $g : X \to \mathbb R$ such......Tue, 01 Sep 2026 07:00:53 -0400https://pop.com.sg/find-first-term-and-common-difference-of-arithmetic-sequence-given-two.htmlhttps://pop.com.sg/find-first-term-and-common-difference-of-arithmetic-sequence-given-two.htmlThe 7th and 11th terms of an arithmetic sequence are 7b + 5c and 11b + 9c respectively. Given these i want to find the first term (term when n = 0) and the common difference. I tried a lot of ...Tue, 01 Sep 2026 06:58:33 -0400https://pop.com.sg/difference-between-matrix-mapping-and-matrix-multiplication.htmlhttps://pop.com.sg/difference-between-matrix-mapping-and-matrix-multiplication.htmlMy textbook says that a matrix mapping is a function f: $\mathbb{R}^n\to\mathbb{R}^m$ such that $f(\vec{x}) = A\vec{x}$ where $A$ is an $m \times n$ matrix So how is a result of a matrix mapping ...Tue, 01 Sep 2026 06:54:56 -0400https://pop.com.sg/how-to-solve-a-triangle-knowing-just-two-legs.htmlhttps://pop.com.sg/how-to-solve-a-triangle-knowing-just-two-legs.htmlGiven a right triangle, if I know two of its legs are $a = 3,5$ and $b = 5,5$, is it possible to solve the triangle with this info? This is, can I determine all of its sides and all of its angles w......Tue, 01 Sep 2026 06:54:49 -0400https://pop.com.sg/what-is-a-negative-number.htmlhttps://pop.com.sg/what-is-a-negative-number.htmlI'm trying to get to an abstract definition of a negative number that would fit in with the basic concept of addition/subtraction. There are questions here about multiplying and dividing negative ...Tue, 01 Sep 2026 06:52:35 -0400https://pop.com.sg/a-cube-abcdefgh-is-given-determine-the-locus-of-all-midpoints-of-segme.htmlhttps://pop.com.sg/a-cube-abcdefgh-is-given-determine-the-locus-of-all-midpoints-of-segme.htmlQuestion is:A cube $ABCDEFGH$ is given. Determine the locus of all midpoints of segments $MN$, where $M$ is any point on segment $AC$ and $N$ any point on segment $FH$. My understanding : Here $M$ ......Tue, 01 Sep 2026 06:44:04 -0400https://pop.com.sg/proof-by-induction-of-bernoulli-039-s-inequality-1-x-n-geq-1-nx.htmlhttps://pop.com.sg/proof-by-induction-of-bernoulli-039-s-inequality-1-x-n-geq-1-nx.htmlI'm asked to used induction to prove Bernoulli's Inequality: If $1+x>0$, then $(1+x)^n\geq 1+nx$ for all $n\in\mathbb{N}$. This what I have so far: Let $n=1$. Then $1+x\geq 1+x$. This is true. ......Tue, 01 Sep 2026 06:42:57 -0400https://pop.com.sg/calculate-angles-and-absolute-value-of-complex-number-matlab.htmlhttps://pop.com.sg/calculate-angles-and-absolute-value-of-complex-number-matlab.htmlhow to Calculate the angles and absolute value of complex number by using matlab commands: the complex number is:$$e^{3+4j}+e^{3-4j}$$ tyy!!...Tue, 01 Sep 2026 06:40:23 -0400https://pop.com.sg/moment-of-inertia-of-ellipse-around-x-axis.htmlhttps://pop.com.sg/moment-of-inertia-of-ellipse-around-x-axis.htmlHow can I calculate the moment of inertia of an ellipse ($\frac{x^2}{a^2}+\frac{y^2}{b^2}=1$) using polar coordinates and the formula: $$I_x=\int_\Omega y^2dx dy$$ I know that the result is suppos......Tue, 01 Sep 2026 06:31:55 -0400https://pop.com.sg/find-the-exact-value-for-cos-alpha-beta-given-the-following-informatio.htmlhttps://pop.com.sg/find-the-exact-value-for-cos-alpha-beta-given-the-following-informatio.htmlSuppose $\alpha$ is interval $$\pi/2 \leq \alpha \leq \pi$$ and $$ \cos(\alpha) = - 1/3 $$ and $\beta$ is in the interval $$0 \leq \beta \leq \pi/2$$ and $$ \sin\beta = 2/5. $$ Use the facts abov......Tue, 01 Sep 2026 06:27:13 -0400https://pop.com.sg/coding-the-power-method-in-python.htmlhttps://pop.com.sg/coding-the-power-method-in-python.htmlI am a very beginner to coding and am trying to write a code for the power method/von Mises iteration for finding the largest eigenvalue (in magnitude) of a given matrix in Python. I'm referencing ...Tue, 01 Sep 2026 06:18:58 -0400https://pop.com.sg/volume-of-ellipsoid-using-triple-integrals.htmlhttps://pop.com.sg/volume-of-ellipsoid-using-triple-integrals.htmlGiven the general equation of the ellipsoid $\frac{x^2}{a^2} + \frac{y^2}{b^2} + \frac{z^2}{c^2} =1$, I am supposed to use a 3D Jacobian to prove that the volume of the ellipsoid is $\frac{4}{3}\pi......Tue, 01 Sep 2026 06:17:59 -0400https://pop.com.sg/integral-of-tan-x.htmlhttps://pop.com.sg/integral-of-tan-x.htmlSo I wanted to find a way to work out any trigonometric integral by simply knowing the integral of the two main trigonometric functions; $\sin(x)$ and $\cos(x)$. This is since any trigonometric fun......Tue, 01 Sep 2026 06:12:29 -0400https://pop.com.sg/why-is-a-circle-not-a-convex-set.htmlhttps://pop.com.sg/why-is-a-circle-not-a-convex-set.htmlWhy is a circle not a convex set? My attempt: I was googling it, but I didn't find the correct answer, as I didn't understand the explanation why a circle is not a convex set as defined here: As ......Tue, 01 Sep 2026 06:10:27 -0400https://pop.com.sg/how-to-tell-the-difference-between-a-parabola-and-a-hyperbola-by-looki.htmlhttps://pop.com.sg/how-to-tell-the-difference-between-a-parabola-and-a-hyperbola-by-looki.htmlIf you are given a curve and you are told that it is either a parabola or a hyperbola, how can you tell definitively which one it is? No coordinates provided. Geometrical method preferred, i.e. v......Tue, 01 Sep 2026 06:07:23 -0400https://pop.com.sg/calculating-centre-of-rotation-given-point-coordinates-at-different-po.htmlhttps://pop.com.sg/calculating-centre-of-rotation-given-point-coordinates-at-different-po.htmlI am trying to work out if the centre of rotation of a measured sphere is actually at 0,0 or slightly offset from the centre. The situation is as follows: I have a machine tool with a table that r......Tue, 01 Sep 2026 05:49:00 -0400https://pop.com.sg/why-is-gradient-noise-better-quality-than-value-noise.htmlhttps://pop.com.sg/why-is-gradient-noise-better-quality-than-value-noise.htmlI have been reading about the mathematics behind Perlin noise, a gradient noise function often used in computer graphics, from Ken Perlin's presentation and Matt Zucker's FAQ. I understand that each ...Tue, 01 Sep 2026 05:43:29 -0400https://pop.com.sg/how-to-solve-int-tan-2-x-sec-x-dx-duplicate.htmlhttps://pop.com.sg/how-to-solve-int-tan-2-x-sec-x-dx-duplicate.htmlHow should one solve the following integral? $$\int \tan^2(x) \sec (x) \ dx$$ I can't think of any substitutions to be made involving $\tan^2(x)=\sec^2 (x)-1$ or $\sec^2(x)=\tan^2(x)+1$, which is ......Tue, 01 Sep 2026 05:27:50 -0400https://pop.com.sg/how-do-i-find-the-original-function-given-the-derivative-closed.htmlhttps://pop.com.sg/how-do-i-find-the-original-function-given-the-derivative-closed.htmlI have no idea how to start these problems. How does one get the original function given the derivative of the function? I would appreciate it if someone would guide me in doing problem 7 at least ......Tue, 01 Sep 2026 05:24:26 -0400https://pop.com.sg/what-is-the-brachistochrone-curve-with-an-initial-velocity.htmlhttps://pop.com.sg/what-is-the-brachistochrone-curve-with-an-initial-velocity.htmlI'm basically trying to solve the opposite of the brachistochrone problem. You know how the brachistochrone is the shortest path between two points A and B, falling under the influence of gravity?......Tue, 01 Sep 2026 05:16:38 -0400https://pop.com.sg/right-group-action.htmlhttps://pop.com.sg/right-group-action.htmlWikipedia says 'The difference between left and right actions is in the order in which a product like $gh$ acts on $x$. For a left action $h$ acts first and is followed by $g$, while for a right a......Tue, 01 Sep 2026 05:12:15 -0400https://pop.com.sg/matrix-is-conjugate-to-its-own-transpose.htmlhttps://pop.com.sg/matrix-is-conjugate-to-its-own-transpose.htmlMariano mentioned somewhere that everyone should prove once in their life that every matrix is conjugate to its transpose. I spent quite a bit of time on it now, and still could not prove it. At the ...Tue, 01 Sep 2026 05:11:57 -0400https://pop.com.sg/let-f-x-x-3-x-2-3x-1-and-h-x-frac-f-x-g-x-where-h-x-is-a-rational-func.htmlhttps://pop.com.sg/let-f-x-x-3-x-2-3x-1-and-h-x-frac-f-x-g-x-where-h-x-is-a-rational-func.htmlLet $f(x) = x^3-x^2–3x–1$ and $h(x)=\dfrac{f(x)}{g(x)}$ where $h(x)$ is a rational function such that (a) it is continuous every where except when $x=– 1$ (b) $\lim_{x\to\infty}=\infty$ (c) $\li......Tue, 01 Sep 2026 05:03:33 -0400https://pop.com.sg/irreducible-dual-representation.htmlhttps://pop.com.sg/irreducible-dual-representation.htmlFor a semisimple Lie Algebra $\mathfrak{g}$ with Cartan Subalgebra $\mathfrak{t}$, let $V(\lambda)$ be the unique irreducible highest weight module with highest weight $\lambda$. I am asked to sho......Tue, 01 Sep 2026 05:02:25 -0400https://pop.com.sg/find-the-angle-of-a-tangent-line-between-two-circles.htmlhttps://pop.com.sg/find-the-angle-of-a-tangent-line-between-two-circles.htmlI have two circles noted as $A$ and $B$. Each of these circles has known positions $\vec{P_A}$ and $\vec{P_B}$ with radii $r_A$ and $r_B$. I need to find the angle ($\theta_A$ or $\theta_B$) shown ......Tue, 01 Sep 2026 04:49:03 -0400https://pop.com.sg/rectangular-parallelepiped-of-greatest-volume-for-a-given-surface-area.htmlhttps://pop.com.sg/rectangular-parallelepiped-of-greatest-volume-for-a-given-surface-area.htmlI am trying to find the rectangular parallelepiped of greatest volume for a given surface area S using Lagrange's method. I tried solving by myself but at x=y=z = a, I am not getting maximum volu......Tue, 01 Sep 2026 04:46:10 -0400https://pop.com.sg/why-is-this-not-a-space-filling-curve.htmlhttps://pop.com.sg/why-is-this-not-a-space-filling-curve.htmlFrom Wikipedia, a space-filling curve is a curve (i.e. a continuous function whose domain is the unit interval $[0,1]$) whose range contains the entire 2-dimensional unit square. Many examples of ......Tue, 01 Sep 2026 04:42:27 -0400https://pop.com.sg/series-expansion-of-square-root-function.htmlhttps://pop.com.sg/series-expansion-of-square-root-function.htmlCould I please know how to expand: $$\sqrt{4-3x^2}$$ I simplified it to $\sqrt{1-\frac34x^2}$ but the question is if I let $x = x^2$, what will the coefficient of, say $x^5$ or $x^{10}$ be in the ...Tue, 01 Sep 2026 04:38:04 -0400https://pop.com.sg/roots-of-unity.htmlhttps://pop.com.sg/roots-of-unity.htmlThe $n$th roots of unity are the complex numbers: $1,w,w^2,...,w^{n-1}$, where $w = e^{\frac{2\pi i} {n}}$. If $n$ is even: The $n$th roots are plus-minus paired, $w^{\frac{n}{2}+j} = -w^j$. Squa......Tue, 01 Sep 2026 04:18:13 -0400https://pop.com.sg/how-to-prove-that-this-simple-graph-does-not-exist.htmlhttps://pop.com.sg/how-to-prove-that-this-simple-graph-does-not-exist.htmlShow that there is no simple graph with 6 vertices in which there are 5 vertices having following degrees 5,5,3,2,1 So, we have 5 vertices (=odd number of vertices) with an even number of degree......Tue, 01 Sep 2026 04:10:36 -0400https://pop.com.sg/finding-a-general-term-for-the-sequence.htmlhttps://pop.com.sg/finding-a-general-term-for-the-sequence.htmlFind a general term in simplest form for the sequence: $$2, 1, -4, 7, -10, 13, -16$$ This is what I tried: $a_n = a_1 + (n - 1)\cdot d$ $a_n = 2 + (n-1)\cdot 3$ $a_n = (-1)^n (2+(n-1)\cdot 3)$ which ...Tue, 01 Sep 2026 04:02:03 -0400https://pop.com.sg/need-help-in-understanding-how-to-find-an-elementary-matrix.htmlhttps://pop.com.sg/need-help-in-understanding-how-to-find-an-elementary-matrix.htmlI read this chapter in my book and thought I understood it, but I don't. I tried working a problem to test my understanding and I just don't know how to get started. Given the following matrices:......Tue, 01 Sep 2026 03:57:37 -0400https://pop.com.sg/saw-horses-with-legs-26-in-height.htmlhttps://pop.com.sg/saw-horses-with-legs-26-in-height.htmlI am building sawhorses with each end cut at a $14^{\circ}$ angle, where one end attaches to the underside of the sawhorse and the other $14^{\circ}$ angle hits the flat floor. My question: if I n......Tue, 01 Sep 2026 03:50:35 -0400https://pop.com.sg/user-ikshu-neithalath-mathematics-stack-exchange.htmlhttps://pop.com.sg/user-ikshu-neithalath-mathematics-stack-exchange.htmlQ&A for people studying math at any level and professionals in related fields...Tue, 01 Sep 2026 03:26:27 -0400https://pop.com.sg/range-of-sin-x-6-cos-x-6.htmlhttps://pop.com.sg/range-of-sin-x-6-cos-x-6.htmlI need to find the range of the function $y = (\sin x)^6+ (\cos x)^6$ I did find the answer but working in a crude way rather than a methodical step by step approach. I give below the steps I used , ...Tue, 01 Sep 2026 03:07:14 -0400https://pop.com.sg/on-the-horizontal-and-slanted-asymptotes-of-rational-functions.htmlhttps://pop.com.sg/on-the-horizontal-and-slanted-asymptotes-of-rational-functions.htmlI am really confused about the horizontal and slant asymptotes of a function. My textbook says that given a rational function: \begin{equation} y=f(x)=\frac{a_{n}x^{n}+a_{n-1}x^{n-1}+\cdot\cdot\cd......Tue, 01 Sep 2026 03:07:06 -0400https://pop.com.sg/what-does-the-derivative-of-a-function-at-a-point-describe-duplicate.htmlhttps://pop.com.sg/what-does-the-derivative-of-a-function-at-a-point-describe-duplicate.htmlI understand that the derivative of a function $f$ at a point $x=x_{0}$ is defined as the limit $$f'(x_{0})=\lim_{\Delta x\rightarrow 0}\frac{f(x_{0}+\Delta x)-f(x_{0})}{\Delta x}$$ where $\Delta x......Tue, 01 Sep 2026 03:06:45 -0400https://pop.com.sg/how-does-one-derive-the-centroid-formula-for-multiple-shapes.htmlhttps://pop.com.sg/how-does-one-derive-the-centroid-formula-for-multiple-shapes.htmlI know how to derive the formula for the centroid of n sets of finite points, each with $m_k$ points and centroid $C_k$. The formula is: $$C=\frac{\sum_{k=1}^{n}C_km_k}{\sum_{k=1}^{n}m_k}$$ However......Tue, 01 Sep 2026 02:57:38 -0400https://pop.com.sg/find-the-sum-or-value-of-following-expression-closed.htmlhttps://pop.com.sg/find-the-sum-or-value-of-following-expression-closed.htmlIn the problem I am not able to derive difference term. I know for solving summation we have to make difference term. How to proceed with this problem? $$\frac{{\displaystyle \sum_{n=1}^{99}} \s......Tue, 01 Sep 2026 02:48:57 -0400https://pop.com.sg/how-to-obtain-more-than-two-abcissas-for-one-ordinate.htmlhttps://pop.com.sg/how-to-obtain-more-than-two-abcissas-for-one-ordinate.htmlI'm reading a book on trigonometry and the author shown that some of the examples contain one abcissa for one ordinate and he showed that a parabola has two abscissas for every positive ordinates, ......Tue, 01 Sep 2026 02:48:20 -0400https://pop.com.sg/inverse-function-of-tanh-x.htmlhttps://pop.com.sg/inverse-function-of-tanh-x.htmlI have a problem while calculating inverse function of tanh(x). I know it is y = sinh(x)/cosh(x) and then I should express x, but I am stuck with that. Will you help me with this? thx a lot...Tue, 01 Sep 2026 02:34:16 -0400https://pop.com.sg/how-to-factor-4x-2-2x-1.htmlhttps://pop.com.sg/how-to-factor-4x-2-2x-1.htmlI want to know how to factor $4x^2 + 2x + 1$? I found the roots using quadratic equation and got $-1 + \sqrt{-3}$ and $-1 - \sqrt{-3}$, so I thought the factors would be $(x - (-1 + \sqrt{-3}))$ an......Tue, 01 Sep 2026 02:28:43 -0400https://pop.com.sg/why-does-the-function-r-theta-graph-a-spiral.htmlhttps://pop.com.sg/why-does-the-function-r-theta-graph-a-spiral.htmlIf $\theta$ denotes an angle (in radians I assume), and $r$ means the distance from the origin, then why does $r = \theta$ make a perfect spiral? I'm not that advanced in math (only in geometry and ...Tue, 01 Sep 2026 02:23:55 -0400https://pop.com.sg/calculus-finding-the-minimum-vertical-distance-between-graphs.htmlhttps://pop.com.sg/calculus-finding-the-minimum-vertical-distance-between-graphs.htmlQuestion:Find the minimum vertical distance between the graphs of $2+\sin x$ and $\cos x$? In order to find out the required distance, what should I do? It seems that there is a problem if I ...Tue, 01 Sep 2026 02:21:06 -0400https://pop.com.sg/how-to-calculate-the-number-of-pieces-in-the-border-of-a-puzzle.htmlhttps://pop.com.sg/how-to-calculate-the-number-of-pieces-in-the-border-of-a-puzzle.htmlIs there any way to calculate how many border-pieces a puzzle has, without knowing its width-height ratio? I guess it's not even possible, but I am trying to be sure about it. Thanks for your help......Tue, 01 Sep 2026 02:14:33 -0400https://pop.com.sg/linear-function.htmlhttps://pop.com.sg/linear-function.htmlI don't get this, need some help, examples and information The linear function $f$ is given by $$f(x) = 3x - 2 ,\quad -2 \leq x \leq 4.$$ Enter the independent variable and the dependent ...Tue, 01 Sep 2026 02:09:19 -0400https://pop.com.sg/lagrange-remainder-is-t-contained-within-an-open-or-closed-interval.htmlhttps://pop.com.sg/lagrange-remainder-is-t-contained-within-an-open-or-closed-interval.htmlAbove is the taylor series centred at a including the lagrange remainder. Can t be equal to a or b?...Tue, 01 Sep 2026 01:45:39 -0400https://pop.com.sg/expected-value-of-an-expected-value.htmlhttps://pop.com.sg/expected-value-of-an-expected-value.htmlI am looking at a proof that $\text{Var}(X)= E((X - EX)^2) = E(X^2) - (E(X))^2$ $E((X - EX)^2) =$ $E(X^2 - 2XE(X) + (E(X))^2) =$ $E(X^2) - 2E(X)E(X) + (E(X))^2)$ I can't see how the second line......Tue, 01 Sep 2026 01:41:55 -0400https://pop.com.sg/how-can-i-get-the-fourier-transform-of-this-generalized-function.htmlhttps://pop.com.sg/how-can-i-get-the-fourier-transform-of-this-generalized-function.htmlwhen function f(x) of the generalized function is $f(x) = x+2$, the fourier transform of the generalized function will be like ($F$ is the Fourier transform) $$\int_{-∞}^{∞}(x+2)(F[k(x)])dx$$ whe......Tue, 01 Sep 2026 01:40:15 -0400https://pop.com.sg/understanding-all-sub-sigma-algebras-of-a-sigma-algebra.htmlhttps://pop.com.sg/understanding-all-sub-sigma-algebras-of-a-sigma-algebra.htmlSuppose I have a set $\Omega$, and I partition $\Omega$ into $4$ sets $A_{1}, A_{2}, A_{3}, A_{4}$. Clearly, the $A_{i}$'s are pairwise disjoint and their union equals $\Omega$. Now, it's also cl......Tue, 01 Sep 2026 01:10:30 -0400https://pop.com.sg/finding-global-max-min.htmlhttps://pop.com.sg/finding-global-max-min.htmlmy task is to figure out the critical points of $f(x,y)=e^y(x^4-x^2+y)$, $\ $$\mathbb{R}^2 \rightarrow \mathbb{R}$, and show which of them is a maximum or minimum. As far as I got, I've shown that ......Tue, 01 Sep 2026 01:09:03 -0400https://pop.com.sg/how-to-compute-the-limit-of-a-rational-function-at-infinity.htmlhttps://pop.com.sg/how-to-compute-the-limit-of-a-rational-function-at-infinity.htmlI am unable to compute the limit, anyone can help ? I only understand some basic knowledge of limit ....Tue, 01 Sep 2026 01:07:05 -0400https://pop.com.sg/units-for-spring-constant-closed.htmlhttps://pop.com.sg/units-for-spring-constant-closed.htmlI have the question: A baby of mass $9.0\;$kg bounces with a period of $1.2\;$s in a baby bouncer. What is the spring constant of the bouncer? I rearranged the equation $T = 2\pi$ $\sqrt{\frac{......Tue, 01 Sep 2026 01:04:36 -0400https://pop.com.sg/name-and-layperson-039-s-explanation-for-an-e8-group-diagram.htmlhttps://pop.com.sg/name-and-layperson-039-s-explanation-for-an-e8-group-diagram.htmlI am taking a risk here, but hoping it will not ignite wrath in the reader. In trying to get an intuition of Lie theory this diagram is all but impossible to ignore: Unfortunately, there are many ...Tue, 01 Sep 2026 01:02:13 -0400https://pop.com.sg/area-of-a-rectangle-by-its-perimeter.htmlhttps://pop.com.sg/area-of-a-rectangle-by-its-perimeter.htmlIf I have a rectangle, with its perimiter 100m how do I show that the area A is given by the quadratic function A = 50x - $x^2$ Where x is the length of one side...Tue, 01 Sep 2026 01:01:46 -0400https://pop.com.sg/definition-of-a-region.htmlhttps://pop.com.sg/definition-of-a-region.htmlDefinition given - Region: Open set with none, some, or all of its boundary points. This seems quite unimportant... and it seems that almost all sets are regions (I can only think of regions, I ......Tue, 01 Sep 2026 00:38:58 -0400https://pop.com.sg/evaluating-frac-1-log-2-100-cdots-frac-1-log-100-100.htmlhttps://pop.com.sg/evaluating-frac-1-log-2-100-cdots-frac-1-log-100-100.htmlFirstly, I am completely aware that logarithms can be rewritten in exponential form. So, I do know at least something about logs. :) However, given the task to simplify and rewrite the expression,......Tue, 01 Sep 2026 00:38:29 -0400https://pop.com.sg/study-materials-to-help-understand-the-generalized-stokes-039-theorem-.htmlhttps://pop.com.sg/study-materials-to-help-understand-the-generalized-stokes-039-theorem-.htmlDear MSE: My goal is to understand the generalized Stokes' theorem both intuitively and rigorously. Could someone give advice or recommend study materials to help understand the generalized Stokes' ...Tue, 01 Sep 2026 00:27:09 -0400https://pop.com.sg/why-is-a-sphere-in-an-n-dimensional-space-called-n-1-sphere.htmlhttps://pop.com.sg/why-is-a-sphere-in-an-n-dimensional-space-called-n-1-sphere.htmlWhy is a sphere in an $n $-dimensional space called $(n-1) $-sphere? Isn't it natural to call a sphere in 3D a 3-sphere, a sphere in 2D (i.e. a circle) a 2-sphere, etc?...Tue, 01 Sep 2026 00:04:31 -0400https://pop.com.sg/what-is-the-concept-of-region-of-convergence-of-z-transform.htmlhttps://pop.com.sg/what-is-the-concept-of-region-of-convergence-of-z-transform.htmlI have been trying to learn Z transform. But I haven't found any good source that will clear my concept about the region of convergence. I have found some keywords like unit circle, but I don't hav......Mon, 31 Aug 2026 23:58:16 -0400https://pop.com.sg/calculating-the-oblique-ness-of-a-line-that-intersects-two-non-paralle.htmlhttps://pop.com.sg/calculating-the-oblique-ness-of-a-line-that-intersects-two-non-paralle.htmlI have a 3d space containing two planes that may have any non-orthogonal relative orientation. In addition I have a line the intersects both planes. I am trying to compute an index of the "oblique-......Mon, 31 Aug 2026 23:49:08 -0400https://pop.com.sg/relation-between-image-and-pre-image-of-a-function.htmlhttps://pop.com.sg/relation-between-image-and-pre-image-of-a-function.htmlWe know that given some function, each element of its domain has only one image in its codomain. But is it possible for an image of some element to have two preimages? For example, the function $f......Mon, 31 Aug 2026 23:45:34 -0400https://pop.com.sg/calculate-periodicity-of-f-x-cos-sin-x.htmlhttps://pop.com.sg/calculate-periodicity-of-f-x-cos-sin-x.htmlGiven the function: $$f(x) = \cos(\sin(x))$$ Calculate the periodicity of $f(x)$. I understand it equals $\pi$ but how would a solid answer look like? Thank you....Mon, 31 Aug 2026 23:36:00 -0400https://pop.com.sg/how-to-solve-a-quartic-equation.htmlhttps://pop.com.sg/how-to-solve-a-quartic-equation.htmlCould someone please explain how to solve this : $x^4 - 10x^3 + 21x^2 + 40x - 100 = 0$ - not the answer only, but a step-by-step solution. I tried to solve it, with the help of khanacademy, but st......Mon, 31 Aug 2026 23:33:02 -0400https://pop.com.sg/how-is-the-phrase-n-bit-number-related-to-the-big-o-notation.htmlhttps://pop.com.sg/how-is-the-phrase-n-bit-number-related-to-the-big-o-notation.htmlWhen it comes to algorithms, you frequently have to evaluate problems like this: Let $x$ be an $n$-bit integer. For each of the following questions, give your answer as a function of $n$. Or a ...Mon, 31 Aug 2026 23:30:41 -0400https://pop.com.sg/proving-trig-identities-complex-numbers.htmlhttps://pop.com.sg/proving-trig-identities-complex-numbers.htmlQuestion: Prove that if $z = \cos (\theta) + i\sin(\theta)$, then $$ z^n + {1\over z^n} = 2\cos(n\theta) $$ Hence prove that $\cos^6(\theta)$ $=$ $\frac 1{32}$$(\cos(6\theta)$ + ......Mon, 31 Aug 2026 23:23:57 -0400https://pop.com.sg/what-is-the-differentiation-operator.htmlhttps://pop.com.sg/what-is-the-differentiation-operator.htmlOn Wikipedia the Differential Operator is described as an operator defined as a function of the differentiation operator. The link that underlies the words "differentiation operator" in fact gives......Mon, 31 Aug 2026 23:21:59 -0400https://pop.com.sg/proving-that-n-over-2-n-converges-to-0-duplicate.htmlhttps://pop.com.sg/proving-that-n-over-2-n-converges-to-0-duplicate.htmlI'm completely clueless on this one. I can easily calculate the limit using L'Hopital's rule, but proving that the series is converging to 0 is far more tricky. $$a_n= {n \over 2^n}$$ Any help?...Mon, 31 Aug 2026 23:15:25 -0400https://pop.com.sg/checking-whether-a-graph-is-planar.htmlhttps://pop.com.sg/checking-whether-a-graph-is-planar.htmlI have to check whether a graph is planar. The given type is $$ e ≤ 3v − 6 .$$ From Wikipedia: Note that these theorems provide necessary conditions for planarity that are not sufficient ...Mon, 31 Aug 2026 23:10:57 -0400https://pop.com.sg/positive-constant-scalar-definition.htmlhttps://pop.com.sg/positive-constant-scalar-definition.htmlIn French when we say "$k$ est une constante positive", that means $k\geq 0$. But I remarked that using the same sentence in English, "$k$ is a positive constant", means that $k>0$. Can one expl......Mon, 31 Aug 2026 23:07:33 -0400https://pop.com.sg/what-039-s-the-connection-between-the-indefinite-integral-and-the-defi.htmlhttps://pop.com.sg/what-039-s-the-connection-between-the-indefinite-integral-and-the-defi.htmlI want to understand the connection between the primitive function or antiderivative and the definite integral. My problem with this is the independent variable called t in the formula for the first ...Mon, 31 Aug 2026 23:01:08 -0400https://pop.com.sg/basic-open-sets-in-the-zariski-topology-are-also-compact.htmlhttps://pop.com.sg/basic-open-sets-in-the-zariski-topology-are-also-compact.htmlLet $A$ be a commutative ring and $X = \text{Spec}(A)$. The closed sets are those of the form $V(E) = \{$ prime ideals $\hat{p} \subset A $ containing $E \}$. And the open sets are the complement......Mon, 31 Aug 2026 22:58:43 -0400https://pop.com.sg/what-is-an-inductive-step.htmlhttps://pop.com.sg/what-is-an-inductive-step.htmlI have a set of problems I am trying to complete. They are all proofs by induction. I have made attempts at most of them and always get stuck doing the inductive step. Can someone explain in laym......Mon, 31 Aug 2026 22:56:12 -0400https://pop.com.sg/lists-versus-sets-in-linear-algebra.htmlhttps://pop.com.sg/lists-versus-sets-in-linear-algebra.htmlI’m currently learning linear algebra from “Linear Algebra Done Right” by Sheldon Axler. The author, in his proofs, makes use of lists of vectors, as opposed to the more conventional usage of sets of ...Mon, 31 Aug 2026 22:42:58 -0400https://pop.com.sg/what-exactly-are-eigen-things.htmlhttps://pop.com.sg/what-exactly-are-eigen-things.htmlWikipedia defines an eigenvector like this: An eigenvector of a square matrix is a non-zero vector that, when multiplied by the matrix, yields a vector that differs from the original vector at mos......Mon, 31 Aug 2026 22:28:50 -0400https://pop.com.sg/when-is-the-order-of-convergence-of-newton-raphson-greater-than-2.htmlhttps://pop.com.sg/when-is-the-order-of-convergence-of-newton-raphson-greater-than-2.htmlIf some function $f$ has a simple root at $x_*$ then I know that the order of convergence of the Newton-Raphson iteration is at least 2. But when is this order strictly greater than 2?...Mon, 31 Aug 2026 22:26:46 -0400https://pop.com.sg/theorem-1-8-viii-proof-of-mathematical-statistics-jun-shao.htmlhttps://pop.com.sg/theorem-1-8-viii-proof-of-mathematical-statistics-jun-shao.htmlI have some questions about the proof of Theorem 1.8(viii) of Jun Shao's Mathematical Statistics. The theorem is stating that If $X_n$ converges to $X$ in distribution, then for any $r > 0$, $li......Mon, 31 Aug 2026 22:21:26 -0400https://pop.com.sg/prove-s-is-a-supremum-iff-for-all-epsilon-0-there-exists-a-in-a-such-t.htmlhttps://pop.com.sg/prove-s-is-a-supremum-iff-for-all-epsilon-0-there-exists-a-in-a-such-t.htmlLet $A\subseteq\Bbb{R}$ is a nonempty set and $s\in \Bbb{R}$ is an upper bound. Prove $s$ is the supremum iff for all $\epsilon>0$ there exists $a\in A$ such that $a>s-\epsilon$. This is impo......Mon, 31 Aug 2026 22:19:29 -0400