https://liberty.ioCurated star highlights with fresh energy.en-usSat, 22 Aug 2026 00:09:25 -0400https://liberty.io/limit-of-x-ln-x.htmlhttps://liberty.io/limit-of-x-ln-x.htmlI am trying to solve $$\lim \limits_{x \to 0}x\ln{x}$$ which according to WolframAlpha (and Wikipedia) equals $0$. I managed to solve it by substituting such that $y = \dfrac{1}{x}$ and then using L'...Sat, 22 Aug 2026 04:02:32 -0400https://liberty.io/solving-x-3-y-3-xy-61-in-integers.htmlhttps://liberty.io/solving-x-3-y-3-xy-61-in-integers.htmlI'm trying to solve the following equation: $$x^3-y^3=xy+61$$ I got: $$(x-y)(x^2+xy+y^2)=xy+61$$ But I can't go any further. I am looking for a solution in integers. I need some hints to proce......Sat, 22 Aug 2026 03:59:19 -0400https://liberty.io/how-is-the-derivative-of-the-cdf-of-a-random-variable-x-its-pdf.htmlhttps://liberty.io/how-is-the-derivative-of-the-cdf-of-a-random-variable-x-its-pdf.htmlFor a continuos random variable $X$, if we have its p.d.f. $f(x)$, then the cumulative densitity function (c.d.f.) of $X$, $F(x)$, is $$F(x) = \int_{-\infty}^{x}f(t)dt$$ We also have $$F'(x) = \fra......Sat, 22 Aug 2026 03:51:03 -0400https://liberty.io/what-is-wrong-with-the-proof-for-ln-2-frac-1-2-ln-2.htmlhttps://liberty.io/what-is-wrong-with-the-proof-for-ln-2-frac-1-2-ln-2.htmlI have got a question which is as follows: Is $\ln(2)=\frac{1}{2}\ln(2)$?? The following argument seems suggesting that the answer is yes: We have the series $1-\frac{1}{2}+\frac{1}{3}-\fra......Sat, 22 Aug 2026 03:51:01 -0400https://liberty.io/variance-of-negative-binomial-distribution-without-moment-generating-s.htmlhttps://liberty.io/variance-of-negative-binomial-distribution-without-moment-generating-s.htmlGiven the discrete probability distribution for the negative binomial distribution in the form $$P(X = r) = \sum_{n\geq r} {n-1\choose r-1} (1-p)^{n-r}p^r$$ It appears there are no derivations o......Sat, 22 Aug 2026 03:50:23 -0400https://liberty.io/reducible-and-irreducible-polynomials-are-confusing-me.htmlhttps://liberty.io/reducible-and-irreducible-polynomials-are-confusing-me.htmlThe definition claims that a polynomial in a field of positive degree is a reducible polynomial when it can be written as the product of $2$ polynomials in the field with positive degrees. Other wi......Sat, 22 Aug 2026 03:38:17 -0400https://liberty.io/logic-behind-pisano-period.htmlhttps://liberty.io/logic-behind-pisano-period.htmlI just came to know about Pisano periods, and it's amazing application in computing modulus of large Fibonacci numbers. I know that a Pisano period periodically starts at $0,1$, but I haven't been ......Sat, 22 Aug 2026 03:16:12 -0400https://liberty.io/find-the-three-consecutive-numbers.htmlhttps://liberty.io/find-the-three-consecutive-numbers.htmlI have my problem here. I can't solve it on my own. I need a solution to prove my answer. Does anybody know this? thanks "Find the three consecutive numbers that have the property that the square ......Sat, 22 Aug 2026 03:15:31 -0400https://liberty.io/how-to-prove-that-sin-180-circ-theta-sin-theta.htmlhttps://liberty.io/how-to-prove-that-sin-180-circ-theta-sin-theta.htmlMi question is: How to prove $$\sin(180^\circ-\theta)=\sin\theta$$ ? Here, sine is defined for any angle such as 'alpha' This is the question mi college teacher asked me to derive it but i could ......Sat, 22 Aug 2026 03:14:25 -0400https://liberty.io/finding-taking-off-speed-help.htmlhttps://liberty.io/finding-taking-off-speed-help.htmlAn athlete jumps a horizontal distance of $7.18$m. He was airborne for $2.29$ seconds. The acceleration due to gravity is taken as $9.81$ms$^{-2}$. Assume that air resistance is negligible. Calcula......Sat, 22 Aug 2026 03:04:53 -0400https://liberty.io/how-to-integrate-frac-1-sqrt-1-x-2-using-substitution.htmlhttps://liberty.io/how-to-integrate-frac-1-sqrt-1-x-2-using-substitution.htmlHow you integrate $\frac{1}{\sqrt{1+x^2}}$ using following substitution? $1+x^2=t$ $\Rightarrow$ $x=\sqrt{t-1} \Rightarrow dx = \frac{dt}{2\sqrt{t-1}}dt$... Now I'm stuck. I don't know how to proceed ...Sat, 22 Aug 2026 02:30:50 -0400https://liberty.io/to-optimize-fenced-area-in-a-semi-ellipse-what-a-b-should-i-choose.htmlhttps://liberty.io/to-optimize-fenced-area-in-a-semi-ellipse-what-a-b-should-i-choose.htmlA fence must enclose an area that borders a straight river. We have a fixed amount of fence, say 100 meters. I want to make the shape a semi-ellipse (half an ellipse). What a/b ratio should I cho......Sat, 22 Aug 2026 02:30:37 -0400https://liberty.io/why-study-quadrilaterals.htmlhttps://liberty.io/why-study-quadrilaterals.htmlMy niece is in the 10th grade, and they have to do lot of theorems related to quadrilaterals. And, I was surprised to know that they have to learn by rote some theorems. This has made her feel that......Sat, 22 Aug 2026 02:19:46 -0400https://liberty.io/solving-arccos-x-arccos-2x-frac-3-pi-4.htmlhttps://liberty.io/solving-arccos-x-arccos-2x-frac-3-pi-4.html$\arccos (x) +\arccos (2x)=\frac{3\pi}{4}$ $\implies \arccos (x)=\frac{3\pi}{4}-\arccos (2x)$ $\implies \cos(\arccos (x))=\cos(\frac{3\pi}{4}-\arccos (2x))$ $\implies x=\cos(\frac{3\pi}{4})\cos(\...Sat, 22 Aug 2026 02:17:37 -0400https://liberty.io/calculating-ln-1-sqrt3.htmlhttps://liberty.io/calculating-ln-1-sqrt3.htmlI distributed the natural logarithm and got $(0 + 0.549)$ [placing the values in a calculator]. However, the answer key states that the answer is $1.0051$. Where did I go wrong?...Sat, 22 Aug 2026 02:17:21 -0400https://liberty.io/use-logarithmic-differentiation-to-find-frac-dy-dx-of-y-ln-x-ln-x.htmlhttps://liberty.io/use-logarithmic-differentiation-to-find-frac-dy-dx-of-y-ln-x-ln-x.htmlI've been asked to find the derivative of $\ln x^{\ln x}$ using a method called "logarithmic differentiation". I'm not familiar with this method of differentiation, and a search on Wikipedia doesn'......Sat, 22 Aug 2026 02:03:03 -0400https://liberty.io/is-the-derivative-of-a-straight-line-the-same-of-a-tangent-line.htmlhttps://liberty.io/is-the-derivative-of-a-straight-line-the-same-of-a-tangent-line.htmlSorry if I'm being too specific and for not showing an example but If you had a derivative of a straight line would the slope of the tangent line be the same as the straight line?...Sat, 22 Aug 2026 02:02:11 -0400https://liberty.io/why-do-we-need-topology-and-what-are-examples-of-real-life-application.htmlhttps://liberty.io/why-do-we-need-topology-and-what-are-examples-of-real-life-application.htmlI am very new to topology. I want to know why we need topology in mathematics. What is its significance? Without topology which mathematics concept cannot be explained. What are its real-life ...Sat, 22 Aug 2026 01:57:03 -0400https://liberty.io/please-help-me-understand-the-definition-of-straight-line-given-by-euc.htmlhttps://liberty.io/please-help-me-understand-the-definition-of-straight-line-given-by-euc.html"A straight-line is (any) one which lies evenly with points on itself." This is how Euclid defines a straight line but I don't know what it really means. Is this saying that any point picked on the ...Sat, 22 Aug 2026 01:54:22 -0400https://liberty.io/prove-3-is-prime-in-the-ring-bbb-z-i.htmlhttps://liberty.io/prove-3-is-prime-in-the-ring-bbb-z-i.htmlI am not sure what is the right phrase for primes in some ring. The definition I was gave is that in a ring $A$, $p\in A$ is prime if $\forall x,y \in A$ such that $p\mid xy$, $p\mid x$ or $p\mid y......Sat, 22 Aug 2026 01:53:45 -0400https://liberty.io/can-all-real-polynomials-be-factored-into-quadratic-and-linear-factors.htmlhttps://liberty.io/can-all-real-polynomials-be-factored-into-quadratic-and-linear-factors.htmlSo I understand how to do integration on rational functions with a linear and a quadratic denominator, and I understand how to do a partial fraction decomposition, but I was wondering what happens ......Sat, 22 Aug 2026 01:53:22 -0400https://liberty.io/examples-of-one-to-one-and-onto-functions-f-mathbb-n-to-mathbb-n.htmlhttps://liberty.io/examples-of-one-to-one-and-onto-functions-f-mathbb-n-to-mathbb-n.htmlGive examples of functions from $\mathbb{N}$ to $\mathbb{N}$ with the following properties: i. one-to-one but not onto ii. onto but not one-to-one iii. both onto and one-t......Sat, 22 Aug 2026 01:46:00 -0400https://liberty.io/nested-convex-curves-inside-curve-strictly-shorter.htmlhttps://liberty.io/nested-convex-curves-inside-curve-strictly-shorter.htmlNote: This question is very similar but different to a previous question [1] of mine. I decided to ask a new question for the sake of clarity. In order to complete a proof, I need to show the foll......Sat, 22 Aug 2026 01:44:56 -0400https://liberty.io/finding-the-9th-derivative-of-frac-cos-5-x-2-1-x-3.htmlhttps://liberty.io/finding-the-9th-derivative-of-frac-cos-5-x-2-1-x-3.htmlHow do you find the 9th derivative of $(\cos(5 x^2)-1)/x^3$ and evaluate at $x=0$ without differentiating it straightforwardly with the quotient rule? The teacher's hint is to use Maclaurin Series......Sat, 22 Aug 2026 01:28:12 -0400https://liberty.io/what-do-the-d-in-sde-notation-mean.htmlhttps://liberty.io/what-do-the-d-in-sde-notation-mean.htmlAn SDE is often written in the form $ dX_t=\mu dt + \sigma dW_t $. What is the meaning of this equation in English? If I had to construct an SDE, I would write something like $ \frac{dX_t}{dt} = \m......Sat, 22 Aug 2026 01:28:11 -0400https://liberty.io/w-such-that-it-contains-at-least-3-ones-is-my-approach-to-the-cfg-righ.htmlhttps://liberty.io/w-such-that-it-contains-at-least-3-ones-is-my-approach-to-the-cfg-righ.htmlSo I was trying to solve the CFG, $$\{w \in (0,1)^* \mid w \text{ contains at least three 1's}\}$$ My approach: I decided that a string can begin with a $0$, end with a $0$, it may begin with a ......Sat, 22 Aug 2026 01:27:16 -0400https://liberty.io/can-an-irrational-number-raised-to-an-irrational-power-be-rational.htmlhttps://liberty.io/can-an-irrational-number-raised-to-an-irrational-power-be-rational.htmlCan an irrational number raised to an irrational power be rational? If it can be rational, how can one prove it?...Sat, 22 Aug 2026 01:25:21 -0400https://liberty.io/dual-vs-reciprocal-basis.htmlhttps://liberty.io/dual-vs-reciprocal-basis.htmlI am confused as to what the differences between the two are. I have just learnt reciprocal vectors, and learnt that they are defined as $u_i v_j = \delta_{ij}$. However, I read that the dual basis......Sat, 22 Aug 2026 01:21:02 -0400https://liberty.io/the-graphs-of-f-f-039-and-f-039-039.htmlhttps://liberty.io/the-graphs-of-f-f-039-and-f-039-039.htmlRecently I have been struggling to solve the following problem: The figure shows the graphs of $f$, $f'$, and $f''$. Identify each curve, and explain your choices. I know the answer, but I have n......Sat, 22 Aug 2026 01:09:32 -0400https://liberty.io/what-is-the-payoff-function-for-games-with-more-than-two-players.htmlhttps://liberty.io/what-is-the-payoff-function-for-games-with-more-than-two-players.htmlFor two player games, with payoff matrices $(A,B)$, let $x \in \Delta_x$ denote the mixed strategy of player $1$, and $y \in \Delta_y$ denote the mixed strategy of player $2$. Then the payoff func......Sat, 22 Aug 2026 01:03:09 -0400https://liberty.io/what-is-the-meaning-of-hermitian.htmlhttps://liberty.io/what-is-the-meaning-of-hermitian.htmlGoogle search-bar gives the definition of Hermitian as: Hermitian: denoting or relating to a matrix in which those pairs of elements that are symmetrically placed with respect to the principal ...Sat, 22 Aug 2026 00:58:18 -0400https://liberty.io/expected-number-of-unpecked-chicks-nyt-article.htmlhttps://liberty.io/expected-number-of-unpecked-chicks-nyt-article.htmlIn this article, the winner of the math competition answered this question correctly: In a barn, 100 chicks sit peacefully in a circle. Suddenly, each chick randomly pecks the chick immediately......Sat, 22 Aug 2026 00:55:55 -0400https://liberty.io/every-polynomial-with-real-coefficients-is-the-sum-of-cubes-of-three-p.htmlhttps://liberty.io/every-polynomial-with-real-coefficients-is-the-sum-of-cubes-of-three-p.htmlHow to prove that every polynomial with real coefficients is the sum of three polynomials raised to the 3rd degree? Formally the statement is: $\forall f\in\mathbb{R}[x]\quad \exists g,h,p\in\ma......Sat, 22 Aug 2026 00:55:25 -0400https://liberty.io/is-the-mle-of-an-exponential-distribution-complete-and-sufficient.htmlhttps://liberty.io/is-the-mle-of-an-exponential-distribution-complete-and-sufficient.htmlI've just prove that the MLE of an exponential distribution is: $$ \hat \theta = \frac{n}{\sum x_i} $$ So I'm trying to show that $\hat \theta $ is complete and sufficient. For sufficiency I've ......Sat, 22 Aug 2026 00:31:27 -0400https://liberty.io/how-you-could-you-change-the-surface-area-formula-for-a-cylinder-to-ca.htmlhttps://liberty.io/how-you-could-you-change-the-surface-area-formula-for-a-cylinder-to-ca.htmlSkateboarders use half pipes for doing tricks. A half-pipe is a half cylinder. Explain how you could manipulate the surface area formula for a cylinder to calculate the curved surface area of the h......Sat, 22 Aug 2026 00:22:34 -0400https://liberty.io/euclidean-distance-proof.htmlhttps://liberty.io/euclidean-distance-proof.htmlHow can I show that the Euclidean distance satisfies the triangle inequality? Where the Euclidean distance is given by: $$d(p,q) = \sqrt{(p_1-q_1)^2 + \cdots + (p_n-q_n)^2}$$ Triangle Inequality......Sat, 22 Aug 2026 00:18:44 -0400https://liberty.io/shortest-distance-from-point-to-a-plane.htmlhttps://liberty.io/shortest-distance-from-point-to-a-plane.htmlQ: Find the shortest distance from the point $A(1,1,1)$ to the plane $2x+3y+4z=5$. I understand that we need to pick a point P on the plane such that we can form a vector PA. Can someone explain ......Sat, 22 Aug 2026 00:18:08 -0400https://liberty.io/how-can-i-prove-ax-bx-for-every-n-times1-column-matrix-x-implies-a-b.htmlhttps://liberty.io/how-can-i-prove-ax-bx-for-every-n-times1-column-matrix-x-implies-a-b.htmlLet $A$ and $B$ be matrices $n\times n$. Suppose $AX=BX$ for every $n\times 1$ column matrix $X$. How can I prove this implies $A=B$?...Sat, 22 Aug 2026 00:16:12 -0400https://liberty.io/lagrange-interpolating-polynomials-error-bound.htmlhttps://liberty.io/lagrange-interpolating-polynomials-error-bound.htmlLet $f(x) = e^{2x} - x$, $x_0 = 1$, $x_1 = 1.25$, and $x_2 = 1.6$. Construct interpolation polynomials of degree at most one and at most two to approximate $f(1.4)$, and find an error bound for the ...Sat, 22 Aug 2026 00:12:30 -0400https://liberty.io/unable-to-evaluate-the-integral-int-csc-left-x-frac-pi-3-right-csc-lef.htmlhttps://liberty.io/unable-to-evaluate-the-integral-int-csc-left-x-frac-pi-3-right-csc-lef.htmlThe Question $$\int \csc\left(x-\frac{\pi}{3}\right)\csc\left(x-\frac{\pi}{6}\right) dx $$ What I Tried- I tried dividing both numerator and denominator by $\sin \pi/6$ but couldn't get too far, w......Sat, 22 Aug 2026 00:09:36 -0400https://liberty.io/what-is-a-principal-minor-of-a-matrix.htmlhttps://liberty.io/what-is-a-principal-minor-of-a-matrix.htmlI was going through the book on operation research by Hamdy A.Taha. It referred to principal minor of a hessian matrix. Can someone explain what is meant by a principal minor? Is it different from ......Sat, 22 Aug 2026 00:07:03 -0400https://liberty.io/a-related-rate-problem-of-a-trough-with-a-trapezoidal-cross-section.htmlhttps://liberty.io/a-related-rate-problem-of-a-trough-with-a-trapezoidal-cross-section.htmlFrom the MIT exercise book (2E-7): A trough is filled with water at a rate of $1$ cubic meter per second. The trough has a trapezoidal cross section with the lower base of length half a meter and ......Fri, 21 Aug 2026 23:59:42 -0400https://liberty.io/the-sixth-number-system.htmlhttps://liberty.io/the-sixth-number-system.htmlIn high school, I learned there are 5 number systems, namely: Natural numbers ($\Bbb N$) Integers ($\Bbb Z$) Rational numbers ($\Bbb Q$) Real numbers ($\Bbb R$) Complex numbers ($\Bbb C$) I ...Fri, 21 Aug 2026 23:59:12 -0400https://liberty.io/discrete-math-understanding-a-proof-involving-the-definition-of-divisi.htmlhttps://liberty.io/discrete-math-understanding-a-proof-involving-the-definition-of-divisi.htmlIn this first course on discrete mathematics, the instructor provided this following solution to a question. The question was asked us to prove the following (the solution is provided as well): My ...Fri, 21 Aug 2026 23:56:54 -0400https://liberty.io/definition-and-example-of-a-partition.htmlhttps://liberty.io/definition-and-example-of-a-partition.htmlPartition of a Set is defined as "A collection of disjoint subsets of a given set. The union of the subsets must equal the entire original set." For example, one possible partition of $(1, 2, 3, 4......Fri, 21 Aug 2026 23:53:44 -0400https://liberty.io/line-integral-work-done-moving-along-a-certain-path.htmlhttps://liberty.io/line-integral-work-done-moving-along-a-certain-path.htmlFind the work done by the force field $F(x,y) = -xi + 6yj$ along the path $C:y = x^3$ from $(0,0)$ to $(6,216)$ I tried parameterizing C which gave me $x(t) = t$ and $y(t) = t^3$ but do I use thos......Fri, 21 Aug 2026 23:52:39 -0400https://liberty.io/similarity-transform-of-a-matrix-preserves-the-determinant.htmlhttps://liberty.io/similarity-transform-of-a-matrix-preserves-the-determinant.htmlIt is posited (in chapter $11$ of the book Numerical Recipes in C) that doing a similarity transform of a matrix $A$ preserves its eigenvalues. This is how a similarity transform is defined: $$A \...Fri, 21 Aug 2026 23:52:37 -0400https://liberty.io/special-derivative-function.htmlhttps://liberty.io/special-derivative-function.htmlI have a function $f(x)$ which I would like to have the derivative with respect to $x$. How can I get the derivative of the following function with respect to $x$? $$f(x) = \log(1-z^{e^{y^{T}x}})$$...Fri, 21 Aug 2026 23:44:29 -0400https://liberty.io/number-of-various-7-character-passwords-that-are-required-to-contain-a.htmlhttps://liberty.io/number-of-various-7-character-passwords-that-are-required-to-contain-a.htmlIn this question on Information Security, different options are given for passwords. One example states that a password contains a mixture of numbers, letters and special characters. Assuming that ...Fri, 21 Aug 2026 23:34:11 -0400https://liberty.io/math-in-horseshoe-puzzle.htmlhttps://liberty.io/math-in-horseshoe-puzzle.htmlWe know that Rubik's Cube is a good demonstration of group theory. Correspondingly, for the horseshoe puzzle as in the picture below, is there a math language for it? Does it demonstrate any math s......Fri, 21 Aug 2026 23:20:28 -0400https://liberty.io/what-can-be-understood-by-a-b-b-a-in-set-theory.htmlhttps://liberty.io/what-can-be-understood-by-a-b-b-a-in-set-theory.htmlWhat can be understood by $\rm A-B = B - A$ in set theory? What does this tell us about the characteristics of $A$ and $B$ and their relationship? I'm quite confused as I am new to set theory....Fri, 21 Aug 2026 23:09:16 -0400https://liberty.io/solving-a-function-where-x-mod-y-represents-the-remainder-after-divisi.htmlhttps://liberty.io/solving-a-function-where-x-mod-y-represents-the-remainder-after-divisi.htmlI would like help solving this practice question. For a function f(x, y) defined below, what is the value of f(775, 527)? Here, x mod y represents the remainder after division of x by y. f(x, y)......Fri, 21 Aug 2026 23:07:40 -0400https://liberty.io/proving-the-symmetry-of-the-ricci-tensor.htmlhttps://liberty.io/proving-the-symmetry-of-the-ricci-tensor.htmlConsider the Ricci tensor : $R_{\mu\nu}=\partial_{\rho}\Gamma_{\nu\mu}^{\rho} -\partial_{\nu}\Gamma_{\rho\mu}^{\rho} +\Gamma_{\rho\lambda}^{\rho}\Gamma_{\nu\mu}^{\lambda} -\Gamma_{\nu\lambda}^{\rho}\...Fri, 21 Aug 2026 23:07:21 -0400https://liberty.io/derivation-of-the-general-forms-of-partial-fractions.htmlhttps://liberty.io/derivation-of-the-general-forms-of-partial-fractions.htmlI'm learning about partial fractions, and I've been told of 3 types or "forms" that they can take (1) If the denominator of the fraction has linear factors: $${5 \over {(x - 2)(x + 3)}} \equiv {A......Fri, 21 Aug 2026 23:04:53 -0400https://liberty.io/how-many-solutions-does-the-system-have-why.htmlhttps://liberty.io/how-many-solutions-does-the-system-have-why.htmlHow can I solve this problem? Let Z be a 4 x 5 matrix. A basis for the null space of Z is {𝐛𝟏} where 𝐛𝟏 ≠ 𝟎. If the system of equations (Z^T)𝐱 = 𝐦 is consistent, how many solutions does the ...Fri, 21 Aug 2026 22:54:33 -0400https://liberty.io/normal-curve-1-5-standard-deviations.htmlhttps://liberty.io/normal-curve-1-5-standard-deviations.htmlFor a normal curve, how much of the area lies within 1.5 standard deviations of the mean? I already know about the 68–95–99.7 rule, and see that it should be between 68% and 95%. I also know that it ...Fri, 21 Aug 2026 22:43:46 -0400https://liberty.io/vandermonde-determinant-from-3-by-3.htmlhttps://liberty.io/vandermonde-determinant-from-3-by-3.htmlI am a novice at math and need help calculating the determinant of the Vandermonde matrix, which is: $V=\begin{bmatrix} 1&a_1&a_1^2&\dots&a_1^{n-1}\\ 1&a_2&a_2^2&\dots&a......Fri, 21 Aug 2026 22:40:49 -0400https://liberty.io/solving-a-system-of-equations-linear-algebra.htmlhttps://liberty.io/solving-a-system-of-equations-linear-algebra.htmlFor one of my homework problems I have the following system of equations. $$ \left\{ \begin{array}{c} x_1+2x_2-2x_3-3x_5+2x_6=-4 \\ -x_4+4x_5+3x_6=-9 \\ x_1+2x_2-5x_5+4x_6=-4 \end{array} \right......Fri, 21 Aug 2026 22:39:06 -0400https://liberty.io/calculate-singular-value-decomposition.htmlhttps://liberty.io/calculate-singular-value-decomposition.htmlI need to find the reduced singular value decomposition of this matrix. $$\begin{pmatrix} -2 & -1 & 2 \\ 2 & 1 & -2 \\ \end{pmatrix} $$ I formed $$ A^TA= \begin{pmatrix} 8 & ......Fri, 21 Aug 2026 22:37:56 -0400https://liberty.io/exponential-and-power-functions-through-two-points.htmlhttps://liberty.io/exponential-and-power-functions-through-two-points.htmlI have a problem where I'm asked to determine the constants of exponential and power functions that go through both points (5, 50) and (10, 1600). I have tried to solve them below, but would apprec......Fri, 21 Aug 2026 22:30:39 -0400https://liberty.io/how-can-three-vectors-be-orthogonal-to-each-other.htmlhttps://liberty.io/how-can-three-vectors-be-orthogonal-to-each-other.htmlIn a scenario, say that: Vectors $\mathbf{U}$, $\mathbf{V}$ and $\mathbf{W}$ are all orthogonal such that the dot product between each of these $(\mathbf{UV}\;\mathbf{VW}\;\mathbf{WU})$ is equal......Fri, 21 Aug 2026 22:25:16 -0400https://liberty.io/giving-a-basis-for-the-column-space-of-a.htmlhttps://liberty.io/giving-a-basis-for-the-column-space-of-a.htmlLet $A = \begin{bmatrix}3&3&3\\3&5&1\\-2&4&-8\\-2&-4&0\\4&9&-1\end{bmatrix}$ Give a basis for the column space of A So what I've done so far is put it in RR......Fri, 21 Aug 2026 22:19:01 -0400https://liberty.io/radial-and-transverse-components-of-velocity-and-acceleration.htmlhttps://liberty.io/radial-and-transverse-components-of-velocity-and-acceleration.htmlI need help with this problem: Find (i) the velocity $\dot r(t)$, (ii) the speed $\Vert \dot r(t)\Vert$, (iii) the acceleration $\ddot r(t)$, (iv) the radial and transverse componentes of velo......Fri, 21 Aug 2026 22:16:45 -0400https://liberty.io/plot-absolute-error-vs-step-size-as-a-log-log-plot-matlab.htmlhttps://liberty.io/plot-absolute-error-vs-step-size-as-a-log-log-plot-matlab.htmlI have approximated up to $t=2$ with step sizes $h=(1/2)^{n}$ for $n=1, ... , 10$ using a Runge-Kutta (order 4) MATLAB code. Then, I wrote down the absolute error of $y(2)$. Now, I want to plot the......Fri, 21 Aug 2026 22:15:02 -0400https://liberty.io/how-to-combine-manipulate-two-summations-into-one-summation-in-general.htmlhttps://liberty.io/how-to-combine-manipulate-two-summations-into-one-summation-in-general.htmlI always struggle to understand what I can and can't do with sums. In fact, even when convergence isn't an issue, I still get confused. What can I do about this problem? For instance, I am currently ...Fri, 21 Aug 2026 22:13:11 -0400https://liberty.io/generalizing-the-abel-theorem-to-higher-order-differential-equations.htmlhttps://liberty.io/generalizing-the-abel-theorem-to-higher-order-differential-equations.htmllet $$y''+p(t)y'+q(t)y=g(t)$$ then By the Abel theorem wronskian equals to : $$W= c e^{\int p(t)dt}$$ Now We want to generalise this to a 3rd order differential equation where : $$y'''+p_1(t)y''......Fri, 21 Aug 2026 22:03:06 -0400https://liberty.io/why-arctan-in-radian-mode-and-arctan-in-degrees-mode-are-different.htmlhttps://liberty.io/why-arctan-in-radian-mode-and-arctan-in-degrees-mode-are-different.htmlSo I've been playing around with my calculator. In degrees mode with my casio calculator, I typed, "$\arctan(1)$" and got an answer of 45. In radian mode with the same parameter, I got, "0.78539......Fri, 21 Aug 2026 22:01:17 -0400https://liberty.io/equation-of-a-rectangle.htmlhttps://liberty.io/equation-of-a-rectangle.htmlI need to graph a rectangle on the Cartesian coordinate system. Is there an equation for a rectangle? I can't find it anywhere....Fri, 21 Aug 2026 21:59:28 -0400https://liberty.io/is-this-the-right-way-to-prove-basic-limit-identities.htmlhttps://liberty.io/is-this-the-right-way-to-prove-basic-limit-identities.htmlFor example let's say I wanted to prove that: $$\lim_{x \to \infty} \frac{1}{x} = 0$$ Normally I would use epsilon-delta for such proofs but then I start out with conditions like: $|\frac{1}{x} ......Fri, 21 Aug 2026 21:56:44 -0400https://liberty.io/integration-of-secant-duplicate.htmlhttps://liberty.io/integration-of-secant-duplicate.html$$\begin{align} \int \sec x \, dx &= \int \cos x \left( \frac{1}{\cos^2x} \right) \, dx \\ &= \int \cos x \left( \frac{1}{1-\sin^2x} \right) \, dx \\ & = \int\cos x\cdot\frac{1}{1-\......Fri, 21 Aug 2026 21:52:12 -0400https://liberty.io/finding-a-regular-16-sided-polygon-whose-vertices-are-lattice-points.htmlhttps://liberty.io/finding-a-regular-16-sided-polygon-whose-vertices-are-lattice-points.htmlFind a regular equilateral and equiangular 16-sided polygon that has vertices that are lattice points. What is the minimum side length? Find the vertices of such a polygon. Find the area of the po......Fri, 21 Aug 2026 21:49:49 -0400https://liberty.io/given-that-g-is-the-centroid-of-delta-abc-ga-2-sqrt-3-gb-2-sqrt-2-gc-2.htmlhttps://liberty.io/given-that-g-is-the-centroid-of-delta-abc-ga-2-sqrt-3-gb-2-sqrt-2-gc-2.htmlGiven that $G$ is the centroid of $\Delta ABC$, $GA = 2\sqrt{3}$ , $GB = 2\sqrt{2}$, $GC = 2$ . Find $[\Delta ABC]$. What I Tried: Here is a picture :- I know the centroid divides each of the med......Fri, 21 Aug 2026 21:44:46 -0400https://liberty.io/prove-the-fhhf-theorem-using-as-much-abstract-non-sense-as-possible.htmlhttps://liberty.io/prove-the-fhhf-theorem-using-as-much-abstract-non-sense-as-possible.htmlThis is my second attempt to solve exercise 1.6H from Vakil: Assume $F$ is a covariant right-exact functor and show that we get a map $HF^i \rightarrow H^iF$. Attempt to a solution: Apply $F$ to ......Fri, 21 Aug 2026 21:33:41 -0400https://liberty.io/what-is-the-general-formula-for-calculating-dot-and-cross-products-in-.htmlhttps://liberty.io/what-is-the-general-formula-for-calculating-dot-and-cross-products-in-.htmlI was writing a C++ class for working with 3D vectors. I have written operations in the Cartesian coordinates easily, but I'm stuck and very confused at spherical coordinates. I googled my question......Fri, 21 Aug 2026 21:18:31 -0400https://liberty.io/taylor-series-analog-for-functionals.htmlhttps://liberty.io/taylor-series-analog-for-functionals.htmlFor a function $f(x)$, it is possible to write it as a taylor series centered around a point $x=a$: $$f(x)=\sum_{n=0}^{\infty}\frac{f^{(n)}(a){(x-a)}^{n}}{n!}=f(a)+f'(a)(x-a)+\frac{f''(a)(x-a)^{2}......Fri, 21 Aug 2026 21:11:26 -0400https://liberty.io/understanding-the-first-step-analysis-of-absorbing-markov-chains.htmlhttps://liberty.io/understanding-the-first-step-analysis-of-absorbing-markov-chains.htmlConsider a time-homogeneous Markov chain $\{X_n\}_{n=0}^\infty$ with the state space state space $S=\{0,1,2\}$ and the following transition probability matrix: \begin{pmatrix} 1 & 0 & 0 ......Fri, 21 Aug 2026 21:10:23 -0400https://liberty.io/how-do-you-find-a-vector-in-the-form-a-b-when-only-the-angle-and-magni.htmlhttps://liberty.io/how-do-you-find-a-vector-in-the-form-a-b-when-only-the-angle-and-magni.htmlHow do you find a vector in the form when only the angle and magnitude are given? Here is an example where an angle of 80 degrees is given along with a magnitude of 3....Fri, 21 Aug 2026 21:05:28 -0400https://liberty.io/determine-the-asymptotic-curves-of-the-catenoid.htmlhttps://liberty.io/determine-the-asymptotic-curves-of-the-catenoid.htmlLet be $X(u,v) = (\cosh v \cos u,\cosh v \sin u, v)$ the parametrization of catenoid. Determine the asymptotic curves of the catenoid. $\textbf{My attempt:}$ $X_u(u,v) = (-\cosh v \sin u,\cosh v \......Fri, 21 Aug 2026 20:51:07 -0400https://liberty.io/two-vectors-are-in-the-same-direction-if.htmlhttps://liberty.io/two-vectors-are-in-the-same-direction-if.htmlAre two vectors in the same direction if their dot product is greater than zero/positive? I know they are orthogonal if their dot product is 0 so they can not be in the same direction. I also read ......Fri, 21 Aug 2026 20:48:39 -0400https://liberty.io/what-is-growth-of-a-function-and-growth-rate-of-a-function.htmlhttps://liberty.io/what-is-growth-of-a-function-and-growth-rate-of-a-function.htmlI'm trying to figure out what is growth of a function and growth rate of a function means and how should I note it. As I understand growth rate of f(n) is dominant term of f(n) when n tends to in......Fri, 21 Aug 2026 20:48:37 -0400https://liberty.io/finding-the-possible-values-of-x.htmlhttps://liberty.io/finding-the-possible-values-of-x.htmlThe lengths of a triangle are 5cm, 7cm and x cm. What are the possible values of x? I don't know where to start with this question as i don't thin enough information has been given. With an ans......Fri, 21 Aug 2026 20:44:16 -0400https://liberty.io/square-brackets-instead-of-parens-for-functions.htmlhttps://liberty.io/square-brackets-instead-of-parens-for-functions.htmlWhy are sometimes square brackets used to apply parameters to functions instead of the usual round parentheses? For instance, in my probability course, they use $\text{P}[X]$ to denote the probabi......Fri, 21 Aug 2026 20:41:47 -0400https://liberty.io/probability-of-a-u-b-u-c.htmlhttps://liberty.io/probability-of-a-u-b-u-c.htmlEvent A and B are independent, A and C are mutually exclusive, B and C are independent. Given: $\Pr(A) = 1/2$ $\Pr(B) = 1/4$ $\Pr(C) = 1/8$ Find $P(A \cup B \cup C)$ ? The teacher gave the answe......Fri, 21 Aug 2026 20:19:49 -0400https://liberty.io/using-the-logarithm-product-rule-with-a-coefficient.htmlhttps://liberty.io/using-the-logarithm-product-rule-with-a-coefficient.htmlI'm trying to solve this logarithmic expression: $\left(\dfrac{\left(\log_6(27)+2\log_6(2)\right)}{\log_6\sqrt[3]{0.25}+\log_6(\frac{1}{3})}\right)^{\!2}$ But after a couple of steps, I got st......Fri, 21 Aug 2026 20:16:28 -0400https://liberty.io/why-is-it-enough-for-a-tangent-plane-to-only-agree-with-the-slope-of-a.htmlhttps://liberty.io/why-is-it-enough-for-a-tangent-plane-to-only-agree-with-the-slope-of-a.htmlThanks for reading. Short Version Why is it enough for the tangent plane to only agree with the slope of a differentiable 2-input function in the $x$ and $y$ directions at a certain point for it to......Fri, 21 Aug 2026 20:05:42 -0400https://liberty.io/definite-integral-area-between-two-trig-functions.htmlhttps://liberty.io/definite-integral-area-between-two-trig-functions.htmlI'm trying to figure out how to find the area between two trig functions. I know the procedure of integration here, finding the difference between two functions and integrating across whatever inte......Fri, 21 Aug 2026 20:04:49 -0400https://liberty.io/what-is-the-difference-between-arg-max-and-max.htmlhttps://liberty.io/what-is-the-difference-between-arg-max-and-max.htmlWhat is the exact difference between $\arg\max$ and $\max$ of a function? Is it right to say the following? $\arg\max f(x)$ is nothing but the value of $x$ for which the value of the function is ...Fri, 21 Aug 2026 20:00:19 -0400https://liberty.io/opposite-of-factorial-duplicate.htmlhttps://liberty.io/opposite-of-factorial-duplicate.htmlPossible Duplicate: Is there a way to solve for an unknown in a factorial? I was just wondering, what would be the opposite of factorial? For example, If I had $n! = 120$. How can I then show ...Fri, 21 Aug 2026 19:51:31 -0400https://liberty.io/cardinality-of-all-cardinalities.htmlhttps://liberty.io/cardinality-of-all-cardinalities.htmlLet $C = \{0, 1, 2, \ldots, \aleph_0, \aleph_1, \aleph_2, \ldots\}$. What is $\left|C\right|$? Or is it even well-defined?...Fri, 21 Aug 2026 19:50:46 -0400https://liberty.io/maximum-spanning-tree.htmlhttps://liberty.io/maximum-spanning-tree.htmlI wonder how to prove that given a Minimum Spanning Tree of a graph, the other spanning tree with the least common edge with Minimum Spanning Tree is always Maximum Spanning tree....Fri, 21 Aug 2026 19:34:07 -0400https://liberty.io/an-interesting-cubic-equation-x-3-3x-1-0.htmlhttps://liberty.io/an-interesting-cubic-equation-x-3-3x-1-0.htmlFor the cubic equation $x^3-3x-1=0$, you can check it has three real roots by calculus knowledge. Let them be $x_1<x_2<x_3$. Now prove that $$x_3^2-x_2^2=x_3-x_1.$$ It is not difficult to ch......Fri, 21 Aug 2026 19:34:03 -0400https://liberty.io/how-can-i-prove-that-every-group-of-n-255-elements-is-commutative.htmlhttps://liberty.io/how-can-i-prove-that-every-group-of-n-255-elements-is-commutative.htmlThere was previous task was same but with $N = 185$. And I prove it by showing that number of Sylow subgroups is 1 for every prime $p\mid N$. But there I have some options $N_5 \in \{1, 51\}$, $N_{......Fri, 21 Aug 2026 19:28:27 -0400https://liberty.io/definition-of-linear-differential-equation.htmlhttps://liberty.io/definition-of-linear-differential-equation.htmlI am a 13 year old self teaching myself Differential Equations from a website and a book, I came across the definition of a Linear Differential Equation but I didn't understand the definition, I lo......Fri, 21 Aug 2026 19:25:53 -0400https://liberty.io/does-there-exist-a-group-isomorphism-from-z-to-zxz.htmlhttps://liberty.io/does-there-exist-a-group-isomorphism-from-z-to-zxz.htmlI know that $\mathbb Z$ and $\mathbb{Z}\times\mathbb{Z}$ have the same cardinality because you can create a bijection between the two. The example I was taught is Cantor's pairing function, which m......Fri, 21 Aug 2026 19:16:11 -0400https://liberty.io/finding-a-joint-probability-density-function-given-marginal-probabilit.htmlhttps://liberty.io/finding-a-joint-probability-density-function-given-marginal-probabilit.htmlX and Y are independent random variables with the following PDFs: $f_{x}(x) = \begin{cases} (1/3)e^{-x/3}, & x \geq\text{ 0} \\ 0, & \text{otherwise} \end{cases}$ $f_{y}(y) = \begin{cases......Fri, 21 Aug 2026 19:13:07 -0400https://liberty.io/how-many-subsets-can-be-made-with-a-set-of-n-items.htmlhttps://liberty.io/how-many-subsets-can-be-made-with-a-set-of-n-items.htmlIf you have a set of $N$ items, how many subsets can you make? For example, for the set existing of $3$ items (Item1, Item2 and Item3) the subsets are Item1, Item2, Item3, Item2+3, Item1+2, Item1+......Fri, 21 Aug 2026 19:09:56 -0400https://liberty.io/understanding-the-formula-for-stereographic-projection-of-a-point.htmlhttps://liberty.io/understanding-the-formula-for-stereographic-projection-of-a-point.htmlI was wondering about the equation of line I can write which can help me finding the coordinates of Point $P'$ in relation with coordinates of points on the sphere that is $P$. Let the $P'(X,Y)$, ......Fri, 21 Aug 2026 19:01:08 -0400https://liberty.io/calculator-tool-to-find-subgroup-of-s-n-generated-by-langle-g-1-dots-g.htmlhttps://liberty.io/calculator-tool-to-find-subgroup-of-s-n-generated-by-langle-g-1-dots-g.htmldoes anyone know an online utility tool that computes the subgroup of the symmetric group generated by generators, for example, I want the program to tell me $$ \langle (13)(24), (13) \rangle = \{......Fri, 21 Aug 2026 18:59:58 -0400https://liberty.io/sketch-the-graph-of-the-integrand-function-and-use-it-to-help-evaluate.htmlhttps://liberty.io/sketch-the-graph-of-the-integrand-function-and-use-it-to-help-evaluate.htmlsketch the graph of the integrand function and use it to help evaluate the integral. integration from(1 , -1) |x|-1 I think I can evaluate the integration f(x) = 1/2 x^2 -x+c but how sketch the ...Fri, 21 Aug 2026 18:44:35 -0400https://liberty.io/how-to-rewrite-logarithmic-equation-in-exponential-form.htmlhttps://liberty.io/how-to-rewrite-logarithmic-equation-in-exponential-form.htmlHow would I rewrite this logarithmic equation: $\ln(37)= 3.6109$, in exponential form? -Thanks...Fri, 21 Aug 2026 18:38:00 -0400