Win At Mathematics

One of our user asked: We can consider $\mathbb{R}^+$ with two operation: +: $\mathbb{R}^+ \times \mathbb{R}^+\to \mathbb{R}^+$ that maps $(a,b)$ to $a+b:=ab$ and $ \cdot : \mathbb{R} \times \mathbb{R}^+ \to \mathbb{R}^+$ that maps $(\lambda,a)$ to $a^\lambda$ . With respect to these operation we have that $\mathbb{R}^+$ is a vector space. We want find an example of linear application from $\mathbb{R}$ to $\mathbb{R}^+$ . We can fix $a\in \mathbb{R}^+$ and we can define $f_a:\mathbb{R}\to \mathbb{R}^+$ such that maps every $\lambda$ to $ f_a(\lambda):=a^\lambda$ . From the rules of powers, the map $f_a$ is a linear map for every $a\in \mathbb{R}^+$ . A natural question can be if all linear maps from $\mathbb{R}$ to $\mathbb{R}^+$ are an exponential maps. The answer is positive because for each linear map $F: \mathbb{R}\to \mathbb{R}^+$ we have that $F(\lambda)=F(\lambda \cdot 1)=(F(1))^\lambda=f_{F(1)}(\lambda)$ so $F=f_{F(1)}$ We can also observe that the m...

One of our user asked: I wonder if someone can help me to find the differentiation with respect to the upper limit of a summation as shown below !! $\frac{d}{dx}\sum\limits_{n=1}^{f(x)} g(n)$ .

One of our user asked: Here are few of the questions from the previous years' exams. I've chosen the ones I'm not sure about. It's a simple TRUE/FALSE task. Would anyone be able to verify my solution? Some of my answers are good, some are just random guess according to my intuition. I don't really need a detailed explanation... Thanks! Domain of $f'$ is contained within domain of $f$ . - TRUE Boundary point of set A is also a cluster point of that set. - TRUE Every increasing sequence and bounded above is convergent. - TRUE Every increasing sequence and bounded below is convergent. - FALSE Every increasing sequence is always bounded below. - TRUE Every sequence is discontinuous function. - FALSE Every sequence is continuous function. - TRUE Every function integrable on $<a, b>$ is continuous on $<a, b>$ . - FALSE Function $f(x) = \ln{|x|}$ is discontinuous at $0$ . - TRUE The continuity is necessary for differentiability. - TRUE Func...

One of our user asked: I don't understand why the last step in the following equations is true. Could someone explain this to me please? Don't think context is important here, but just in case it's from a proof of a bound on the error of the explicit Euler method.

One of our user asked: Suppose $y'' + f(x)y = 0$ where $M \geq f(x) \geq m > 0$ on some interval $[a,b]$ , then the number zeros $N$ of a non trivial solution is $\lfloor\frac{(b-a)\sqrt{m}}{\pi}\rfloor \leq N \leq \lceil\frac{(b-a)\sqrt{M}}{\pi}\rceil$ Simple. Now suppose I have an equation $y''+4y'+\frac{8x+\sin(x)}{x+1}y = 0$ and I want to estimate the number of zeros of a non trivial solution. I can't use the theorem as is, because the ODE is not in the correct form, to fix this, we can multiply by $e^{2x}$ and get $y''e^{2x}+4y'e^{2x}+\frac{8x+\sin(x)}{x+1}ye^{2x} = 0$ Now if we let $ye^{2x} = z$ we have an ODE $z'' + (\frac{8x+\sin(x)}{x+1}-4)z = 0$ which is in the correct form How did the professor know to multiply by $e^{2x}$ ? Is there a method to this or was this just a lucky guess

One of our user asked: I have been all over the internet, but I just can't make sense of this stuff. I have done my best to learn from my textbook and different websites, but this is confusing for me. I haven't taken any calculus in years, and I'm jumping in headfirst. If anyone can help me understand how to prove these using Cartesian Tensor Notation, I would really appreciate it! First identity: ∇ x ( ∇ x a ) = ∇(∇ . a ) - ( ∇^2 ) a Second identity: ∇ . ( a b ) = a . ∇ b + b ( ∇ . a ) Third identity: ∇ . ( f δ ) = ∇ f Fourth identity: δ : ∇ a = ∇ . a Thanks everyone

One of our user asked: I am attempting to prove that $$J=\int_0^\infty\frac{\log^2(x)}{x^2+1}\mathrm dx=\frac{\pi^3}8$$ With real methods because I do not know complex analysis. I have started with the substitution $x=\tan u$ : $$J=\int_0^{\pi/2}\log^2(\tan x)\mathrm dx$$ $$J=\int_0^{\pi/2}\log^2(\cos x)\mathrm dx-2\int_{0}^{\pi/2}\log(\cos x)\log(\sin x)\mathrm dx+\int_0^{\pi/2}\log^2(\sin x)\mathrm dx$$ But frankly, this is basically worse. Could I have some help? Thanks.

One of our user asked: $$M=\{f\in C[0,2\pi],\int_{0}^{2\pi}f(x)sinxdx=\pi,\int_{0}^{2\pi}f(x)sin2xdx=2\pi\} $$ $a,b\in \mathbb R, g\in M, g(x)=asinx+bsin2x,x\in [0,2\pi]$ I've read on the answers that $a=1,b=2$ and I don't know how to calculate them. Can somebody explain me,please? By the way, the problem is to determine $ \int_{0}^{2\pi}(g(x))^2 dx$ so you have to first get the constants $a$ and $b$ .

One of our user asked: Why is Euler's number $\mathtt 2.71828$ and not for example $\mathtt 3.7589$ ??? I know that e is the base of natural logarithms, I know about areas on hyperbola xy=1 and I know it's formula : $$e =\sum_{n=0}^\infty \frac{1}{n!}$$ And I also know it has many other characterizations. But, why is e equal to that formula (which sum is approximately $\mathtt 2.71828$ ) ??? I googled that many times and every time it ends in having "e is the base of natural logarithms", I don't want to work out any equations using e without understanding it perfectly. Thanks in advance.

One of our user asked: I want to calculate and evaluate the following integral: $$\frac{B}{2 D}\int_{0}^{\infty} xe^{\frac{-Bx}{D}} erfc(\frac{x+x_{0}-Bt}{2\sqrt{Dt}})$$ My idea was to integrate by parts by setting: $$u= erfc(\frac{x+x_{0}-Bt}{2\sqrt{Dt}}), du=-\frac{1}{\sqrt{Dt}} e^{-(\frac{x+x_{0}-Bt}{2\sqrt{Dt}})^2}$$ $$dv=xe^{-\frac{Bx}{D}},v=e^{-\frac{Bx}{D}}(\frac{Bx}{D}x+1)*\frac{D^2}{B^2}$$ I have calculated further and got some results, but I am not sure if my thinking is right, or even if there is some easier or more efficient way to do this analytically or numerically,(for example by expanding the error function as infinite series). Any tips would be appreciated. And of course, erfc is the complementary error function with: $$erfc(\frac{x+x_{0}-Bt}{2\sqrt{Dt}})=\frac{2}{\sqrt{\pi}}\int_{\frac{x+x_{0}-Bt}{2\sqrt{Dt}}}^{\infty} e^{-z^2} dz $$

One of our user asked: . Find the first 4 terms of the Taylor series expansion of the error function with respect to x=0. The error function is defined as: enter image description here `

One of our user asked: The series of functions $$ \sum_{n=0}^{+\infty} \frac{(-x)^n}{1+n!} $$ is pointwise convergent for any $x\in \mathbb{R}$ , thus it defines a function $f:\mathbb{R}\rightarrow \mathbb{R}$ . Is there a way to evaluate the limit $\lim_{x\rightarrow +\infty} f(x)$ ?

One of our user asked: I am wondering why finding a function that is harmonic on the sphere and that respect some conditions on the frontiere of the sphere is important ? This is called the Dirichlet problem, and I don't understand why we are interesting in this problem ? Why are we want the function to be harmonic ? I know what it means for a function to be harmonic, but I don't understand what is the "advantage" of being harmonic. Moreover does the solution of the Dircihlet helps solving real-life/physic problems ? Thank you !

One of our user asked: The following that $$\int_0^1 f(x) \,\mathrm dx=\lim_{n\rightarrow\infty}\sum_{k=1}^{n-1}f(\frac{k}{n})\frac{1}{n} \, $$ is well-known fact! But if $$\lim_{n\rightarrow\infty}\frac{r_n}{n}=\alpha,$$ then is it true that $$\int_0^{\alpha} f(x) \,\mathrm dx=\lim_{n\rightarrow\infty}\sum_{k=1}^{r_n-1}f(\frac{k}{n})\frac{1}{n} \, $$ ? If this is true, why? Help me with big mercy!!

One of our user asked: Find a step function s such that $$\int_{0}^{2} s(x) dx=5 \quad \int_{0}^{5} s(x) dx=2$$ The given answer is $$s(x)=\dfrac{5}{2} \quad \text{if} \quad 0 \leq x < 2$$ $$s(x)=-1 \quad \text{if} \quad -2 \leq x \leq 5$$ I don't understand how does one arrive to this solution.. even graphically trying to understand it I didn't come to a solution. Can someone please help me figure out how?

One of our user asked: I have a vector function $f: \mathbb{R}^n \to \mathbb{R}^n$ defined with components $$ f_i(a) = \sum_{j=1}^n \sin(a_i - a_j) $$ which I want to integrate from ${\bf{\alpha}}^0$ to ${\bf{\alpha}}^1$ where ${\bf{\alpha}}^k = [\alpha^k_1, \ldots, \alpha^k_n]$ for $k \in \{1,2\}$ . So the problem looks like $$ \int_^0}^^1} f(a)^{\top} {\rm d}\, a. $$ I thought that I could integrate as below $$ \int_{\alpha^0}^{\alpha^1} \sum_{i=1}^n \left\{ \sum_{j=1}^n \sin(a_i - a_j)\right\} {\rm d}a_i $$ by expanding the inner product in the integrand. I think that I can then write the integral as $$ \int_{(\alpha_1^0, \ldots, \alpha_n^0)}^{(\alpha_1^1, \ldots, \alpha_n^1)} \sum_{i=1}^n \left\{ \sum_{j=1}^n \sin(a_i - a_j)\right\} {\rm d}a_i = \sum_{i=1}^n \int_{\hat{\alpha}_i^0}^{\hat{\alpha}_i^1} \left\{ \sum_{j=1}^n \sin(a_i - a_j)\right\} {\rm d}a_i $$ where $\hat{\alpha}_i^0$ treats every component of $a$ as fixed $\alpha_j^0$ for $j \neq i$ , which I think would...

One of our user asked: I am reading "Calculus" by Takeshi Saito. In this book, Saito adopts the following axiom of $\mathbb{R}$ . I like this axiom. I think it is easy to understand what this axiom is saying. But I cannot find a book in which this axiom of $\mathbb{R}$ is adopted. Are there any other books which adopt this axiom of $\mathbb{R}$ ? Axiom 1.1.1: 1. If $a$ is a real number, then there exists an integer $n$ such that $n \leq a \leq n+1$ . 2. If $\{a_n\}$ is a sequence such that $a_i \in \{0, 1\}$ for all $i \in \{1, 2, \cdots\}$ , then there exists a real number $b$ such that $$\sum_{n=1}^m \frac{a_n}{2^n} \leq b \leq \sum_{n=1}^m \frac{a_n}{2^n}+\frac{1}{2^m}$$ for all $m \in \{0, 1, 2, \cdots\}$ . By Axiom 1.1.1.1, if $a$ is a real number, there exists a unique integer such that $m \leq a < m+1$ and we define this $m$ as $[a]$ . $[a]+1$ is the smallest integer which is greater than $a$ . Proposition 1.1.2: Let $a, b$ be real numbers...

One of our user asked: This follow my previous post here , where Song has proven that $\forall b>1,\lim\limits_{x\to 1^{-}}\frac{1}{\ln(1-x)}\sum\limits_{n=0}^{\infty}x^{b^n}=-\frac{1}{\ln(b)}$ , that is to say : $$\forall b>1,\sum\limits_{n=0}^{\infty}x^{b^n}=-\log_b(1-x)+o_{x\to1^-}\left(\log_b(1-x)\right)$$ (The $o_{x\to1^-}\left(\log_b(1-x)\right)$ representing a function that is asymptotically smaller than $\log_b(1-x)$ when $x\to1^{-}$ , that is to say whose quotient by $\log_b(1-x)$ converges to $0$ as $x\to1^{-}$ , see small o notation ) So we have here a first asymptotical approximation of $\sum\limits_{n=0}^{\infty}x^{b^n}$ . I now want to take it one step further and refine the asymptotical behaviour, by proving a stronger result which I conjecture to be true (backed by numerical simulations) : $$\sum\limits_{n=0}^{\infty}x^{b^n}=-\log_b(1-x)+O_{x\to1^-}\left(1\right)$$ (The $O_{x\to1^-}\left(1\right)$ representing a function that is asymptotically bounded wh...

One of our user asked: I'm studying for my calculus 1 exam and came across this sample question from the professor's collection: Calculate: $\lim\limits_{n\ \rightarrow\ \infty} \frac{1}{2\log(2)}+\frac{1}{3\log(3)} + \dots + \frac{1}{n\log n}$ (hint: separate into blocks) Unfortunately the sample questions don't include answers and I'm at a loss as to how to proceed; I'd really appreciate some help. Thanks!

One of our user asked: I have the manifold with boundary $M:= \lbrace (x_1,x_2,x_3) \in \mathbb R^3 : x_1\geq 0, x_1^2+x_2^2+x_3^2=1\rbrace \cup\lbrace (x_1,x_2,x_3) \in \mathbb R^3 : x_1= 0, x_1^2+x_2^2+x_3^2\leq1\rbrace$ and I need to find the boundary of this manifold. I think it is $\lbrace (x_1,x_2,x_3) \in \mathbb R^n : x_1= 0, x_2^2+x_3^2=1\rbrace$ , the other option is that the boundary is the empty set? I think the first is right? Am I wrong?