# Tagged Questions

For basic questions about limits, derivatives, integrals, and applications, mainly of one-variable functions.

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Hi I am attempting problem 8 (Chapt2 Evans PDE). Again I found the solution on the internet. enter link description here I understood much of everything of the proof except for one line. " Since $x=\... 2answers 239 views ### How to prove Raabe's Formula [duplicate] For quite some time, I've been trying to prove Raabe's Formula, or in other words: $$\int_a^{a+1} \ln\bigg(\Gamma(t)\bigg)dt=\dfrac{1}{2}\ln(2\pi)+a\ln(a)-a$$ This is how I tried: $$I(s)=\int_a^{... 5answers 62 views ### Show that this limit is related to Euler number I am calculating the limit \lim_{n \rightarrow \infty} \left( \frac{n!^{\frac{1}{n}}}{n} \right)= \frac{1}{e}. I got this limit from wolframalpha, but don't know how to show this.wolframalpha 1answer 124 views ### How to integrate \int^{\infty}_{-\infty} e^{-2\pi^2/x^2} dx? I am wondering how can i integrate this quantity above? Here it is again,$$\int^{\infty}_{-\infty} e^{-2\pi^2/x^2}dx.$$Thanks a lot. 1answer 29 views ### Calculating the volume of a solid of revolution about a line. A figure is formed by revolving the region bounded by f(x) = \cos{(x)} and g(x) = \sin{(x)} from 0 to \dfrac{\pi}{4} about the line y=-1. This figure is formed by integration of two ... 5answers 136 views ### f,g diffirentiable function at point (x_0, y_0) how to show that fg diffirentiable function at point (x_0, y_0)? I guess there is pretty simple way of showing the statement below.. I tried using definition but it seem complicated. Suppose f, g: \Bbb R^{2} \to \Bbb R. Prove if f, g are differentiable at ... 1answer 34 views ### finding infimum find the infimum and supremum of E=\{x \in \mathbb{R}:x=\frac{2}{n}+(-1)^n, n\in \mathbb{N}\} Max(E)=2 therefore it is also the Sup(E) Let assume that there is -1<m: m\in E so -1 is ... 0answers 79 views ### Determine null, extreme and inflection points of function f(x)=\frac{x+e^x}{x-e^x} This function has a null point, but I can't compute it from equation f(x)=0 which gives$$\frac{x+e^x}{x-e^x}=0x+e^x=0$$How to compute this equation? Extreme points can be computed from ... 1answer 348 views ### Volume of the intersection of two cylinders I have two infinite cylinders of unit radius in \mathbb{R}^3, whose axes are skew lines. Say that the axis of one is centered on the x-axis, and the axis of the other is determined by the two ... 1answer 65 views ### What is the definition of a gradient? It has been a while since I have done any vector calculus, is this statement true? \nabla f(x,y,z) = 0 \iff \dfrac{\partial f}{\partial x} + \dfrac{\partial f}{\partial y} + \dfrac{\partial f}{\... 1answer 52 views ### There are two periodic functions f(x) and g(x), provide an example when f(x)*g(x) is unbounded, and f(x)+g(x)=0 has infinitely many solutions There are two periodic functions f(x) and g(x) which are defined on \mathbb{R}, provide an example when f(x)\cdot g(x) is unbounded, and f(x)+g(x)=0 has infinitely many solutions ? 1answer 43 views ### help with wrong result for v(x) = \frac{\sqrt[3]{x-1}}{(x+2)^2} I need to differentiate this:$$v(x) = \frac{\sqrt[3]{x-1}}{(x+2)^2}$$I used this formula:$$ \frac{f'(x)g(x) - f(x)g'(x)}{g^2(x)}$$Where:$$ f'(x) = \frac{1}{3\sqrt[3]{(x-1)^2}}$$and$$ g'(x) ... 3answers 677 views ### To prove a sequence is Cauchy [duplicate] I have a sequence:$ a_{n}=\sqrt{3+ \sqrt{3 + ... \sqrt { 3} } } $, it repeats$n$-times. and i have to prove that it is a Cauchy's sequence. So i did this: As one theorem says that every ... 1answer 242 views ### Classifying peak and valley *regions* of a histogram I've been playing with a few ways of classifying contiguous regions of a histogram as: 1) peak, 2) valley, or 3) in-between bit. Global thresholding has worked minimally well for me so far, but I'm ... 4answers 159 views ### To check the convergence of an integral (2) I tried to check if this integral is convergent:$\int _{-\infty }^{\infty }\left(\frac{\sin\left(x\right)\ln\left|x\right|}{1+x^2}\right)dx\:$so, there are 3 points to check:$\pm\infty$and$0$. ... 2answers 63 views ### To check the convergence of an integral I tried to find out if this integral is convergent or divergent, $$\int _0^{\frac{\pi}{2}}\left(\frac{\ln\left(\sin x\right)}{\sqrt{x}\:}\right)\:dx$$ I know that the problematic point is near$x=0$,... 1answer 28 views ### Multivariable calculus: what principle is this step based on? The background is that I was asked to solve the following problem using Green's formula$L$is a Jordan curve (smooth and closed) which encloses the origin point in$xOy$plane. Caculate this ... 2answers 54 views ### Finding the derivative of an integral with variable limits:${\mathrm{d} \over \mathrm{d}x}\int_{x}^{x^2}{1 \over -2y}e^{-5xy^{2}}\mathrm{d}y$? How do you compute the derivative $${\mathrm{d} \over \mathrm{d}x}\int_{x}^{x^2}{1 \over -2y}e^{-5xy^{2}}\mathrm{d}y$$ where the integral has variable limits? 3answers 55 views ### what is the value of$\int \sin(x)\cos(x)dx$?$\frac{\sin^2(x)}{2}$or$\frac{-\cos^2(x)}{2}$or$\frac{-\cos(2x)}{4}\int \sin(x)\cos(x)dx = \frac{\sin^2(x)}{2}$because $$\frac{d}{dx}\frac{\sin^2(x)}{2}=\sin(x)\frac{\sin(x)}{dx}=\sin(x)\cos(x)$$ but also $$\frac{d}{dx}\frac{-\cos^2(x)}{2}=-\cos(x)\frac{\cos(x)}{... 1answer 34 views ### Finding the volume of a cone with and oblique base. The base of S is an elliptical region with boundary curve 9x^2+4y^2=36. Cross-sections perpendicular to the x-axis are isosceles right triangles with hypotenuse in the base. The base of S is ... 1answer 25 views ### How proceeded author from (3) to get a_n<f(n)-f(0)<1 We have f:(-1,\infty)\rightarrow\mathbb{R}, f(x)=\frac{x}{x+1} and$$a_n=\sum_{k=1}^n f(k)-\int_0^1 f(x)dx$$We need to prove that a_n is bounded. I don't need another method, I want to help ... 1answer 74 views ### If k is a non-zero constant, determine by inspection the indefinite integral of \int e^{kx} dx. I have to solve this exercise: If k is a non-zero constant, determine by inspection the indefinite integral of \int e^{kx} dx. By inspection, I guess it means that it should be solved by ... 4answers 231 views ### Why doesn't using the approximation \sin x\approx x near 0 work for computing this limit? The limit is$$\lim_{x\to0}\left(\frac{1}{\sin^2x}-\frac{1}{x^2}\right)$$which I'm aware can be rearranged to obtain the indeterminate \dfrac{0}{0}, but in an attempt to avoid L'Hopital's rule (... 2answers 68 views ### Problem with understanding a Differential in Multivariable Calculus I have just started with Partial Differentiation and the book from where I'm learning (Mathematical Methods in the Physical Sciences) had the following problem on approximations using differentials ... 4answers 52 views ### f is integrable & continuous over [a,b] , \int_{a}^{b}f(x)dx \geq 0 for any subinterval (\alpha,\beta) of (a,b), then f \geq 0 in [a,b] Some known things about this problem are: if f(c) < 0, a < c < b, then f(x) < f(c)/2 in some neighborhood of c, but I am not exactly sure how to use this to get to my goal of ... 2answers 43 views ### If f is integrable on [a,\ b] and \int_a^b f(x) \mathrm dx >1, then there exists a point c in (a,\ b) such that f(c) > \frac{1}{b-a} So far for this problem, to my understanding, for something to be integrable means that U(p,\ f) - L(p,\ f) < \epsilon but not sure how exactly to move beyond there to show that there exists a ... 1answer 51 views ### Prove that e^x-x\geq 1. [duplicate] Please prove that$$e^x - x$$is always bigger than or equal to 1 1answer 59 views ### Evaluate \lim_{\alpha \to \infty} e^{-t\sqrt{\alpha}}(1-\frac{t}{\sqrt{\alpha}})^{-\alpha} How does one show$$\lim_{\alpha \to \infty} e^{-t\sqrt{\alpha}}\left(1-\frac{t}{\sqrt{\alpha}}\right)^{-\alpha} = e^{t^2 / 2}?$$Not homework, this is from this proof that the gamma distribution ... 3answers 71 views ### prove continuity Let f:\Bbb R \to \Bbb R satisfy the property f(x+y)=f(x)+f(y) for all x,y in \Bbb R I have to show that 1)f(0)=0 , f(-x)=-f(x), for all x in \Bbb R, and f(x-y)=f(x)-f(y) y in ... 1answer 297 views ### Second Mean Value Theorem for Integrals Meaning The Second Mean Value Theorem for Integrals says that for f (x) and g(x) continuous on [a, b] and g(x)\ge 0$$\int_a^bf(x)g(x)\,dx=f(a)\int_a^cg(x)\,dx+f(b)\int_c^bg(x)\,dx$$I have a ... 1answer 32 views ### Generalized linear combination of probability density functions I am working with linear non-unity combinations of independent variables in the equation form of:$$Y_i=\sum_{j=1}^N a_{ij} X_j ~~~~\forall~ a_{ij} \in \mathbb{R}, a_{ij}\neq 1$$I am aware of the ... 1answer 72 views ### double integral problem \iint e^{\frac{x}{x+y}}dxdy I'm trying to integrate$$\iint e^{\frac{x}{x+y}}dxdy$$where y \leq (1-x) and 0 \leq x,y \leq 1. I tried to define new variables as u=x and v=x+y, but I can't solve this either. I have ... 1answer 76 views ### Multiple Integrals$$\int _{ 5 }^{ 20 }{ \int _{ 5 }^{ 20 }{ \int _{ 5 }^{ 20 }{ \int _{ 5 }^{ 20 }{ \ln(w+x+y+z) }\ dw\; dx\; dy\; dz } } }$$Unfortunately I cannot think of how to approach this problem. The only ... 2answers 84 views ### Infinite Series for Arctan [closed]$$ \sum_{k=1}^{\infty}\arctan\left(\frac{1}{k^2}\right) $$Does anyone know how to determine if this infinite series diverges or converges and if it converges, what its value is? 1answer 97 views ### Why does a differential form represent a vector field? I'm trying to learn the Divergence/Stoke's theorem and I can't wrap my head around the meaning of a differential form in this context. What does it mean that a differential form represents a vector ... 2answers 45 views ### How to solve \lim_{x\to-\infty} \frac{\left|x + 1\right|e^{-x}}{x} ? I'm trying to solve this limit$$\lim_{x\to-\infty} \frac{\left|x + 1\right|e^{-x}}{x} $$but I get stuck with$$\lim_{x\to-\infty} -\frac{xe^{-x} + e^{-x}}{x} $$I've tried to transform it in ... 1answer 75 views ### Is this simple calculus proof formal enough and correct? There is function f differentiable at x=0 and f'(0) = m > 0, f(0) = 0. I need to prove that there is K > 0 and \delta > 0 that for every 0<x<\delta : f(x) > Kx. So I ... 1answer 54 views ### compute \nabla f for a function over a cone Let D be the cone D=\{rt:r>0, t\in\Omega\} with \Omega\subset S^{n-1}. I want to show that$$ \frac1{r^2}\int_{B_r}\frac{|\nabla f(x)|^2}{|x|^{n-2}} dx= C(n,g)r^{2(a-1)} $$where C(n,g) is ... 1answer 49 views ### Differential of a tricky function I have a function that I'm strugling to take the differential of.$$F(t) = F(t-a)G(t).$$My attempt is the following:$$ dF(t) = F(t-a)dG(t) + G(t) dF(t-a)) $$but I have a feeling something is not ... 1answer 62 views ### Sum on integration and binomial theorem. If (1+x)^n = \sum_{r=0}^n \binom{n}{r}x^r and$$\sum_{r=0}^n \frac{(-1)^r}{(r+1)^2} \binom{n}{r} = k\sum_{r=0}^n \frac{1}{r+1}$$Then prove that$$k=\frac{1}{n+1}.$$2answers 110 views ### Calculate definite integral \int_0^{\pi/2} 3\sin x\cos x/(x^2-3x+2)\; dx [closed] Please help to calculate definite integral$$ \int_0^{\pi/2} \frac{3\sin x\cos(x)}{x^2-3x+2} \, dx. $$I feel that there is a trick somewhere, but I cannot understand where? 4answers 108 views ### Limit (e^x+x)^{1/x}, when x\to 0 Can I expand e^x in the limit$$\large{\lim _{x\to 0}(e^x+x)^{1/x}},$$just as 1+x according to the Taylor expansion? I mean is it normal to think about limits of a form (1+x+o(x))^{1/x} just as ... 4answers 58 views ### Translating basic limit intuition to epsilon delta definition Early on in precalculus/calculus I learned that a$$\lim_{x\to c} ~ f(x)$$does not exist if$$\lim_{x\to c^{+}} ~ f(x) \neq \lim_{x\to c^{-}} ~ f(x)$$I'm having trouble understanding how to ... 1answer 58 views ### A problem about prism with triangular bases Consider a prism with triangular base . The total area of the three faces containing a particular given vertex is$k$. Then is the maximum possible volume of the prism$\sqrt {\dfrac {k^3} {54} } $? ... 1answer 45 views ### ode and area of triangle Question: find a curve$x$so that the area bounded between it's tangent at some point$t$and the time axis on the interval between the point of contact of$x$and it's tangent ($t$), and the ... 0answers 56 views ### Minimal boundary conditions for divergence theorem I've noticed that some domain conditions of questions here were only supposed to be finite dimensional and bounded. And then the divergence theorem was applied in the answers. But if I'm not mistaken, ... 3answers 118 views ### If a function is defined on the interval$(a, b)$, is the derivative necessarily defined at$a$and$b\$?

I am asked to prove something that assumes this. But is it true that derivative is necessarily defined at the "edges" of the domain of the definition of its function? Does it matter if the original ...