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

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3
votes
3answers
107 views

integral of $\sin(\ln(x))\,dx$

I tried to calculate this integral: $$\int \sin(\ln(x))\,dx$$ but it seems my result is wrong. My calculation is: $$\int \sin(\ln(x)) \,dx = \left| \begin{array}{c} u=\ln x \\ du=(1/x)dx\\ dx=...
0
votes
1answer
89 views

Why does a line integral not depend on the parametrization you use?

I have a question about my calculus course: Why is it true that a line integral over a certain functiondoes not depend on the parametrization you use?. For example, take a function $f(x,y,z)$ of 3 ...
9
votes
3answers
159 views

difficult problem in riemman integrals

Could anyone help me with the following problem? Because i have stuck. problem Let $f:[a,b]\rightarrow [0,\infty)$ be continuous and not the zero function. Prove that $$\lim_{n\to \infty} \frac{\int\...
1
vote
1answer
141 views

Evans PDE, Problem 8 Chapter 2 clarification on $|x-y|$

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=\...
3
votes
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^{...
1
vote
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
1
vote
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.
0
votes
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 ...
2
votes
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 ...
0
votes
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 ...
1
vote
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 $$\frac{x+e^x}{x-e^x}=0$$ $$x+e^x=0$$ How to compute this equation? Extreme points can be computed from ...
9
votes
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 ...
0
votes
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}{\...
0
votes
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 ?
1
vote
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) ...
4
votes
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 ...
2
votes
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 ...
1
vote
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 $0$. ...
2
votes
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 $x=0$,...
0
votes
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 ...
0
votes
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?
1
vote
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)}{...
0
votes
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 ...
0
votes
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 ...
1
vote
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 ...
9
votes
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 (...
1
vote
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 ...
2
votes
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 ...
1
vote
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 ...
-1
votes
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
2
votes
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 ...
1
vote
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 $...
2
votes
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 ...
3
votes
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 ...
5
votes
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 ...
1
vote
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 ...
-1
votes
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?
3
votes
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 ...
-1
votes
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 ...
2
votes
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 ...
0
votes
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 ...
1
vote
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 ...
0
votes
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 $$k=\frac{1}{n+1}.$$
-2
votes
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?
1
vote
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 ...
2
votes
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 ...
-3
votes
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} } $ ? ...
-1
votes
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 ...
0
votes
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, ...
1
vote
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 ...