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

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0
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0answers
10 views

Prove that $[f(x+1)-f(x)] = 0$

I know this is a very elementary level question. But, I still need t0 understand this in term of mean value theorem. So here it goes: Suppose $f$ is differentiable on $(0,\infty)$ and lim$_{x \to ...
0
votes
0answers
16 views

Explicit expression of a sequence

In my research, I am stack in finding the explicit expression of the sequence $(x_k)_{k \geq 1}$ defined as two consecutive terms of this sequence are : $1/(3*k+1)$ and $1/(3*k).$ The same thing for ...
0
votes
0answers
9 views

Solving for a function inside an integral

Is there a way to solve for $f(x)$ when $$ g(x)=\int_0^x dx' W(x,x') f(x') $$ If it weren't for the x-dependence in $W(x,x')$, I could write for example, $$ f(x)=\frac{1}{W(x)}\frac{\partial ...
0
votes
0answers
16 views

Prove an upper bound

Let $O$ be the origin, and $K$ be a convex polygon in $\mathbb{R}^2$ with edges $K_1, \dots K_n$. Let $\nu_i$ be the unit outer normal of each $K_i$. Suppose $O \notin K$. Prove that there exists a ...
3
votes
6answers
67 views

How do you show that $\displaystyle\lim_{x\to 0}\frac{\sin(x)}{\sqrt{x\sin(4x)}} $does not exist?

How can I show that $\displaystyle\lim_{x\to 0}\frac{\sin(x)}{\sqrt{x\sin(4x)}} $does not exist ?
0
votes
0answers
20 views

5 point derivative filter

n many of the paper it is said that 5 point derivative filter transfer function is given by $$H(z) = \left(\frac{1}{8T}\right)(-z^{-2} - 2z^{-1} + 2z + z^{2}).$$ But no one has given detailed ...
4
votes
3answers
77 views

Why is $\int_{0}^{\pi}{1\over 1-\sin x}dx=2\int_{0}^{\pi\over 2}{1\over 1-\sin x}dx$?

Why is $\int_{0}^{\pi}{1\over 1-\sin x}dx=2\int_{0}^{\pi\over 2}{1\over 1-\sin x}dx$, or to be accurate: why is $\int_{\pi\over 2}^{\pi}{1\over 1-\sin x}dx=\int_{0}^{\pi\over 2}{1\over 1-\sin x}dx$? ...
3
votes
4answers
30 views

Unknown both as a exponent and as a term in an equation

Let's say I have an equation $e^{x-1}(x+1)=2$. According to Solving an equation when the unknown is both a term and exponent it's impossible to solve this using elemetary functions. If so, then how do ...
0
votes
2answers
49 views

Differentiate the Function: $h(x)=\ln\ (x\sqrt{x^2-1})$

$h(x)=\ln\ (x\sqrt{x^2-1})$ $\frac{dy}{dx}\frac{1}{y}=\frac{1+1(x^2-1)(2x)}{2(x+\sqrt{x^2-1})}$ $\frac{dy}{dx}\cdot\frac{1}{y}\ (y)=(y)\frac{2(2x^3-2x)}{2(x+\sqrt{x^2-1})}$ $\frac{dy}{dx} =\frac ...
1
vote
5answers
132 views

Limit notation, tried factorising.

I am trying to calculate the following: $$\lim_{x \rightarrow -1} \frac{(2-5x)(x+3)}{x+2}$$ I tried factorizing to cancel of numerator values with denominator values but it was futile I tried ...
0
votes
0answers
19 views

Find two linearly independent solutions of a Legendre equation about $x=0.$

Here is the statement of the problem: Consider the Legendre Equation $$ (*)\qquad (1-x^2)y''-2xy'+2y=0 $$ (a) Find two linearly independent solutions about $x=0$, solving completely any relevant ...
1
vote
0answers
14 views

reversibility scalar conservation law

I am reading here and there (see for instance Denis Serre systems of conservation laws 1- p.36), the following for which I can't spot the mistake I am making that prevents me from arriving to the same ...
2
votes
2answers
97 views

Evaluate the improper integral $\int_{0}^{\infty}{f(x)-f(2x)\over x}dx$, where $\lim_{x \to \infty} f(x) = L$

Find $$\int_{0}^{\infty}{f(x)-f(2x)\over x}\, \mathrm{d}x$$ if $f\in C([0,\infty])$ and $\lim\limits_{x\to \infty}{f(x)=L}$. I tried denoting $\displaystyle \int{f(x)\over x}dx=F(x)$, but I don't ...
4
votes
1answer
104 views

Definite integral with logarithm and arctangent inside of arctangent

How to prove $$\int_0^1 \left[ \frac{2}{\pi }\arctan \left(\frac 2 \pi \arctan \frac{1}{x} + \frac{1}{\pi }\ln \frac{1 + x}{1 - x}\right) - \frac{1}{2} \right]\frac{\mathrm{d}x} x = \frac{1}{2} \ln ...
1
vote
1answer
29 views

Following a simplification of expression

I am struggling with this expression: In particular I get stuck with the simplification from the first to the second line. As far as I can see they replace $\text{m$\ell $}=\mu$. Does the new ...
-2
votes
1answer
30 views

Orthogonal vectors and potential

given the potential $ψ(x;y)$, such that $dψ=−u_2dx+u_1dy$, why are $∇ψ=(−u_2;u_1)$ and $ψ(x;y)=c$ orthogonal vectors ? $c \in \mathbb{R}$ is a constant, and $\mathbf{u}(x; y) = (u_1(x;y); u_2(x;y))$, ...
0
votes
3answers
54 views

Differentiate the Function $g(x)= \ln\ \frac{a-x}{a+x}$

$$g(x)= \ln\ \frac{a-x}{a+x}$$ $$\frac{dy}{dx}\ =\frac{d}{dx}\ \ln \frac{a-x}{a+x}$$ $$g'(x) = \frac{1}{\frac{a-x}{a+x}}\cdot\frac{1}{1}\ \ln\ \frac{a+x}{a-x}$$ $$g'(x)= \frac{a+x}{a-x}$$ This ...
1
vote
3answers
77 views

Proof of there is no limit at $x=0$ for $f(x)=\sin(\frac{1}{x})$

I've seen a few questions posted before about mine, but this is a bit different. The original form of the question can be found here: http://librarun.org/book/10452/159. It says that prove by ...
-1
votes
0answers
32 views

$ΔV$ and $Δh$ of a cone.

Find the exact volume $V_0$ in terms of $\pi$ . Then derive an approximate formula for the height error $Δh$ in terms of the volume error $ΔV$. $d = 3$, $h = 6$ $$V=\pi r^2\frac h3$$ $r=\frac h4$ ...
7
votes
1answer
88 views

A tough integral:$\int_0^{+\infty}\left( \frac1{\log(x+1)-\log x}-x-\frac12\right)^2 dx$

I would like to prove the convergence of $$I=\int_0^{+\infty}\left( \frac1{\log(x+1)-\log x}-x-\frac12\right)^2 dx$$ then obtain a closed form of $I$. Convergence is ensured by the fact that $x ...
0
votes
1answer
21 views

Optimization Example with some constraints !?

We want to optimize the following function: $f(x,y)=x^2+3y^2+2xy+2$ with constraint $-2 \leq x< 2$ $-2 <y<2$ $3y^2+x \leq 10$ Who Can Help me for the above example from my note? My TA‌ ...
0
votes
4answers
91 views

why $ \sum_{k=0}^{\infty} x^{2k} = \frac{1}{1-x^2}\\$

Why $$ \sum_{k=0}^{\infty} x^{2k} = \frac{1}{1-x^2}\\$$ I know that $$ \sum_{k=0}^{\infty} x^k = \frac{1}{1-x}\\$$ can I use the above to derive the first result?
0
votes
5answers
54 views

Finite set of points of $R^n$ is compact

In order to show that a finite set of points of $R^n$ is compact, I just need to show that the set is closed and bounded. First of all, since it's a finite set, I can Always pick the greatest ...
2
votes
3answers
62 views

Differentiate the Function: $f(x)=\ln (\sin^2x)$

$$\begin{align}f(x)&=\ln (\sin^2x)\\ f'(x)&=\frac{1}{\sin^2x}\cdot 2(\sin x)(\cos x)\\ &=\frac{2(\sin x)(\cos x)}{\sin^2x}\\ &=\frac{2\ \ (\cos x)\ }{\sin x}\\ &=2\cot x ...
2
votes
0answers
44 views

Gamma function still hard for me

During my study I find a form for gamma function it was $\Gamma (x) = \lim_{n\to\infty} \frac{n! n^{x-1}}{x(x-1)(x-2)........(x+n-1)}$ And then by simplify this limit I get $$\lim_{n\to\infty} ...
2
votes
3answers
86 views

Differentiate the Function: $ f(x)= x\ln x\ - x $

$ f(x)= x\ln x - x $ Wondering if my answer is right. Here is my process. I will simply find the derivative by using the product and difference rule. $x \frac{d}{dx}[\ln x]+ \ln ...
2
votes
1answer
58 views

What is the d used in calculus?

I know the letter d is commonly used in calculus represents a derivative. Does this d act as a variable that can be simplified or as a function of another variable?
0
votes
1answer
33 views

Existence of such a function

I am supposed to construct a function $f \in C_c^1((-\frac{3R}{4},\frac{3R}{4}))$ such that $f|_{(-\frac{R}{2},\frac{R}{2})}=1$ and $|f'(x)| \le \frac{4}{R}$ for almost all $x \in (-R,R)?$ I ...
0
votes
1answer
37 views

Proof inequality using Lagrange Multipliers

Is it possible: $a,b,c$ are non-negative real numbers for which holds that $a+b+c=3.$ Prove the following inequality: $$ 4\ge a^2b+b^2c+c^2a+abc $$ Is it possible using Lagrange Multipliers. I ...
0
votes
2answers
38 views

A general method for integration of rational function.

$\int\frac {x^3}{1+x^5}$ ATTEMPT: I did the following substitution: Let $x=\frac{1}{t}.$ $dx=\frac{-1}{t^2}dt.$ substituting back: $I=\int\frac{-1}{1+t^5}dt$ which doesn't seems a simpler ...
-2
votes
0answers
30 views

Limit of a recursive sequence containing log [on hold]

Let $\alpha$ be a real number. Consider the following recursive formula: $a_1=1$ and $$a_n=1-\alpha . \sum_{i=1}^{n-1}{a_i\over{i.\log(n-i+1)}} \: \: \: \:for\:\:n\ge2$$ Note that the logarithm is ...
6
votes
0answers
50 views

The quadratic and cubic versions of a tough intregral

In this post, Proving that $\int_0^1 \frac{\log \left(\frac{1}{t}\right) \log (t+2)}{t+1} \, dt=\frac{13}{24} \zeta (3)$, it's proved that $$I_1=\int_0^1 \frac{\log \left(\frac{1}{t}\right) \log ...
1
vote
3answers
43 views

Finding $\lim\limits_{n\to \infty}({1\over n+1}+{1\over n+2}+…+{1\over n+n})$ using integrals [duplicate]

Finding $\lim\limits_{n\to \infty}\left({1\over n+1}+{1\over n+2}+\dots+{1\over n+n}\right)$. I tried many things but it would work out. I am now studying calculus 2 (In my country the first calculus ...
0
votes
3answers
39 views

Can you simplify this expression?

This is a Bayes formula incorporating 2 random variables. The final expression seems a bit tricky to simplify the exponents and I'm still not so confident with my algebra (pardon me ;)). Can you have ...
0
votes
1answer
32 views

$f,g \in R(T)$ such that $\hat{f} \cdot n^{2/3} = \hat{g}$ prove that $f$'s Fourier series converges absolutely.

Can someone help me by checking my solution. Is there a shorter More elegant solution ?(i'm almost sure you can some how express $f$'s Fourier series using $|\hat{g}|^2$ + constant, i saw someone do ...
3
votes
0answers
34 views

Origin of the Integral (Theory Behind It - How it came about)?

How exactly was the integral derived? Like similarly to how the difference quotient explains where the derivative came from, what can we use to explain the origins of the integral? Like how does ...
3
votes
4answers
336 views

Am I using the chain rule correctly?

I'm supposed to find $y'$ and $y''$ of this function: $$y=e^{\alpha x} \sin\beta x$$ This is what I have done so far: $$y'=e^{\alpha x}\sin\beta x\cdot \alpha x'\sin\beta x\cdot \sin'\beta x \cdot ...
1
vote
1answer
35 views

Integral Test question

So this is the problem: http://postimg.org/image/5g815zgk5/ I am getting $\lim_{b\to\infty} 2\sec^{-1}(2b) - 2\sec^{-1}2$ Now what? What do I do with $\sec^{-1}(2b)$? What happens to a trig function ...
0
votes
4answers
84 views

Limit of $\{a_n\}$, where $a_{n+1} = \sqrt{2+a_n}$

I am struggling with this question: Let $\{a_n\}$ be defined recursively by $a_1=\sqrt2$, $a_{n+1}=\sqrt{2+a_n}$. Find $\lim\limits_{n\to\infty}a_n$. HINT: Let $L=\lim\limits_{n\to\infty}a_n$. ...
1
vote
1answer
40 views

Analysis for Engineering : Practical Applications

I don't know much more about Analysis than what I've read about it on Wikipedia, although I have just begun reading Introduction to Calculus and Analysis I, by Richard Courant. My understanding is ...
-2
votes
0answers
40 views

Finding all points on $y=x^2$ for which the normal line goes through the point $(0,3)$. [on hold]

Find the coordinates of all points of the parabola $y=x^2$ for which the normal line goes through the point $(0,3)$. Give exact answers using radicals if necessary. No decimals.
1
vote
2answers
39 views

Finding limit points for these sets

Here's my resoning for finding limit points for some sets. Could you guys read it and see if it's all good? <3 $$\{(x,y)\mid \ x^2+y^2<1\}$$ For this set, its kinda simple to see that every ...
2
votes
1answer
27 views

Order of Rate of Growth

How would you put these functions in order of rate of growth from the greatest to the smallest? $f(x) = \log_2 x$, $g(x) = x^x$, $h(x) = x^2 $, $k(x) = 2^x$ I took the derivatives and ended up with ...
10
votes
4answers
422 views

Evaluating limit (iterated sine function)

The limit is $$\lim_{x\rightarrow0} \frac{x-\sin_n(x)}{x^3},$$ where $\sin_n(x)$ is the $\sin(x)$ function composed with itself $n$ times: $$\sin_n(x) = \sin(\sin(\dots \sin(x)))$$ For $n=1$ the ...
1
vote
0answers
21 views

Matrices derivative

I have a linear product of matrices, I did solve most of it, however, I stop at this component $(X^T W^T D W X)^{-1}$. Given that $X$ is $n \times p$ matrix and $D$ is $n\times n$ matrix. $W$ is a ...
-3
votes
1answer
21 views

How can I find the area of this region? 11 [on hold]

Find the area of the region of the function y=x^2 +2, given [0,1].
2
votes
3answers
70 views

Show that $f$ is bounded.

Let $-\infty<a<b<\infty$. Suppose $f$ is continuous on $[a,b]$. Show that $f$ is bounded on $[a,b].$ We are supposed to use intermediate value theorem for this problem. But, I don't ...
0
votes
0answers
59 views

Show that if $f$ is differentiable as to function $x\mapsto ||x||$ with $x\in R$,then $f'(0)=0$ [on hold]

Let $f\in C^{\infty}(Ω)$ for some open set $Ω \subset R^n$ that contains $0$. Show that if $f$ is differentiable as to function $x\mapsto ||x||$ with $x\in R$,then $f'(0)=0$. I found this problem in a ...
-1
votes
2answers
28 views

Determine subsequence of sequence [on hold]

I know the formal definition of a subsequence, but can't figure out how to find them for some particular sequence. Could someone show some of the methods for finding them? Thanks for replies.
2
votes
2answers
65 views

How can I calculate this limit?

$\lim _{ m\rightarrow \infty }{ \left( \lim _{ n\rightarrow \infty }{ \cos ^{ 2n }{ \left( \pi m!x \right) } } \right) } $ Attempt : since $\cos ^{ 2 }{ x=\frac { 1+cos2x }{ 2 } } $ so we can ...