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

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1answer
39 views

The existence of $y$ such that $f(x+y)=f(x)+f(y)$

For a real-valued $C^1(\mathbb{R})$ function with $f(0)=0$. $f'$ is strictly increasing from $-\infty$ to $\infty$ as $x$ goes from $-\infty$ to $\infty$. Prove that for any $x \neq 0$, there exists ...
-8
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0answers
38 views

Real analysis Textbook 2 [on hold]

$f(x)\in L_1(E)$ , where $E\subset \mathbb{R}^n$ is a Lebesgue measurable set.Assume $f(x)>0$ , and show that $$\lim\limits_{n\to-\infty} \int |f(x)|^{1/n}\,dx=|E|$$
1
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1answer
22 views

convergent subsequences etc.

It is well known that: A sequence of real numbers converges $\{a_n\}$ to p, if and only if every subsequence $\{a_{n_k}\}$ converges to p. I am wondering if this similar statement holds.: ...
0
votes
1answer
30 views

Question about convergence proof, why has he chosen the parameter this way

In this proof he says that $n > 2k$, but would it work if $n \ge k$, if not, why? If $p>0$ and $\alpha$ is real, then $\displaystyle\lim_{n\to\infty}\frac{n^\alpha}{(1+p)^n}=0$. Proof: ...
1
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3answers
43 views

Arriving at the value of Limits

According to the formal definition of limits, Let $f(x)$ be a function defined on an open interval $D$ that contains $c$, except possibly at $x=c$. Let $L$ be a number. Then we say that $$\lim_{x ...
0
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1answer
22 views

Distance between 2 points in 3D space (in spherical polar coordinates)

In 3D Euclidean space, we know that distance between 2 points: $a=(x_1,y_1,z_1)$ and $b=(x_2,y_2,z_2)$ is $s^2=(x_2-x_1)^2+(y_2-y_1)^2+(z_2-z_1)^2)$from metric $ds^2=dx^2+dy^2+dz^2$. But I was ...
2
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1answer
29 views

Natural progression in a curriculum for self-study of analysis

Would you list what is a natural and effective progression to self-study topics in analysis in order to gain a broad knowledge of the enormous corpus of knowledge that modern analysis involves. As a ...
0
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1answer
35 views

Help with a 3-body problem

If I have three particles with masses $ m_1, m_2, m_3$ with their respective position vectors $ x_1, x _2, x_3 $ and their speeds $ v_1, v_2, v_3 $ how could I find a parametric function that would ...
2
votes
3answers
72 views

Differentiate Archimedes's spiral

I read that the only problem of differential calculus Archimedes solved was constructing the tangent to his spiral, $$r = a + b\theta$$ I would like to differentiate it but I don't know much about ...
1
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3answers
54 views

Induction on nth polynomial proof.

Question: Prove by induction that $ 1+r+r^2+\cdots+r^n = \dfrac {1-r^{n+1}} {1-r} $ where $ r \in \mathbb{R} $ When $n$ is odd, this is really easy as the right side breaks down to $\dfrac ...
1
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2answers
45 views

How to solve for multiple unknowns using substitution?

$R_1$, $R_2$, $R_3$, $R_4$, $R_5$ and $V_6$ suppose to be 'known' values. $$\frac{V_{n_1}}{R_1} + \frac{V_{n_1}-V_{n_3}}{R_2} + i_6 = 0$$ $$ \frac{V_{n_2}-V_{n_3}}{R_4} + \frac{V_{n_2}-V_{n_4}}{R_3} ...
2
votes
2answers
53 views

show that inequality holds for $n \ge 10$

I want to prove that for $n \ge 10$ holds: $$(n+1)^{\sqrt{n+1}}<n^{\sqrt{n+2}}$$ I know that holds $(n+1)^{{n+1}}<n^{{n+2}}$ which can be proven by induction, but here I don't know how to deal ...
3
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1answer
44 views

How to solve this problem except calculate it directly

$$ \int_0^\infty 1-\left(1-e^{-2w}\right)^8 dw $$ I really don't know how to solve this problem except calculate it directly.
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0answers
18 views

Calculus: Visualizing the bounded region to solve volume

Find the volume of the solid that results when the region bounded by $x=1-y^2$ and the y-axis is revolved around the y-axis. What does this region look like? How can you bind the region around the ...
0
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1answer
21 views

Integral of function and its derivative, multi-dimension

Is the following statement true? If so, how can I prove it? $f = f(x,a)$ $\int \left(f(x,a_1)\int f(x,a_2)dx\right)dx = \frac{1}{2} \int f(x,a_1) dx \int f(x,a_2) dx$
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0answers
35 views

trouble with sequence and series question [on hold]

$x^2f''(x)+xf'(x)+(x^2-1)f(x)=0$ show that $f(x) = \displaystyle\sum_{n=0}^{\infty} \dfrac{(-1)^n}{n!(n+1)!2^{2n+1}}x^{2n+1}$ Dont know to approach this one
2
votes
1answer
39 views

A matrix version of L'Hopital's Rule?

Is there a version of L'Hopital's Rule for matrix calculus? For example: let $A$ be a symmetric $n\times n$ positive definite matrix and $b$ be an $n\times 1$ vector. As $b$ converges to $0_{n\times ...
-1
votes
1answer
33 views

What are some applications of real analysis? [duplicate]

What are some applications of real analysis? Can someone post a simple example of how real analysis can solve such problems?
2
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1answer
43 views

Limit of a function

I am trying to find the limit (If it does exist) $\lim_{n\rightarrow\infty}\left(1-|\mathcal{X}|^{-\alpha n}\right)^{2^{nC}\left(1-|\mathcal{X}|^{-\alpha n}\right)}$, where $0<\alpha<1$, ...
2
votes
2answers
34 views

Evaluate fourier coefficient of $f(t)=t$.

Evaluate the Fourier coefficient of $f(t)=t$. $$\hat{f}(n) = \frac{1}{2\pi}\int_0^{2\pi} te^{-int}dt$$ I'd be glad for help with this calculation. My integration skills need an improvement. My ...
3
votes
2answers
53 views

if $f(z),\overline {f(z)}$ are analytic then they are constant

I'm trying to prove this "theorem": if $f(z),\overline {f(z)}$ are analytic in some open set $\Omega \subseteq \mathbb C$, then $f(z)$ is a constant. Hint: Use Cauchy-Riemann equations to show that ...
2
votes
1answer
38 views

Integral of the convolution of two functions: $\int_{-\infty}^{\infty} (f*g)(x)dx$

There is this proof for the integral of convolution between two functions: $$\begin{align}\int_{-\infty}^{\infty} (f*g)(x)dx&=\int_{-\infty}^{\infty}\left [ ...
1
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1answer
44 views

How to simplify this inequality

I have the following inequality where $i$, $N$ and $p$ are constants, $j$ is a variable and $p_j$ is the chance that 'event' $j$ is happening: $$i\geq -pi+((1-p)\cdot \sum ^N _{j=0}(j\cdot p_j))+\sum ...
0
votes
2answers
41 views

examine if series is convergent

I have problem with $$ a_n=\sum_{n=2}^{\infty}(-1)^{\left\lfloor{\frac{n^3+n+1}{3n^2-1}} \right\rfloor}\cdot\frac{\ln(n)}{n}$$ I'd like to use here a dirichlet's test I know how to show ...
0
votes
1answer
41 views

Cauchy Schwartz inequality and absolute value

Here's the inequality: $$|\langle u,v\rangle|\le\|u\|\cdot\|v\| $$ Why on the LHS there's an absolute value? We know that $\langle u,v\rangle \ge 0$ Isn't it redundant?
0
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0answers
22 views

Show that $\Phi^*_t\mu=\mu \iff \mathrm{div}_{\mu}X=0$.

Let $\mu$ be a non-vanishing $1$-form on $\mathbb{R}^n$. Given a smooth vector field X on $\mathbb{R}^n$, we define the divergence of $X$ wrt $\mu$, denoted by $\mathrm{div}_{\mu}X$, by $L_X \mu = ...
1
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1answer
96 views

I want to learn math from zero

I finished high school 2 years ago and now I'm stuck in a university in Turkey. I am interested in learning precalculus, discrete mathematics, physics and chemistry. Question: I need to learn math ...
1
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2answers
35 views

Implicit Differentiation

If $$y=(12s^4-4s^3+12s^2)/4s$$ then $$\frac{dy}{ds} =9s^2-2s+3$$ but its a multiple choice question that says $9s^2-2s+3$ for all $s$ or another choice that says $9s^2-2s+3$ for all $s$ that do ...
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2answers
34 views

Slope of Line tangent

The slope of the line tangent to the curve implicitly defined by $$y^2+(1-xy)^3=0$$ at $(2,1)$ is? $a)\ -\frac{3}{2}$ $b)\ -\frac{3}{4}$ $c)\ \ \ \ \ \ 0$ $d)\ \ \ \ \ \frac{3}{4}$ $e)\ \ ...
1
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1answer
20 views

Question on first differentiate Equation

How can Differentiate the following function please help ? 1) $u=e^{\frac{-1}y}$ my answer is $u=-e^\frac{-1}{y}$ .Is my solution is correct please ?
0
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0answers
20 views

On solving min-max optimization problems in original ways (that is, avoiding the frenzy of differentiation)

As I see from the students I'm tutoring, once faced with a min-max problem, the average student is taken by the frenzy of differentiation. I would like to show that sometimes it is better to use ...
0
votes
1answer
26 views

Using $e^{ix}$ instead of sine and cosine in contour integration

A while ago I asked: Evaluation of $\int_{0}^{\infty} cos(x)/(x^2+1)$ using complex analysis. Instead of using $\cos(z)$ an answerer said that is valid to use $e^{ix}$ How is this valid? I dont ...
1
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2answers
25 views

Trying to show property of fractional part

We have this sequence $$\left\{\{\sqrt n\}\right\}_{n=1}^\infty\;,\;\;\{\sqrt n\}:=\;\text{the fractional part of}\;\;\sqrt n$$ The exercise is to prove it doesn't have a limit, and we get several ...
0
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1answer
29 views

plotting $\frac{-\pi}{2}<x<\frac{\pi}{2} $ and $ 0<y<1$ under mapping $w=\sin(z)$

i need to plot this $\frac{-\pi}{2}<x<\frac{\pi}{2} $ and $ 0<y<1$ under $w=\sin(z)$ mapping so what i did is $ y=0 , \frac{-\pi}{2}<x<\frac{\pi}{2} => -1<u<1 , v=0 $ $ y=1 ...
-1
votes
1answer
38 views

Suppose all we know about y=f(x) is that it is continuous for all x and f(4)=5. Which must be true?

Suppose all we know about $y=f(x)$ is that it is continuous for all $x$ and $f(4)=5$. Which must be true? a. $f'(4)=5$ b. Every number x is in the domain of f c. The function is increasing near x=4 ...
0
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1answer
33 views

Combinatorial techniques, methods, and ideas in (“undergraduate”) real analysis

This question is dual to Probabilistic techniques, methods, and ideas in ("undergraduate") real analysis: I would like to collect some examples of combinatorial arguments to undergraduate ...
2
votes
1answer
23 views

Power series for inverse of truncated power series of $e$

Let $T_n(x)=\sum_{k=0}^{n}\frac{x^k}{k!}$. I'm looking at the function $$f(x)=\frac{1}{T_n(x)}$$ and I would like to find the power series of this particular function. I know I can use Maclaurin ...
2
votes
4answers
122 views

Calculate the limit : $\lim_{x \to 0}\frac{x-\sin{x}}{x^3}$ WITHOUT using L'Hopital's rule

I was given a task to find $$\lim_{x\to0}\frac{x-\sin{x}}{x^3}$$ at my school today. I thought it was an easy problem and started differentiating denominator and numerator to calculate the limit but ...
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1answer
21 views

Objects sliding on a frictionless surface

An object with the mass 2 kg slides on a frictionless surface. When the velocity of the object is h, the object is subjected to a force (air resistance) that's $5v^2$ N. Apparently the equation is ...
1
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1answer
75 views

use comparison test to show divergence or convergence

I'm not sure if my reasoning is correct. a) $\displaystyle \sum_{n=2}^{\infty} \frac{\ln^5{(2n^7+13)+10\sin(n)}}{n\cdot \ln^6{(n^\frac{7}{8}}+2\sqrt{n}-1)\cdot\ln{\ln{(n+(-1)^n}})} = ...
5
votes
3answers
82 views

Evaluation of $\int_{0}^{\infty} cos(x)/(x^2+1)$ using complex analysis.

Evaluate: $$\int_{0}^{\infty} \frac{\cos(x)}{x^2 + 1} dx$$ Using only complex analysis. $$I = \int_{0}^{\infty} \frac{\cos(x)}{x^2 + 1} dx = (\frac{1}{2})\int_{-\infty}^{\infty} \frac{\cos(x)}{x^2 ...
0
votes
3answers
37 views

fractional part of the square of natural number

How can if prove that the sequence :$$a_n\:=\left\{\sqrt{n}\right\}\left(fractional\:part\:of\:\sqrt{n}\right)\:=\:\:\sqrt{n}\:-\:\left[\sqrt{n}\right]$$ is bounded from above by 1? So far i try ...
2
votes
0answers
25 views

Choosing a contour to integrate over.

What are the guidelines for choosing a contour? For example to integrate a real function with a singularity somewhere. What type of contour from Square, keyhole, circle, etc should be chosen for ...
0
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0answers
23 views

Limit with natural logorithm [duplicate]

Here's a limit I've been stuck on for more than an hour : $$\lim_{x\to0}\frac{\ln(x+1)-x}{x^2}$$ I tried many different ways but I always get something like $\frac00 + 1$ or something like that, ...
0
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0answers
35 views

Reference request: calculus of variations

I am searching for a good book to self-study calculus of variations. It should be fairly complete; build up gradually from the very basics; offer detailed explanations; have some emphasis on ...
0
votes
1answer
32 views

Limit on a continuous differential equation

Let $f$ be a continuously differentiable function and let $L=\lim_{x\to\infty}(f(x)+f'(x))$ be finite. Does this imply that if $$\lim_{x\to\infty} f'(x)$$ exists, then it is equal to $0$?
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votes
1answer
36 views

Equivalent condition for continuity of a function

Let $g: [0,+\infty) \rightarrow \mathbb{R}$ be a continuous function and let $f: [0,+\infty) \rightarrow \mathbb{R}$ be defined by \begin{equation} f(t) = \inf \{ s \geq 0 \,|\, g(s) > t\}. ...
0
votes
0answers
42 views

Where is the maximum of the Gaussian?

The maximum of the envelope $\exp\left(-\frac{x^2}{2\sigma^2}\right)$ is supposed to be at $x=0$. Why is this the case?
4
votes
4answers
147 views

Solutions of $z^6 + 1 = 0$

Solve: $$z^6 + 1 = 0$$ That lie in the top region of the plane. We know that: $$(z^2 + 1)(z^4 - z^2 + 1) = 0$$ $$z = -i, i$$ We need to solve: $$((z^2)^2 - (z)^2 + 1) = 0$$ $$z = \frac{1 \pm ...
4
votes
3answers
191 views

Integration without complex analysis on rational-improper integral

Evaluate: $$\int_{0}^{\infty} \frac{1}{x^6 + 1} \,\mathrm dx$$ Without the use of complex-analysis. With complex analysis it is a very simple problem, how can this be done WITHOUT complex analysis? ...