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

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1answer
43 views

Calculating Legendre Transform

Let $\Omega,\Omega^*$ be bounded domain in $\mathbb{R}^n$, $B_r(y_0)$ is a small ball of radius $r$ center at $y_0\in\Omega^*$, in $\Omega^*$ define a function $\psi(y)=-\sqrt{(r^2-|y-y_0|^2)}$ on ...
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2answers
52 views

Continuity of set inclusion function

Let $A_1$, $A_2$, $A_3$, $\cdots$ be a sequence of nonempty subsets of $[0,1]$. For $x \in [0,1]$, set $a_i (x) = 1$ if $x\in A_i$ and $0$ otherwise. Define $f(x) = (0.a_1 (x) a_2 (x) a_3 (x) \cdots ...
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0answers
30 views

Change of variables to flatten the boundary

It is known that one can perform a change of variables to flatten a $C^2$ domain $\Omega$, that is, for any point $x \in \partial \Omega $, there is a $C^2$ diffeomorphism $\psi$ which maps a ...
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1answer
33 views

The one-sided limits at each point of discontinuity exist

A function is partially continuous if it has at most a finite number of discontinuities and furthermore the one-sided limits at each point of discontinuity exist and they are finite. Does the ...
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1answer
34 views

conditions for Convergence of sequence of functions

Suppose $\{f_n\}$, $n \in \mathbb{N}$ is a sequence of a positive real-valued functions defined on $[0, T]$ and continuous on $(0, T)$. If {$f_n$} satisfies the following conditions : $f_n( iT/2^n ) ...
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1answer
43 views

Extrema and inflection points for $x^3 - 3x^2 + kx$

My girlfriend has a problem with her math task. I did all this stuff years ago when so I am pretty behind and clueless what to do. She has following function: $x^3 - 3x^2 +kx $$ Her tasks are ...
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0answers
27 views

Closure of a linear subspace of $C([a,b])$

Given the space $C([a,b])$ (the collection of all real-valued, continuous (with respect to the metric $d(x,y)=|x−y|)$ functions defined on the interval $[a,b]⊆R$), along with the uniform norm and the ...
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1answer
42 views

Uniform convergence for sequence of functions

Is $f_n(t)$ uniform convergent in $(0,\infty)$ for $$f_n(t)=\frac{\sin(nt)}{n\sqrt t}?$$ I tried and proved it when $t\ge1$, but got confused when it comes to $0$.
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2answers
109 views

Limit involving complicated integral

$$\lim_{x\to\infty} \sqrt{x} \int_0^\frac{\pi}{4} e^{x(\cos t-1)}\cos t\ dt$$ I attempted to work out the integral part, but it did not work well because of the existence of the e part. So whether ...
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1answer
82 views

Why does $u$-substitution not work here?

$$ \int{\frac{1}{2y}dy} $$ Method 1: $$\int{\frac{1}{2y}dy} = \frac{1}{2}\int{\frac{1}{y}dy} = \frac{1}{2}\ln|y|+C$$ Method 2 ($u$-substitution): $$\int{\frac{1}{2y}dy} = \int{\frac{1}{u}dy} = ...
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2answers
71 views

Is the series uniform convergent in $(0,\infty)$?

For $$f(x)=\sum_{n=1}^{\infty}\frac{1}{1+n^2x}$$ And is it bounded in $(0,\infty)$?
2
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1answer
77 views

How prove this interesting identity $(y_{1})^2\cdot y_{2}\cdot y_{3}=x^2_{1}\cdot x_{2}\cdot x_{3}$

let $0<x_{1}<x_{2}<x_{3}$, and there exsit $a$ such $$\begin{cases} y_{1}=x_{1}-\ln{x_{1}}=\dfrac{x^2_{1}}{ax_{1}+\ln{x_{1}}}\\ ...
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2answers
61 views

Question on asymptotes

Consider a function $f: \mathbb{R} \to \mathbb{R}$ that has an asymptote at $- \infty$ of the type $y=\lambda x + \beta$. According to trigonometry $\lambda=\tan{\theta}$ for a very small value of x ...
2
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0answers
26 views

help finding this surface integral

Given that $$S : z = xe^y, \hspace{1mm} 0 \leq x\leq 1, \hspace{1mm} 0 \leq y\leq 1 $$ Find $$\int\int_S \left(x^2+y^2+z^2\right)dS$$ upto four decimal places
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5answers
66 views

Limit of trig identity

So I've come across bit of a stump for myself, and I'm hoping I can get some help. So I know from the squeeze theorem, that limit of $x\sin (\frac1x)$ is equal to zero as $x \to 0$, and that makes a ...
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0answers
44 views

a step function 2

Let $f$ be a function defined by $f: [0,1]\longrightarrow \mathbb{R},\quad t \longmapsto \begin{cases} \lfloor \frac{1}{t} \rfloor & t\in (0,1] \\ 0 & t=0 \end{cases} $ $f$ is it a step ...
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1answer
43 views

Question on surjectivity, involving a function and its derivative

Yesterday I came across this question. It seems very hard to answer without finding an answer to the next question first: Is there a function $f:\mathbb{R}\to\mathbb{R}$ such that both $f$ and ...
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6answers
66 views

What is the difference between two statements of $\varepsilon-N$ definition?

Here is a homework question, TRUE/FALSE: $$\lim_{n\to\infty}a_n=a\Longleftrightarrow$$ $\forall\varepsilon>0,\ \exists N\in\mathbb{Z^+},\ \text{whenever}\ ...
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6answers
2k views

What is the meaning of infinitesimal?

I have read that an infinitesimal is very small, it is unthinkably small but I am not quite comfortable with with its applications. My first question is that is an infinitesimal a stationary value? It ...
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2answers
84 views

How do we solve such an equation

I've been reading about the inverse function theorem and i tried to solve this problem that seems quite elementary: Find where $f:\mathbb{R}^2 \rightarrow \mathbb{R}^2; f(x,y)=(2xy,x^2-y^2)$ is ...
2
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2answers
67 views

Given $\frac {a\cdot y}{b\cdot x} = \frac CD$, find $y$.

That's a pretty easy one... I have the following equality : $\dfrac {a\cdot y}{b\cdot x} = \dfrac CD$ and I want to leave $y$ alone so I move "$b\cdot x$" to the other side $$a\cdot y= \dfrac ...
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3answers
94 views

How do you actually calculate inverse $\sin, \cos, $ etc. ?

I started to wonder, how does one actually calculate the $\arcsin, \arccos, $ etc. without a calculator? For example I know that: $$\arccos(0.3) = 72.54239688^{\circ}$$ by an online calculator, but ...
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1answer
78 views

Showing $\int e^{x^2} dx = \frac12 \sqrt\pi \operatorname{erfi}(x) + C$

Showing $\int e^{x^2} dx = \frac12 \sqrt\pi \operatorname{erfi}(x) + C$ I have seen this identity, but how is it reached? What parts of it make sense and what part is just made into erf?
4
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1answer
103 views

Is there a generalization of the fundamental theorem of algebra for power series?

Given the similarity between polynomials and power series, I was wondering if there is any generalization of the fundamental theorem of algebra for power series. I understand that it doesn't make much ...
2
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1answer
32 views

A question on multivariate integral

Let $f:\mathbb{R} \rightarrow \mathbb{R}$ be a given function. Suppose $\boldsymbol{f}:\mathbb{R}^{N} \rightarrow \mathbb{R}^{N}$ is the vector version of $f$, e.g., ...
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2answers
135 views

$dy\over dx$ is one things but why in integration we can treat it as 2 different terms [duplicate]

when i am learning differentiation, my lectuer tell us that the deriative $dy\over dx$ is one things, it is not the ration between dy and dx. However when i learn about integrating, sometime we need ...
3
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2answers
810 views

Why does the taylor series of $\ln (1 + x)$ only approximate it for $-1<x \le 1$?

I'm looking for an intuitive understanding instead of a formal proof. Thanks for the help.
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1answer
81 views

Weighted Singular Value Decomposition

Lemma: $\forall A\in R^{n\times n}$ and a diagonal matrix $\forall W\in R^{n\times n}$ with $ w_{11}\geq w_{22}\geq ...\geq w_{_{nn}} >0$. The singular value decomposition of A denoted by: $A=XM ...
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0answers
29 views

Show that $\{w^{1/2}\phi_n\}$ is an orthonormal set in $L^2(D)$ if $\{\phi_n\}$ is an orthonormal set in $L^2_w(D)$

As mentioned in the title, my problem is: Show that $\{w^{1/2}\phi_n\}$ is an orthonormal set in $L^2(D)$ if $\{\phi_n\}$ is an orthonormal set in $L^2_w(D).$ So I know that: ...
1
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2answers
61 views

Why $N$ is not the function of $\varepsilon$?

Here is a calculus question: $N$ is not the function of $\varepsilon$ in $\varepsilon-N$ definition. I cannot understand why? In my opinion, every time we try to find a suitable $N$ which based on ...
0
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1answer
41 views

Proof that the unit tangent vector has length $1$?

I want to prove that $|dr/dS| = 1$. I know that this can be proven using the chain rule, I'm just not sure how. Any help would be appreciated.
3
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4answers
120 views

How to show that $x_n=-\sqrt{n} + n\ln\Big(1+\frac{1}{\sqrt{n}}\Big)$ is decreasing?

I am a non-mathematician who knows some elemententary calculus ans I want to prove that the sequence $(x_n)$ given by $$ x_n=-\sqrt{n} + n\ln\Big(1+\frac{1}{\sqrt{n}}\Big) $$ is decreasing. Is there ...
0
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1answer
44 views

a step function

Let $f$ be a function defined by $f: [0,1]\longrightarrow \mathbb{R},\quad t \longmapsto \lfloor tn \rfloor \quad \forall n \in \mathbb{N}^{*}$ Show that $f$ is step function indeed, let ...
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3answers
44 views

Convergence or Divergence using Limits

Use a comparison test to determine whether or not the following improper integrals converge or diverge $$\int_{2}^\infty \frac{1}{\ln(x)}dx.$$ I'm stuck thinking of a function to compare it to.
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2answers
25 views

Convergent or Divergent using Limits

Using limits evaluate the following improper integral or show it is divergent $$\int_{-\infty}^\infty x^3e^{-x^4}dx$$ I did $$\lim_{c\to-\infty}\int_{c}^0 x^3e^{-x^4}dx+\lim_{b\to\infty}\int_{0}^b ...
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4answers
76 views

Integrate using substitution

Integrate using U substitution only$$\int x(5x-1)^{19}$$I used $u=x^2$, and got $$ \frac{x(5x-1)^{20}}{100} + C$$which happend to be wrong. Please help, I'm stuck.
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0answers
42 views

Calculate the first partial derivatives

Calculate the first partial derivatives of the given functions at $(0,0)$ by using the definition 1) $$\begin{align}f(x,y) & = \frac{2x^3 -y^3}{x^2 +3y^2}, & \text{ if } (x,y) \neq (0,0) \\ ...
3
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1answer
94 views

Proving $\cos x < 1 - \frac{x^2}{2} +\frac{x^4}{24}$

I wish to prove the following inequality for $x\ne 0$: $$\cos x < 1 - \frac{x^2}{2} +\frac{x^4}{24}$$ Using the fact that I already prove: $$\cos x > 1 - \frac{x^2}{2}$$ My try: $\cos x = 1 - ...
4
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2answers
50 views

$u_{n}>u_{n+1}>0$ and $u_{2}+u_{4}+u_{8}+u_{16}…$diverges. prove that $\sum \frac{u_{n}}{n}$ diverges.

$u_{n}>u_{n+1}>0$ and $u_{2}+u_{4}+u_{8}+u_{16}.....$diverges. prove that $\sum \frac{u_{n}}{n}$ diverges. The only thing i found is that $\left | \frac{u_{2^{n+1}}}{2u_{2^n}} \right |>1$.
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0answers
29 views

One-side-discontinuous Riemann-Stieltjes integrator

Let $g:[a,b]\to\mathbb R$ be the indicator function of $(c,b\,]$, with $a<c<b$. Show that a function $f:[a,b]\to\mathbb R$ is Riemann-Stieltjes integrable with respect to $g$ if and only if it ...
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3answers
65 views

Composing a function with its antiderivative

What does it mean that, given a function and its antiderivative, if I make the composition $ g(h(x))=h(g(x)) = \alpha(x) $? I mean, I was thinking that $ \alpha(x) $ could be some sort of identity ...
2
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1answer
172 views

Integrate $\int^{\ln(2)}_0 (3e^u - e^{2u} - 2)\sin(nu)du$

I'm having trouble integrating this function $$\begin{equation} \begin{split} f(x) & = \int^1_0x(1-x)\sqrt{1+x}\sqrt{1+x}\sin(n \ln(1+x))/[(1+x)^2] = \\ & = ...
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0answers
20 views

Differential calculus on locally convex spaces

For real finite dimensional vector spaces $V,W,Z$, i know that a map $f:V \times W \to Z$ is smooth if the maps $f(v,.)$ and $f(.,w)$ are for every $v\in V, w \in W$. Does the same thing hold for ...
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3answers
98 views

Give an example about limits: $\lim\limits_{x\to0}f(x^{2})$ exists but $\lim\limits_{x\to0}f(x)$ does not.

Can someone help me find an example where $\lim\limits_{x\to0}f(x^{2})$ exists but $\lim\limits_{x\to0}f(x)$ does not. Thanks.
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2answers
56 views

How to prove this series to be convergent?

For$$\sum_{n=1}^{\infty}ne^{-na}$$ I have no idea about it.Is there any way to deal with it?thx~
1
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1answer
101 views

Prove that all constant functions are in a linear subspace of $C([a,b])$.

Given the space $C([a,b])$ (the collection of all real-valued, continuous (with respect to the metric $d(x,y)=|x-y|$) functions defined on the interval $[a,b]\subseteq \mathbb{R}$), along with the ...
2
votes
2answers
47 views

An object is travelling in a straight line. Its distance, s meters, from a fixed point at time t seconds is given by the expression

$$s=t^3−t^2−6t$$ a) Find ds/dt when t=3 and interpret this result. b) Find d^2s/dt^2 when t=3 and interpret this result. c) Find the time in seconds when the velocity is 2m/s (d) Using the ...
2
votes
3answers
140 views

Differentiate the function into the simplest form

My question: $y=\sin^{2}(x)$ My attempt: Is $\sin^{2}x$ the same as $(\sin(x))^2$? By rearranging the function I came up with the following. $$ \begin{align} u = \sin(x), \ & y=u^2 \\ ...
1
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3answers
59 views

Finding the equation of the tangent

I was working on the following question: The curve $C$ has equation $y=(x+3)^2$ and the point $A$, with $x$ coordinate $-5$, and lies on $C$. $a)$ Find the equation of the tangent line to $C$ at ...
2
votes
1answer
67 views

Volume and stationary values

I don't understand how to do any of the above questions, would someone be able to give me some tips on what functions i'm supposed to use?