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

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1
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4answers
37 views

Find $\lim_\limits{x\to 1-0}{\ln x\cdot \ln(1-x)}$.

Find $\lim_\limits{x\to 1^-}{\ln x\cdot \ln(1-x)}$. I can't even start because I don't really know what $x\to 1^-$ means. If you know what it means it would really help me. I would as well if you ...
1
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0answers
21 views

Equivalent of $\int_0^{\pi/2}\cos^n(\sin(x))dx$

Let $\displaystyle u_n=\int_0^{\pi/2}\cos^n(\sin(x))dx$. How can I find an equivalent of $u_n$ when $n\to\infty$ ?
3
votes
1answer
21 views

If $f$ and $g$ are uniformly continuous on $\Bbb{R}$ then $f\circ g$ is uniformly continuous on $\Bbb{R}$

Prove or disprove: If $f$ and $g$ are uniformly continuous on $\Bbb{R}$ then $f\circ g$ is uniformly continuous on $\Bbb{R}$. I think there's something crooked in my attempt. I would like to know what ...
1
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2answers
15 views

Expression for Taylor's formula with a remainder

Assume $f$ has a continuous second derivative $f~''$ in some neighborhood of $a$.Then, for every $x$ in this neighborhood, we have $f(x) = f(a) + f~'(a)(x-a) + E_1(x)$ , where $E_1(x) = \int_a^x ...
-1
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0answers
16 views

Arc length of a curve in Elliptic Co-Ordinates

I have a homework question where I must find the arc length of a curve in elliptic co-ordinates given the parametric equations $$h = h(t)$$, $$g = g(t)$$ I simply said $x=a Cosh[u] Cos[v]$ and $y = ...
0
votes
4answers
34 views

Given $\int_0^x (x-t+1)g(t)\,\mathrm{d}t = x^4 + x^2,$ Find $g(x)$

(Stanford Math Tournament 2012 #7) A differentiable function $g$ satisfies $$\int_0^x (x-t+1)g(t)\,\mathrm{d}t = x^4 + x^2,$$ Find $g(x) \, \forall x \geq 0.$ My attempt: First distribute the ...
1
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4answers
61 views

Limit problems and quandaries: finding $\lim_\limits{n\to \infty } {({n^2-n\over n^2+1})^{n+10} }$.

Find $\lim_\limits{n\to \infty } {({n^2-n\over n^2+1})^{n+10} }$. What I did is: $\lim_\limits{n\to \infty }{({n^2-n\over n^2+1})^{n+10}}=\lim_\limits{n\to \infty } {({n^2+1-1-n\over ...
1
vote
1answer
28 views

Prove that $\lim_{n \rightarrow \infty} \int_0^1 f_n(x)dx \ne \int_0^1\lim_{n \rightarrow \infty}f_n(x) dx$

If $f_n(x)=nxe^{-nx^2}~\forall~n=1,2,\cdots$ and $x$ real, show that $$\lim_{n \rightarrow \infty} \int_0^1 f_n(x)dx \ne \int_0^1\lim_{n \rightarrow \infty}f_n(x) dx$$ Attempt: By the $Mn$ Test, it ...
1
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2answers
24 views

Does this supremum equal infinity?

This is a generalization of the previous question Does this infinum tend to infinity? Let $f:\mathbb{R}^2\to\mathbb{R}$ be a continuous function satisfying $$\sup ...
0
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0answers
8 views

Courant. Real numbers determined by nested sequences of rational intervals.

In his book Introduction to Calculus and Analysis vol.1, page 95 Courant writes: Every nested sequence of intervals with real end points contains a real number. To prove this, consider closed ...
0
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0answers
46 views

A lower bound for $\log\left( \frac{a+x^2}{b+x^2}\right)$

I am looking for a tight lower bound for $$f(x)=\log\left( \frac{a+x^2}{b+x^2} \right)$$ $x>0$ and $1<b<<a$. I didn't check for convexity analytically, but I plotted this function ...
1
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0answers
16 views

Rodrigues formula Associated Laguerre polynomial

Could you find the rodriguez formula of $$L_n^{\beta }\left(x^2\right)$$ knowing that $$\frac{\left(e^x x^{-\beta }\right) \frac{\partial ^n\left(e^{-x} x^{\beta }\right)}{\partial ...
1
vote
2answers
16 views

Indefinite integrals with rati0nal and polynomial functions and Substituion

I am totally confused with the substitution method of evaluating indefinite integrals, especially those with rational functions and polynomials. I have 2 cases, which if I made to understand, would ...
2
votes
4answers
33 views

How to evaluate the limit $\lim_\limits{x\to 0+ } \frac{1}{\sqrt{x}}\left ( \frac{1}{\sin x} - \frac{1}{x}\right )$?

$$\lim_{x\to 0^+ } \frac{1}{\sqrt{x}}\left ( \frac{1}{\sin x} - \frac{1}{x}\right ) =\ ?$$ I rearranged it as $$\lim_{x\to 0^+ } \frac{x-\sin x}{x\sqrt{x}\sin x} = \lim_{x\to0^+ } \frac{x-\sin ...
1
vote
1answer
27 views

nth derivative of ${1\over x}$. A problem. [on hold]

$f(x)=f^{(0)}(x)=x^{-1}$, $f^{(1)}(x)=-x^{-2}$, $f^{(2)}(x)=2x^{-3}$. Therefore, $f^{(n)}(x)=(-1)^{n}n!x^{-n-1}$. Except I see in some places that the expression is different, using, for example, ...
3
votes
2answers
176 views

Limit at Infinity: $ \lim\limits_{n\rightarrow\infty}n\left(1-\frac{1}{\ln(n)}\right)^n$

Maple says that this limit is zero but I can't prove it. Any help or tips would be appreciated. $\displaystyle\lim_{n\rightarrow\infty}n\left(1-\frac{1}{\ln(n)}\right)^n$
0
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2answers
53 views

How do I solve $\int_{0}^{\infty} \frac{\ln(x)}{1+x^{2}}\,dx$?

If we first split the integral into two: $$\int_{1}^{\infty} \frac{\ln(x)}{1+x^{2}}\,dx$$ and $$\int_{0}^{1} \frac{\ln(x)}{1+x^{2}}\,dx$$ Let $x = 1/u$ and $dx = -1/u^2 du$, then we have: ...
0
votes
1answer
18 views

Velocity of an oscilating particle

I am working on an assignment where I have an equation in which I am to calculate X. The equation, which describes the velocity of a oscilating particle, is as follows. $v_x = 52cos(700t+0.2\pi)$ ...
0
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0answers
14 views

Questions on curvilinear asymptotes

I just saw curvilinear asymptote which sort of fascinated me. A little bit of thinking raised two questions for which I couldn't get the the answer by googling. Is there a general method to find a ...
0
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0answers
24 views

What is the proof of this procedure

This wikipedia article describes the general procedure for finding the Asymptote of algebraic curves without mentioning any proof. I tried googling but it produced no relevant results. where can I get ...
0
votes
0answers
19 views

Does isometry preserve volume on open sets?

Suppose there are two open sets $A,B$. $h$ is an isometry. And the function $h$ maps $A$ to $B$; $h(A)=B$. I need to show that isometry is volume preserving. My idea is that if the continuous ...
0
votes
4answers
55 views

2003 Putnam A-1 Help needed about sequences

Okay so for $n=1$ there is only one way. For $n=2$ you have, $1+1, 2 + 0$ for $n=3$ you have: $1+1+1, 1+ 2, 3 + 0$ three ways. So $P(n): n$ ways, we must prove the $P(n+1): n + 1$ statement is ...
0
votes
1answer
29 views

Using set theory to prove a function problem

I begin with: $$A = \{a \le x < x_0 | f(x) = 0 \}$$ $$B = \{x_0 < x \le b | f(x) = 0 \}$$ Let $c = \sup A$ and let $d = \sup B$ First to prove $f(x) > 0$ for $x \in (c, d)$ I will ...
1
vote
2answers
31 views

Existence of function with a hard derivative.

I had the question: does there exist diferentiable function $\;f(x)\;$ in the reals such that for $\;x\neq 0\;$ we have $$f'(x)=\frac{e^{1/x}+1}{e^{1/x}}$$ I know that $\;f'(0)\;$ exists because it ...
0
votes
1answer
41 views

what is going on here?

Suppose we have a function $f(x), D:( -\infty,0)\cup (0,\infty)$ and for which $$f'(x) = \frac{x^3-1}{x^3} $$ Apparently there is only one point of extremum here, $x=1$, however upon reviewing the ...
2
votes
0answers
64 views

Question about proof particular L'Hospital's case

My brain is not exactly understanding a particular proof for the L'Hospital's case when $x$ goes to infinity. The author considers $\lim\limits_{x\to+\infty}\frac{f(x)}{g(x)}$ where he subs $t=1/x$ It ...
1
vote
1answer
45 views

${f \text{ is differentiable on } I \iff f_{\left|\ [a,b]\right.} \text{ is differentiable }\ \forall a,b \in I}$

Let $f\in \mathbb{R}^{I}$ $I$ interval of $\mathbb{R}$ Show that $${f \text{ is differentiable on } I \iff f_{\left|\ [a,b]\right.} \text{ is differentiable }\ \forall a,b \in I}$$ in ...
1
vote
2answers
55 views

Solving $\int^{\pi}_{ 0.5\pi} \frac{dx}{1-\cos x}$

Evaluate $$\int^{\pi}_{ 0.5\pi} \frac{dx}{1-\cos x}$$ This is my attempt: $$\int^{\pi}_{ 0.5\pi} \frac{dx}{1-\cos x} = ...
0
votes
2answers
27 views

derivative and integral as opposite operations

Consider: $$\lim_{y\to\infty} \left( \int_0^y f(t)dt \right)' = \lim_{y\to\infty} f(y)$$ So the integral and the derivative cancel each other, but why is it happened to be that it equals to the ...
1
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3answers
37 views

Find $f^{(n)}(1)$ where $f(x)={1\over x(2-x)}$.

Find $f^{(n)}(1)$ where $f(x)={1\over x(2-x)}$. What I did so far: $f(x)=(x(2-x))^{-1}$. $f'(x)=-(x(2-x))^{-2}[2-2x]$ $f''(x)=2(x(2-x))^{-3}[2-2x]^2+2(x(2-x))^{-2}$. It confuses me a lot. I know I ...
0
votes
0answers
21 views

To Prove that The Level Set Of AConstant Rank Map is a Manifold

Let $f:\mathbf R^n\to\mathbf R^m$ be a smooth function of constant rank $r$. Let $\mathbf a\in \mathbf R^n$ be such that $f(\mathbf a)=\mathbf 0$. Then $f^{-1}(\mathbf 0)$ is a manifold of ...
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votes
1answer
16 views

Finding the stationary points of a function

I have a question that I need help with. How do I find the stationary points of the following function? $$y = \frac{4x^3}{(x-1)^2}$$ I differentiated the function and got $$\begin{align} y' ...
0
votes
0answers
25 views

indefinite integral problem: help needed

What will be the integral with respect to $t$ of: $$\frac{dA}{dt} = cx(t)y(t),$$ where $c$ is a constant and $x$ and $y$ are functions of time ($t$). Is there any other method besides inegration by ...
5
votes
4answers
49 views

Prove that $\int_0^1 \frac{\ln x}{x-1} dx$ converge.

Prove that $\int_0^1 \frac{\ln x}{x-1} dx$ converges. We cannot apply Abel's/Dirichliet's tests here (For example, Dirichliet's test demands that for $g(x)=\ln x$, $\int_0^1 g(x)dx < \infty$ ...
0
votes
0answers
15 views

considering the elliptic coordinates (u,v), x=lcoshucosv and y=lsinhusinv. l is a dimensionful constant, what it the dimension of l?

u is greater than or equal to zero. v is greater than or equal to zero and less than or equal to 2pi. Could anyone work out this question please?
1
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0answers
14 views

Prove there is an $a>0$ such that $\forall x\in [0,1]$, $f(x)>x+a$.

Let $f$ be continuous on $[0,1]$ and $f(x)>x\space \space \forall x \in [0,1]$. Prove there exists an $a>0$ such that $f(x)>x+a\space \space \forall x \in [0,1]$. It is really important ...
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3answers
60 views

How to solve: $\int _0 ^1 x (1+x)^n dx$?

Solve: $\int _0 ^1 x (1+x)^n dx$? Original question: find $ \sum _{r=1}^n [(^nC _r)/(r+2)] $ In order to solve this question, I planned to integrate $x(1+x)^n$, this gives a wrong answer : ...
1
vote
2answers
26 views

Choosing path to show limit does not exist

I'm trying to show that the limit as $(x,y)$ go to $(0,0)$ for the function $f(x,y) = sin( x + y )/( |x| + |y|)$ does not exist. I initially tried the path $y=2$ and $y=1$, but I don't think I can use ...
0
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0answers
18 views

A few queries of the method of variation of parameters

I've been reviewing my knowledge on the technique of variation of parameters to solve differential equations and have a couple of queries that I'd like to clear up (particularly for 2nd order ...
4
votes
5answers
90 views

Solve $\int_0^\infty \frac{\ln x}{x^2+4} \,\mathrm{d}x$

(Stanford Math Tournament 2012 #8) I tried rewriting the denominator as $4\left(\frac{x}{2}^2 + 1\right)$ and then integrating by parts, but that got me nowhere... I then tried the substitution $x = ...
2
votes
1answer
51 views

Find $\lim_\limits{n\to \infty}\{en!\}$.

Find the limit $\lim_\limits{n\to \infty}\{en!\}$. $Attempt:$ $\lim_\limits{n\to \infty}\{en!\}=\lim_\limits{n\to \infty}\{(1+{1\over 1!}+{1\over 2!}+{1\over 3!}+...+{1\over n!}+...)n!\}$. The ...
2
votes
2answers
382 views

Splitting an integral

Why is the following equality true? $$ \int_1^{2e} \left| \ln x - 1 \right| dx = \int_1^e(1-\ln x) dx + \int_e^{2e} (\ln x - 1) dx$$
5
votes
3answers
39 views

Finding $\lim_{x\to +\infty}(\frac{x+\ln x}{ x-\ln x})^{\frac{x}{\ln x}}$

Find $\lim_{x\to +\infty}(\frac{x+\ln x}{ x-\ln x})^{\frac{x}{\ln x}}$. I tried using l'Hospital rule with the continuity of $e$ function. Also tried using Taylor expansion with no success. What ...
0
votes
2answers
22 views

Find x,y,z where multiplication of them equals 36 and sum equals to the square of the sum of two of them

I need to find three numbers x, y, z where: 1) the multiplication of all these numbers equals 36 2) the sum of these three equals to the square of the sum of the two. The question goes if there enough ...
1
vote
1answer
36 views

Show how $\frac{\partial}{\partial x} \left[\int_0^x (x-t)g(t)\,\mathrm{d}t\right] = \int_0^x g(t)\,\mathrm{d}t$

It has something to do with the second part of the Fundamental Theorem of Calculus right? I've always had trouble with this theorem ever since I learned it several years ago :\ Would somebody please ...
0
votes
0answers
12 views

Point on ellipse after walking a distance on the perimeter [duplicate]

I've the equation of an ellipse. Given a point (x,y) on the ellipse and a length L , I want to find the coordinates (x1,y1) of the point where I'd end up after taking a walk of length L from (x,y), ...
0
votes
0answers
19 views

Derivative of $Ad(c(t))X$

Let $G=SO(3)$ and $V=\{c'(0)|c:(-\epsilon,\epsilon)\to G, c\in C^{\infty} , c(0)=1\}$. For $g\in G$, define $Ad(g): V\to V$ by $Ad(g)(X)=gXg^{-1}$. The book says ...
1
vote
3answers
29 views

$\delta-\epsilon$ Question on Ordered Field $\mathbb{R}$

I got came across this question with the $\delta-\epsilon$ definition of a limit, but I do not know how to use it to solve the context of this problem: Problem: Let $f:\mathbb{R}\to\mathbb{R}$ be ...
1
vote
3answers
54 views

Explanation for $\lim_{x\to2} e^{\frac{1}{x-2}}$

I can't find out why is the limit from the left side = 0 and from the right = Infinity?
0
votes
1answer
31 views

How to bound the biggest eigenvalue of $\sum_{i=1}^{n}x_ix_i^T$?

My question is to bound the biggest eigenvalue of $A=\sum_{i=1}^{n}x_ix_i^T$, where $x_i\in\mathbb{R}^d$ is a column vector. My idea is, to bound the biggest eigenvalue of $A$, i.e. $\|A\|_2$. I can ...