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

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0
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0answers
9 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
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1answer
25 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 ...
1
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0answers
20 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
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1answer
27 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
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2answers
45 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
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2answers
23 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 ...
0
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3answers
32 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
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0answers
14 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 ...
-1
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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
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0answers
22 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
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4answers
47 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
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0answers
6 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
13 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 ...
-1
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3answers
55 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
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2answers
22 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 ...
1
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0answers
17 views

Values of a satisfying the inequality [on hold]

if $ 1-\cos x=\frac {\sqrt3}{2} |x| +a$ has no solution then find the complete set of values of $'a'$.Here is the question i got struck.
0
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0answers
11 views

Basic inequality problem

Here is my problem if $ 16-x^2> |x-a|$ is to be satisfied by atleast one negative value of $x$, then i have to find complete set of values of $'a'$ .Please provide me hint to solve this ...
0
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0answers
17 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
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5answers
83 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
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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
218 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
38 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
20 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
35 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
10 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
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0answers
15 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
28 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
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3answers
52 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
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1answer
30 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 ...
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0answers
19 views

This is a 2+1 D problem. I know a little about contour integration. Please suggest how may I proceed. [on hold]

Kindly check it.I don't know what to do with this delta and etc etc $\int \frac {e^{-i\vec{k}.( \vec{x}'-\vec{x})}e^{-ik^{\circ}x_{\circ}}\delta(k^{\circ})}{k^2-\mu^2}d^2kdk^{\circ}$
2
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2answers
14 views

Question about Speeding and Calculus

At 6.00 am, a driver picked up a fare card at the entrance of a tollway. At 10:30 am, the driver pulled up to a toll booth 250 miles away. After computing the fare, the toll booth operator issues a ...
0
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1answer
20 views

How can a function be closed and bounded on the interval [-infinity,infinity]?

How can a function be closed and bounded on the interval [-infinity,infinity]? To me the word infinity implies that it would be with out bound. Perhaps I'm getting to caught up in semantics?
5
votes
2answers
90 views

Arc length contest! Minimize the arc length of $f(x)$ when given 3 conditions.

Contest: Give an example(s) of a continuous function $f$ that satisfies three conditions: $f(x) \geq 0$ on the interval $0\leq x\leq 1$; $f(0)=0$ and $f(1)=0$; the area bounded by the graph of $f$ ...
0
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3answers
28 views

Help with Infinite series + Integral Test + Improper Integrals

I am having some trouble with the infinite series $\displaystyle\sum_{n=2}^{\infty}\frac{1}{n\ln^2n}$ . I used the integral test and simplified it to $\int_{\ln 2}^b - \frac 1{\ln(n)}$ (implied ...
5
votes
2answers
26 views

Conceptual question on substitution in integration [duplicate]

In calculus we learn about the substitution method of integrals, but I haven't been able to prove that it works. I mainly don't see how manipulations of differentials is justified, i.e how $dy/dx = ...
0
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0answers
4 views

Set containment problem misunderstanding

I just wanted to know if I was misinterpreting this question, my interpretation given after the question below: Given An <= Bn and Cn <= Dn, and as An - Bn -> 0 and Cn - Dn -> 0 as ...
2
votes
1answer
13 views

Proof that the velocity vector is tangential to the path?

In calculus class my teacher asserted that the velocity vector is tangential to the path a point takes. I have tried to prove this but have gotten stuck. I computed $\dfrac{v_y}{v_x}$ to be ...
0
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3answers
72 views

How $\int_{-\infty}^{\infty}\frac{dx}{1+x^2}$ exists?

How $$\int_{-\infty}^{\infty}\frac{dx}{1+x^2}$$ exists? It is difficult question to me. i have tried to evaluate by using fact that $$\int_{-\infty}^{\infty} f(x) \ dx =\int_{-\infty}^{0} f(x)\, dx ...
2
votes
3answers
54 views

What is the integral of $\frac{\sqrt{x^2-49}}{x^3}$

I used trig substitution and got $\displaystyle \int \dfrac{7\tan \theta}{343\sec ^3\theta}d\theta$ Then simplified to sin and cos functions, using U substitution with a final answer of: ...
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0answers
14 views

Is product of 2 positive concave function with different domain concave? [on hold]

Suppose f(x) and g(y) are two positive concave functions. h(x,y) = f(x)g(y) Is h(x,y) concave as well?
3
votes
5answers
84 views

Explain why the integral $\int_{-\infty}^\infty x \,dx$ does not exist

Why is it that $$\int_{-\infty}^\infty x \,dx$$ does not exist, but $$\lim_{N \to \infty} \int_{-N}^{N} x\,dx$$ does exist? I was thinking that it involves the fact that in the second case, the ...
0
votes
2answers
40 views

Finding all continuous functions so that $f^n(x)=x$ for some $n$.

I came up with this problem in class but I can't seem to solve it. I need to find all the functions $f$ with domain and codomain $\mathbb R$ such that there is an $n$ such that $f^n(x)=x$ for all $x$, ...
3
votes
1answer
64 views

Prove the sequence determined by $a_{n+1}={a_n\over \sin a_n}$ is convergent, and found its limit.

Let $\{a_n\}$ be a sequence defined by $0<a_1<{\pi \over 2}$, $a_{n+1}={a_n\over \sin a_n}$. $Attempt:$ $a_1>0$ and $\sin a_1>0$ and therefore the sequence begins positive and remains ...
1
vote
2answers
25 views

Rate of change of angle formed by $y=\sqrt[3]{x}$ and the origin.

the problem I am working on is this. Point $P(x,y)$ is on the curve $y=\sqrt[3]{x}$. The angle formed between the $x$-axis and the line that connects the origin and $P$ is $\theta$. As ...
2
votes
0answers
35 views

how to calculate this line integral $\int_{0}^{2\pi} (16\sin^2 3t +16\cos^2 4t)\sqrt{(144\cos^2 3t +256\sin^2 4t)}dt$

I am working on a line integral to calculate the amount of chocolate to cover a pretzel. the density of the pretzel is given by this formula $\lambda=3(x^2+y^2)$ and the parameter equation of a ...
1
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0answers
19 views

Proving the following multivariable limit using the definition. [on hold]

I am trying to prove that $ \lim_{(x, y) \to (0, 0)} x + 2xy + 2y + 6 = 6$ using the definition of a multivariable limit, but I am having no luck. Could someone please help me? Thanks!
1
vote
2answers
31 views

What is the indefinite integral of $x^2\sqrt{1+x^2}$

I get this but I don't know if it is correct. I used a reduction formula for $\tan^{2n}(x)\sec^{3}(x)$. Any help would be appreciated. My Final Answer: $$\frac{\sqrt{x^2+1} x}{8}+\frac{\sqrt{x^2+1} ...
2
votes
0answers
14 views

Find a vector equation and parametric equations for the line segment that joins $P$ to $Q$. [duplicate]

Find a vector equation and parametric equations for the line segment that joins $P$ to $Q$. Here $P(1,-1,7)$ and $Q(7,5,1)$. I have tried to find $r(t)$ by using the formula $r(t)=p+t(p-q)$ but ...
1
vote
0answers
21 views

Integral involving rapidly decreasing functions [on hold]

Let $F,\varphi\in S(\mathbb{R}^n)$, the Schwartz space of rapidly decreasing functions. Is this enough to guarantee that the integral $$\int_{\mathbb{R}^n} F(x)\varphi(x)dx$$ is well-defined? Why or ...
2
votes
0answers
23 views

How to evaluate this (Fourier) integral? [duplicate]

Does somebody know how to evaluate $$\int_{\mathbb{R}^n}\frac{e^{i\langle\xi,x\rangle}}{\|\xi\|_2^2}d\xi$$ for some given $x\in\mathbb{R}^n$ and $n\in\{1,2,3\}$?