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

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-1
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1answer
10 views

Estimating values using tangent line?

How do you do this type of question and what would be correct answers in this case? Thank you all in advance!
4
votes
0answers
28 views

Can we interchange the Integral and Summation when a limit is $\infty$?

I was trying to Evaluate the Integral: $$\Large{I=\int_1^{\infty} \frac{\ln x}{x^2+1} dx}$$ $$\color{#66f}{{\frac{1}{x^2+1} = \frac{1}{x^2\left(1+\frac{1}{x^2}\right)}=\frac{1}{x^2}\cdot ...
3
votes
1answer
42 views

Solve $(\alpha,\beta)$ for $\lim_{n\to\infty} \frac{\sqrt[n^2]{1!2!\cdots n!}}{n^\alpha} = \beta$

Find the ordered pair $(\alpha,\beta)$ with non-infinite $\beta \ne 0$ such that $$\lim_{n\to\infty} \frac{\sqrt[n^2]{1!2!\cdots n!}}{n^\alpha} = \beta$$ My approach: $$\ln (1!2!\cdots n!) = (n)\ln ...
0
votes
3answers
33 views

What is the integral of $\frac{\sqrt{x^2 +4}}{x}dx$

I use trig substitution then get to this step but then I get stuck: $\int \frac{2\sec ^3\theta}{\tan \theta}d\theta$ anything I do seems to further complicate it. Thanks in advance.
1
vote
3answers
62 views

Evaluating $\int_0^1 \frac {x^3}{\sqrt {4+x^2}}\,dx$

How do I evaluate the definite integral $$\int_0^1 \frac {x^3}{\sqrt {4+x^2}}\,dx ?$$ I used trig substitution, and then a u substitution for $\sec\theta$. I tried doing it and got an answer of: ...
0
votes
0answers
13 views

Setting up a double integral in terms of x and y to find flux

I am presented with the following problem, and it wants me to set up the double integral in terms of x and y, but I have no idea on how to continue solving this one, any ideas? Set up a double ...
2
votes
3answers
41 views

Is it possible to find the area of a shape from its perimeter?

Is it possible to find the area of a free form shape knowing the perimeter? An example would be a clover leaf shape. If the perimeter is 96 how would I know what the area would be?
0
votes
0answers
13 views

Can this multivariable function exist?

(3) Is there a function of two variables whose z = 0 level curve consists exactly of the circles $x^2$ + $y^2$ = 4 and $x^2$ + $y^2$ = 10? If so, what is an example? If not, why not? I initially ...
0
votes
0answers
12 views

How to understand the meaning of 'Oblivious' in Oblivious Subspace Embedding?

For the definitions of Oblivious Subspace Embedding and Subspace Embedding, please refer to the 1st page of paper http://arxiv.org/pdf/1308.3280v1.pdf.
2
votes
1answer
20 views

Finding the limit of a recursively defined sequence

how do I go about finding the limit of ${a_n}$? Let $a_n$ be defined recursively by $$a_1 =1$$ $$a_{n+1}=(1+2a_n)^{1/2}$$
2
votes
1answer
32 views

Integral inequality $\int_0^x{f(t)^3 dt \leq \left( \int_0^x f(t) dt\right)^2} :\forall x>0$

Let $f(0) = 0$ and $0<f'(x)\leq1$ for all $x \geq0$, then prove: $$\int_0^x{f(t)^3 dt \leq \left( \int_0^x f(t) dt\right)^2} :\forall x>0$$ The hint I was given was "differentiate, factor and ...
1
vote
1answer
33 views

Integration of a function defined by its graph, the union of semi-circles and a line segment

I don't understand how to do this problem and I would someone to help me with it.Please step by step for me. I just started on integration so this problem is a bit too hard for me due to my lack of ...
1
vote
0answers
24 views

Find $\lim_\limits{x\to \infty}{\sqrt[3]{x^3+ax^2+bx+c}-x}$. [duplicate]

Find $\lim_\limits{x\to \infty}{\sqrt[3]{x^3+ax^2+bx+c}-x}$. As I understand, I should use Taylor series, but I don't know how. What should I translate into Taylor series, to what extent, etc. This is ...
0
votes
0answers
8 views

Instantaneous relative growth

I'm studying Fisher information and the function $\frac{d}{dx} \ln(f(x))$ thus $\frac{f'(x)}{f(x)}$ appears. I'm trying to interpret what this is quantifier could represent in a function. Thank you.
1
vote
1answer
32 views

How to integrate $12x^3(3x^4+4)^4 $ in a nice way

How would I antidifferentiate $12x^3(3x^4+4)^4 dx$ ? I guess it is possible to multiply it all out, and then do term by term, but is there a more efficient solution?
-1
votes
0answers
28 views

peicewise fuctions with limits [on hold]

$$f(x)= \begin{cases} 0 & \text{if } x \text{ is rational}\\1 & \text{if } x \text{ is irrational}\end{cases}$$ and $$g(x)= \begin{cases} 0 & \text{if } x \text{ is rational}\\x & ...
6
votes
1answer
54 views

Find $\lim_\limits{n\to \infty}\left({1\over \sqrt{n^2+1}}+{1\over \sqrt{n^2+2}}+\cdots+{1\over \sqrt{n^2+n}}\right)$

Find $\lim_\limits{n\to \infty}\left({1\over \sqrt{n^2+1}}+{1\over \sqrt{n^2+2}}+\cdots+{1\over \sqrt{n^2+n}}\right)$. I do know it is bounded by $1$. I tried using the sandwich rule with no ...
-1
votes
1answer
30 views

sequences and convergence [on hold]

1) Explain why the function $f(x)=\ln(1+x^2)$ is increasing on $[0,\infty)$. 2) Define the sequence $\{a_n\}$ by $a_1=1$ and $a_{n+1} = \ln(1+a_n^2)$, $n\geq1$. Show that $\{a_n\}$ is decreasing. 3) ...
1
vote
4answers
60 views

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

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 ...
2
votes
1answer
38 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
vote
2answers
33 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
votes
0answers
18 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
37 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 ...
2
votes
5answers
77 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
31 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
vote
2answers
25 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
votes
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
votes
0answers
49 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
vote
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
18 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
37 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
32 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, ...
4
votes
2answers
184 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
votes
2answers
54 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. $v_x = 52cos(700t+0.2\pi)$ ...
1
vote
0answers
17 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 ...
1
vote
0answers
37 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
25 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. Any hint would be appreciated! Thanks ...
0
votes
4answers
56 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
30 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
32 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
65 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
46 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
57 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
vote
3answers
38 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
23 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
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' ...