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

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1
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0answers
6 views

Following a simplification of expression

I am struggling with this expression: In particular I get stuck with the simplification from line 1 to line 2. As far as I can see they replace $\text{m$\ell $}=\mu$. Does the new absolute value of ...
-1
votes
0answers
6 views

Orthonormal vectors and potential

given the potential $ψ(x;y)$, such that $dψ=−u_2dx+u_1dy$, why are $∇ψ=(−u_2;u_1)$ and $ψ(x;y)=c$ orthonormal vectors ? $c$ is a constant. Thanks!
0
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3answers
26 views

Differentiate the Function $g(x)=\ln \frac{a-x}{a+x}$

$g(x)= \ln \frac{a-x}{a+x}$ $\frac{dy}{dx}\ =\frac{d}{dx}\ \ln \frac{a-x}{a+x}$ $g'(x)\frac{1}{\frac{a-x}{a+x}}\cdot\frac{1}{1}\ \ln \frac{a+x}{a-x}$ $g'(x)= \frac{a+x}{a-x}$ This answer is ...
1
vote
2answers
31 views

Proof of there is no limit at $x=0$ for $f(x)=sin(\frac{1}{x})$

I've seen a few questions posted before about mine, but this is a bit different. The original form of the question can be found here: http://librarun.org/book/10452/159. It says that prove by ...
0
votes
0answers
12 views

$ΔV$ and $Δh$ of a cone.

Find the exact volume $V_0$ in terms of $\pi$ . Then derive an approximate formula for the height error $Δh$ in terms of the volume error $ΔV$. $d = 3$, $h = 6$ $$V=\pi r^2\frac h3$$ $r=\frac h4$ ...
4
votes
0answers
24 views

A tough integral:$\int_0^{+\infty}\left( \frac1{\log(x+1)-\log x}-x-\frac12\right)^2 dx$

I would like to prove the convergence of $$I=\int_0^{+\infty}\left( \frac1{\log(x+1)-\log x}-x-\frac12\right)^2 dx$$ then obtain a closed form of $I$. Convergence is ensured by the fact that $x ...
0
votes
1answer
8 views

Optimization Example with some constraints !?

We want to optimize the following function: $f(x,y)=x^2+3y^2+2xy+2$ with constraint $-2 \leq x< 2$ $-2 <y<2$ $3y^2+x \leq 10$ Who Can Help me for the above example from my note? My TA‌ ...
1
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4answers
49 views

why $ \sum_{k=1}^{\infty} x^{2k} = \frac{1}{1-x^2}\\$

Why $$ \sum_{k=0}^{\infty} x^{2k} = \frac{1}{1-x^2}\\$$ I know that $$ \sum_{k=0}^{\infty} x^k = \frac{1}{1-x}\\$$ can I use the above to derive the first result?
0
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5answers
43 views

Finite set of points of $R^n$ is compact

In order to show that a finite set of points of $R^n$ is compact, I just need to show that the set is closed and bounded. First of all, since it's a finite set, I can Always pick the greatest ...
1
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3answers
51 views

Differentiate the Function: $f(x)=\ln (\sin^2x)$

$$\begin{align}f(x)&=\ln (\sin^2x)\\ f'(x)&=\frac{1}{\sin^2x}\cdot 2(\sin x)(\cos x)\\ &=\frac{2(\sin x)(\cos x)}{\sin^2x}\\ &=\frac{2\ \ (\cos x)\ }{\sin x}\\ &=2\cot x ...
2
votes
3answers
83 views

Differentiate the Function: $ f(x)= x\ln x\ - x $

$ f(x)= x\ln x - x $ Wondering if my answer is right. Here is my process. I will simply find the derivative by using the product and difference rule. $x \frac{d}{dx}[\ln x]+ \ln ...
2
votes
1answer
56 views

What is the d used in calculus?

I know the letter d is commonly used in calculus represents a derivative. Does this d act as a variable that can be simplified or as a function of another variable?
0
votes
1answer
32 views

Existence of such a function

I am supposed to construct a function $f \in C_c^1((-\frac{3R}{4},\frac{3R}{4}))$ such that $f|_{(-\frac{R}{2},\frac{R}{2})}=1$ and $|f'(x)| \le \frac{4}{R}$ for almost all $x \in (-R,R)?$ I ...
0
votes
1answer
30 views

Proof inequality using Lagrange Multipliers

Is it possible: $a,b,c$ are non-negative real numbers for which holds that $a+b+c=3.$ Prove the following inequality: $$ 4\ge a^2b+b^2c+c^2a+abc $$ Is it possible using Lagrange Multipliers. I ...
0
votes
2answers
36 views

A general method for integration of rational function.

$\int\frac {x^3}{1+x^5}$ ATTEMPT: I did the following substitution: Let $x=\frac{1}{t}.$ $dx=\frac{-1}{t^2}dt.$ substituting back: $I=\int\frac{-1}{1+t^5}dt$ which doesn't seems a simpler ...
-2
votes
0answers
18 views

Limit of a recursive sequence containing log [on hold]

Let $\alpha$ be a real number. Consider the following recursive formula: $a_1=1$ and $$a_n=1-\alpha . \sum_{i=1}^{n-1}{a_i\over{i.\log(n-i+1)}} \: \: \: \:for\:\:n\ge2$$ Note that the logarithm is ...
4
votes
0answers
42 views

The quadratic and cubic versions of a tough intregral

In this post, Proving that $\int_0^1 \frac{\log \left(\frac{1}{t}\right) \log (t+2)}{t+1} \, dt=\frac{13}{24} \zeta (3)$, it's proved that $$I_1=\int_0^1 \frac{\log \left(\frac{1}{t}\right) \log ...
1
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3answers
39 views

Finding $\lim\limits_{n\to \infty}({1\over n+1}+{1\over n+2}+…+{1\over n+n})$ using integrals [duplicate]

Finding $\lim\limits_{n\to \infty}\left({1\over n+1}+{1\over n+2}+\dots+{1\over n+n}\right)$. I tried many things but it would work out. I am now studying calculus 2 (In my country the first calculus ...
0
votes
3answers
36 views

Can you simplify this expression?

This is a Bayes formula incorporating 2 random variables. The final expression seems a bit tricky to simplify the exponents and I'm still not so confident with my algebra (pardon me ;)). Can you have ...
0
votes
1answer
28 views

$f,g \in R(T)$ such that $\hat{f} \cdot n^{2/3} = \hat{g}$ prove that $f$'s Fourier series converges absolutely.

Can someone help me by checking my solution. Is there a shorter More elegant solution ?(i'm almost sure you can some how express $f$'s Fourier series using $|\hat{g}|^2$ + constant, i saw someone do ...
3
votes
0answers
33 views

Origin of the Integral (Theory Behind It - How it came about)?

How exactly was the integral derived? Like similarly to how the difference quotient explains where the derivative came from, what can we use to explain the origins of the integral? Like how does ...
3
votes
4answers
329 views

Am I using the chain rule correctly?

I'm supposed to find $y'$ and $y''$ of this function: $$y=e^{\alpha x} \sin\beta x$$ This is what I have done so far: $$y'=e^{\alpha x}\sin\beta x\cdot \alpha x'\sin\beta x\cdot \sin'\beta x \cdot ...
1
vote
1answer
35 views

Integral Test question

So this is the problem: http://postimg.org/image/5g815zgk5/ I am getting $\lim_{b\to\infty} 2\sec^{-1}(2b) - 2\sec^{-1}2$ Now what? What do I do with $\sec^{-1}(2b)$? What happens to a trig function ...
0
votes
3answers
64 views

Limit of $\{a_n\}$, where $a_{n+1} = \sqrt{2+a_n}$

I am struggling with this question: Let $\{a_n\}$ be defined recursively by $a_1=\sqrt2$, $a_{n+1}=\sqrt{2+a_n}$. Find $\lim\limits_{n\to\infty}a_n$. HINT: Let $L=\lim\limits_{n\to\infty}a_n$. ...
1
vote
1answer
38 views

Analysis for Engineering : Practical Applications

I don't know much more about Analysis than what I've read about it on Wikipedia, although I have just begun reading Introduction to Calculus and Analysis I, by Richard Courant. My understanding is ...
-2
votes
0answers
39 views

Finding all points on $y=x^2$ for which the normal line goes through the point $(0,3)$. [on hold]

Find the coordinates of all points of the parabola $y=x^2$ for which the normal line goes through the point $(0,3)$. Give exact answers using radicals if necessary. No decimals.
1
vote
2answers
39 views

Finding limit points for these sets

Here's my resoning for finding limit points for some sets. Could you guys read it and see if it's all good? <3 $$\{(x,y)\mid \ x^2+y^2<1\}$$ For this set, its kinda simple to see that every ...
2
votes
1answer
25 views

Order of Rate of Growth

How would you put these functions in order of rate of growth from the greatest to the smallest? $f(x) = \log_2 x$, $g(x) = x^x$, $h(x) = x^2 $, $k(x) = 2^x$ I took the derivatives and ended up with ...
9
votes
4answers
336 views

Evaluating limit (iterated sine function)

The limit is $$\lim_{x\rightarrow0} \frac{x-\sin_n(x)}{x^3},$$ where $\sin_n(x)$ is the $\sin(x)$ function composed with itself $n$ times: $$\sin_n(x) = \sin(\sin(\dots \sin(x)))$$ For $n=1$ the ...
1
vote
0answers
18 views

Matrices derivative

I have a linear product of matrices, I did solve most of it, however, I stop at this component $(X^T W^T D W X)^{-1}$. Given that $X$ is $n \times p$ matrix and $D$ is $n\times n$ matrix. $W$ is a ...
-3
votes
1answer
20 views

How can I find the area of this region? 11 [on hold]

Find the area of the region of the function y=x^2 +2, given [0,1].
2
votes
3answers
69 views

Show that $f$ is bounded.

Let $-\infty<a<b<\infty$. Suppose $f$ is continuous on $[a,b]$. Show that $f$ is bounded on $[a,b].$ We are supposed to use intermediate value theorem for this problem. But, I don't ...
0
votes
0answers
47 views

Show that if $f$ is differentiable as to function $x\mapsto ||x||$ with $x\in R$,then $f'(0)=0$ [on hold]

Let $f\in C^{\infty}(Ω)$ for some open set $Ω \subset R^n$ that contains $0$. Show that if $f$ is differentiable as to function $x\mapsto ||x||$ with $x\in R$,then $f'(0)=0$. I found this problem in a ...
-1
votes
2answers
28 views

Determine subsequence of sequence [on hold]

I know the formal definition of a subsequence, but can't figure out how to find them for some particular sequence. Could someone show some of the methods for finding them? Thanks for replies.
2
votes
2answers
64 views

How can I calculate this limit?

$\lim _{ m\rightarrow \infty }{ \left( \lim _{ n\rightarrow \infty }{ \cos ^{ 2n }{ \left( \pi m!x \right) } } \right) } $ Attempt : since $\cos ^{ 2 }{ x=\frac { 1+cos2x }{ 2 } } $ so we can ...
1
vote
0answers
27 views

Functions linearly independent and linearly independent gradients?

Let $F_1,...,F_n: \mathbb{R}^n \rightarrow \mathbb{R}$ be a set of $C^{1}$ functions. Is it true that they are linearly independent on a joint level set $\Omega:= \{ p \in \mathbb{R}^n; ...
-4
votes
3answers
110 views

How can you evaluate $\lim_{x \to 2} \frac{x^3-8}{x-2}$ [on hold]

How can I find the following limit? $$\lim_{x \to 2} \frac{x^3-8}{x-2}$$
1
vote
2answers
66 views

Approximating $\tan61^\circ$ using a Taylor polynomial centered at $\frac \pi 3$ : how to proceed?

Here's what I have so far... I wrote a general approximation of $f(x)=\tan(x)$ , which then simplified a bit to this: $$\tan \left(\frac{61π}{180}\right) + ...
1
vote
2answers
63 views

Proving inequalities using Calculus

In general how do you prove inequalities using calculus, I believe it is using maxima or minima right? For example $$a^2b+b^2c+c^2a \le 3, \qquad a,b,c \ge 0,\quad a+b+c=3.$$ How would you use ...
1
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4answers
40 views

When can I use the natural log to help solve an integral?

Why is it okay to do this: $\int \frac{1}{x-2}dx = \ln(x-2)$ but not this: $\int \frac{1}{1-x^2}dx = \ln(1-x^2)$
7
votes
4answers
229 views

Integral involving a trig. term

I came across the following integral. $$ \int\frac{dx}{1+\sin x} $$ I have no idea how to solve it! I went for the obvious substitution of $u=1+\sin x$, but then you get an annoying $\cos x$ kicking ...
2
votes
6answers
62 views

Prove that $\{(x,y)\mid xy>0\}$ is open

I need to prove this using open balls. So the general idea is to construct a open ball around a point of the set. A point $(x,y)$ such that $xy>0$. Then we must prove that this ball is inside the ...
2
votes
2answers
79 views

Need help with $e^x=1/x$

I've tried everything. I expressed $x$ and I got $x=\ln{1\over x}$, and don't know what to do. Original question is to find $e^x-{1\over x}=0$. There is a solution I've typed it in Wolfram
1
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0answers
29 views

How to manipulate the bound on the summation

$$ B_n^{f^2}(x) = \sum_{k=1}^n\sum_{j=0}^{n-k} 2^{k-j} {j+k \choose j} \frac{d^j}{df^j}[f^k] B_{n,j+k}^f(x) $$ I am looking to have the bounds switched, can someone show me exactly how this is done? ...
0
votes
2answers
29 views

Multivariable function as a set of functions

Consider a function $f:\mathbb{R}^n \to \mathbb{R}^m$. I've understood that it can be seen as: $f_i = (f_1,f_2,\ldots ,f_m)$, where $f_i: \mathbb{R}^n\to \mathbb{R}$. What are $f_i$ exactly? ...
2
votes
4answers
107 views

What is wrong in my $f'(x)$?

We have $f:\mathbb{R}\rightarrow\mathbb{R}, f(x)=\frac{x^2-x+1}{x^2+x+1}$ and we need to find $f'(x)$. Here is all my steps: ...
5
votes
2answers
198 views

Evaluating the indefinite integral $\int\log\!\left(x+\sqrt{x^2-1}\right)\!dx$

I came across the following integral, and I don't know how to solve it. $$ \int\log\left(x+\sqrt{x^2-1}\right)dx $$ I tried the "obvious" substitution of $x=\sec\theta$, which gives you: $$ ...
3
votes
5answers
72 views

Differentiate the Function $f(x)= \sqrt{x} \ln x$

Differentiate the Function $f(x)= \sqrt{x} \ln x$
2
votes
0answers
33 views

Advanced calculus: Solving quaternion differential equations

I have a system of two differential equations $$\frac{\partial X(t)}{\partial t}=a_1 A X(t)+a_2X(t) B+a_3 C Y(t)+a_4Y(t) D+a_5$$ $$\frac{\partial Y(t)}{\partial t}=b_1 E X(t)+b_2X(t) F+b_3 G ...
0
votes
2answers
70 views

Integral of $x/(2x-1)$

I'm not sure how to do this, I'm also new to math.stackexchange so please excuse any novice mistakes. So anyways, here is a question I have on a summer assignment for Calculus BC (this is review from ...