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

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0
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0answers
12 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
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2answers
23 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 ...
-1
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0answers
15 views

Limit of a recursive sequence containing log

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
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0answers
29 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
31 views

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

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
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3answers
33 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
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0answers
23 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
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0answers
31 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
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4answers
276 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
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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 ...
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3answers
61 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
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1answer
35 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
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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.
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2answers
37 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
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1answer
24 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 ...
8
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4answers
241 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
17 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
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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
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3answers
68 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
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0answers
44 views

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

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 ...
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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
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2answers
62 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
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0answers
25 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
108 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
65 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
60 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 ...
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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
61 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
78 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
192 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
70 views

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

Differentiate the Function $f(x)= \sqrt{x} \ln x$
2
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0answers
31 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 ...
4
votes
1answer
52 views

Prove a sequence converges using sub-sequences

Let there be a sequence $a_n$ The following sub-sequences converge: $a_{n^3},a^3_{2n+3}-a^3_{2n+4},a^2_{2n+3}-a^2_{2n+4},a_{2n+15}$ Prove: $a_n$ converges I think it has something to ...
0
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1answer
50 views

Determine the roots of equation if possible

How to determine the roots of equation using numerical methods? I have this particular equation: $$\arctan(e^x)=\ln \left(\sqrt{\frac{e^{2x}}{e^{2x}+1}}\right)$$ In my solution I have that this ...
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4answers
45 views

Prove that $f(x)=m$ has three distinct real roots for $m\in(0,8)$

We have $f:\mathbb{R}\rightarrow\mathbb{R},f(x)=x^5-5x+4$ and we need to show that $\forall m\in(0,8)$, $f(x)=m$ has three distinct real roots. How can I prove it?
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1answer
64 views

How to calculate this derivative $D^{\alpha}f(x)$?

Let $v\in\mathbb{R}^n$ be a fixed vector, and $f$ a function given by $f(x)=\cos(x\bullet v)$, where $x\bullet y$ is the dot product. What is the derivative $D^{\alpha}f(x)$ for $x\in\mathbb{R}^n$ ...
0
votes
2answers
54 views

Convergence of $\int_0^\infty x^\alpha \cos e^x \, dx$

I tried to solve whether this integral is convergent or not and whether that convergence is conditional or absolute for a given $\alpha$. $$\int _0^{\infty }\:\:x^{\alpha \:}\cos\left(e^x\right)\, ...
3
votes
1answer
32 views

The value of an integral of a piecewise defined function

I have been given $$f(x)=\begin{cases} 4 & 2\le x<5\\ 3 & x=5\end{cases}$$ and I want to find the value of $$\int_2^5 f(x) dx.$$ I proceeded as follows: $$\int_2^5 f(x) dx = 4(5-2) = ...
0
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2answers
31 views

Calculus: simpler way of showing that derivative is negative?

I want to show that $\frac{1-(1-\beta)^N}{\beta}$ is strictly decreasing in $\beta$ for $\beta \in (0,1)$ and $N \geq 2$. My approach so far is as follows: I take the derivative with respect to ...
0
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2answers
23 views

Show that $\frac {\partial B} {\partial T} =$ $\frac{c}{(\exp\frac{hf}{kT}-1)^2}\frac{hf}{kT^2}$

Find an expression for $\frac {\partial B} {\partial T}$ applied to the Black-Body radiation law by Planck: $$B(f,T)=\frac{2hf^3}{c^2\left(\exp\frac{hf}{kT}-1\right)}$$ I get an answer of $\frac ...
0
votes
2answers
63 views

Integral of cos(1/x) dx

Is the following integral expression correct (neglecting the constant of integration)? $$ \int\cos\left(\frac{1}{x}\right)dx = x^2\sin\left(2x\right) $$ When I take the derivative, it returns to the ...
2
votes
2answers
45 views

Why is the integral starts from $0$?

Consider $$f(x) = \sum_{n=0}^\infty \frac{(-1)^n}{3n+1} x^{3n+1}$$ It's a power series with a radius, $R=1$. at $x=1$ it converges. Hence, by Abel's thorem: $$\lim_{x\to 1^-} f(x) = ...
16
votes
1answer
221 views

Proving that $\int_0^1 \frac{\log \left(\frac{1}{t}\right) \log (t+2)}{t+1} \, dt=\frac{13}{24} \zeta (3)$

Are we aware of an elementary way of proving that? $$\int_0^1 \frac{\log \left(\frac{1}{t}\right) \log (t+2)}{t+1} \, dt=\frac{13}{24} \zeta (3)$$ Of course, with the help of Mathematica it can be ...
0
votes
1answer
48 views

Prove the limit is $e^\alpha$

prove that $\lim_{n \to \infty} \left(1+{\alpha\over n}\right)^n=e^\alpha$ $$\lim_{n \to \infty} \left(1+{\alpha\over n}\right)^n=\lim_{n \to \infty} \left(\left(1+{\alpha\over n}\right)^{n\cdot ...
1
vote
2answers
39 views

Find maximum of a function

I want to find the maximum of a function. $$ d = \frac{35}{3} + \frac{7}{3}\sin( \frac{2\pi}{365}t ) $$ I don't know if I applied the chain rule correctly. $$ w = \frac{2\pi}{365}t $$ $$ w' = ...