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

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1answer
18 views

Double integral and integration by parts

Let $f:[0,b]\to[0,d]$ be a continuous bijection. If $h:[0,d]\to \mathbb{R}$ is a Riemann integrable function, how to prove that $$\int_{0}^b\left(\int_{0}^{f(x)}h(y)dy\right)dx = \int_0^d ...
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0answers
7 views

Looking for an alternative solution for optimal control problem

Let's say we have the following function ; $\intop_{0}^{\infty}\int_{0}^{N}V\left(C(t,\tau\right)dtd\tau$ and we want to maximise it according to the following constraint ; ...
1
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0answers
28 views

Does there exist such function?

Fix an integer value $k\geq 1$. Let $[0,1]$ the unit interval and let $s\in [0,1]$. Does there exist a function $f$ (which depends on $k$ of course but not on $s$) such that $$\int_s^1 \left( ...
0
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1answer
42 views

how can I give an elementary proof of Maximum Modulus Theorem for polynomials?

how can I give an elementary proof of Maximum Modulus Theorem for polynomials? I got proof, but not elementary. This question in this book Conway.
1
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0answers
21 views

how to calculate $f(D(0,\delta) - \{ 0 \})$ with $f(z)=z\sin(\frac{1}{z})$?

how to calculate $f(D(0,\delta) - \{ 0 \})$ with $f(z)=z\sin(\frac{1}{z})$ ?. I know that zero is an essential singularity, and so $f(D(0,\delta)-\{ 0 \})$ is dense in $\mathbb{C}$. This question ...
2
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1answer
21 views

Improper integral: is it convergent?

Is this integral finite? $$\int_s^t \frac{dx}{x^{1/2} - s^{1/2}}$$ where $s,t \in (0,\infty)$. More generally, I have the following integral $$\int_s^t ...
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1answer
37 views

$\int_y^x \cos(e^t)\,dt=(\sin(e^t))]_y^x$

I have this problem: Originally, I had $F(x,y)=\int_y^x \cos(e^t)\,dt$. I want to find the partial derivate of x. So this is the solution given to me $F_x(x,y)=\frac{d}{dx} \int_{t=y}^{t=x} ...
3
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1answer
161 views

Find the limit of recursive sequence, if it exists (Analysis, calculus)

My goal is to to test this recursive sequence if it's convergent and if yes, find the limit. $$a_1=3,\:a_{n+1}=\frac{7+3a_n}{3+a_n}$$ I know how to do this with normal sequences, but this is the ...
1
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1answer
43 views

Prove whether a particular function is concave

Given the following equation: $$V(w) = - \frac{\alpha}{2} \left[ y_1(w) + y_2(w) + \int _{-\infty}^{+\infty} \vert y_1(w) - y_2(w) - x\vert f_{T1}(x)dx\right] \\- \beta \int _{w - y_1(w)} ^{+\infty} ...
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2answers
5 views

between what two disjoint sections we can do a Union in order to get this group of solutions?

between what two disjoint sections we can do a Union in order to get this group of solutions? $0<|x+6|\leq{0.4}$ in other words, in what values should I fill the blankets: (____),(____) $\cup$ ...
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1answer
21 views

polynomial solution of second order differential equation

Find the polynomial solution $$u_n(x) = x^n + a_1x^{n-1}+...+a_n$$ of the differential equation $$u_n'' + xu_n' - nu_n = 0$$ satisfied by u_n(x). Note that this is entry-level calculus, so in my ...
1
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1answer
53 views

Limit problem involving cosine

Simply, how would you calculate $$\lim_{x \to 0}x^3\cos\left(\frac{1}{x}\right)$$ I gave this problem by accident to my calculus 1 class.
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5answers
99 views

Find $\int_{ - \infty }^{ + \infty } {\frac{1} {1 + {x^4}}} \;{\mathrm{d}}x$

How can we find the integral: $$\int_{ - \infty }^{ + \infty } {\frac{1} {1 + {x^4}}} \;{\mathrm{d}}x$$ I tried to find and got it to be $\cfrac{\pi}{\sqrt2}$. Am I correct? Please help me with an ...
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0answers
22 views

Find supremum and infimum

I have problem with finding sup and inf of set: $\displaystyle \frac{k^2-n}{k^3+n^2}$ where $k,n \in N$ Is taking first n=1 and later k=1 good move ? Because I dont't have other idea
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3answers
33 views

L'Hospital's Rule to find limit

I am asked to find $\lim\limits_{x \rightarrow \infty} {\left(\frac{8x}{8x+4}\right)}^{5x}$. Could anyone help me with figuring out how to start this problem? Thanks!
3
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3answers
48 views

Solve for $\int \sqrt{x}(\sqrt{x}-2x)^2 dx $

$\int \sqrt{x}(\sqrt{x}-2x)^2 dx $ so I solved this using U-substitution where $u= \sqrt{x}$ so my $du2\sqrt{x}=dx$ then it will be $2 \int u^2(u-2u^2)^2$ and just expand then distribute the $u^2$ so ...
2
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3answers
31 views

Finding maximum points by constrain optimization (multivariable calculus)

Find the maximum value of the function $f(x,y)=x^2+y^2+2x+y$, on the closed disc (the circle together with the region inside the circle) of radius 2, centred at the origin. What i tried I know that ...
2
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2answers
31 views

Calculate an integral that has a sum within.

Im trying to calculate this integral: $\displaystyle \int_{0}^{\pi} \sum_{n=1}^{\infty} \frac{n \sin(nx)}{2^n}$ The only thing I have been able to do is switch the integral and the sum, and in the ...
8
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0answers
472 views

Integral Contest

Before you answer this OP, please read all the terms and conditions below. Thank you... Today I hold an unofficial little contest on brilliant.org. Now, I will hold it here on Math S.E. It's just for ...
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1answer
31 views

Why have we made a function to be many to one and not one to many? [on hold]

We have allowed function to only relate many to one but not one to many. Why haven't we included sin(x) to be a function? Is it just for simplicity? Also, I've seen someone quote a function not even ...
0
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1answer
17 views

find the differential dy

I know this is right is it not? I did the work and I am almost positive I did it right.
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0answers
12 views

f is continious and g is defined as the integral of f. Now how can I show that g'(1)=f(1)

Im trying to solve a question I had on my exam but Im not sure how I should go about solving it. I dont even know in what direction I should look if I want to solve it. I am hoping someone here has a ...
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0answers
15 views

bibliography for weak solutions of ODE's

Some one could recommend to me some bibliography about weak solutions of ODE's, and solutions of ODE's that are not lipschitz??
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0answers
67 views

Real analytic methods for the following integral [duplicate]

A few days back, the following integral was posted $$\int_0^1 x^x(1-x)^{1-x}\sin(\pi x)\,dx=\frac{\pi e}{24}$$ The integral was answered using complex analysis tools but I am interested in other ...
0
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1answer
31 views

summation of series by telescoping series method (feedback needed)

i am stuck i did the first part by cancelling out terms since its a telescoping series. But I do not know how I can proceed any further . Please help. I am not sure of whatever i have done so far. so ...
2
votes
3answers
126 views

Find if this series converges and if so find its value

I need help I cant understand how we can solve this. I am confused when the log came in. I listed the first few terms but i do not know how to proceed further. all I know is that the sequence is ...
2
votes
1answer
38 views

Calculate the inverse of $h(x)=f(2x)$

I have to calculate the inverse $f^{-1}(x)$ of $y=f(x)=2x-1$ and it is simple for this kinds of functions Let $x=f(y)=2y-1$ $x+1=2y$ $\displaystyle\frac{x+1}{2}=y$ We now have the inverse ...
0
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2answers
21 views

Subadditive sequence implies $\lim s_n/n$ exists.

How do I prove that for a subadditive sequence the next limit $$\lim s_n/n$$ exists where $s_n$ is subadditive? PS I can see that $s_n/n$ is subadditive as well, but I don't see how to use it here.
0
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1answer
32 views

Convergence of a non explicit sequence

I was wondering if someone could help me prove this: Let $\lbrace a_n \rbrace$ be a sequence in $\mathbb{R}$ with ${a_n} \rightarrow a $ as $n \rightarrow \infty$. If $a>0$ prove that there exists ...
3
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1answer
29 views

feedback on my solution (integration)

I need help in this problem. I managed to find the answer for this problem by using mathmatica but cannot do the working for it. i have done most of it but i am stuck on the last part.
3
votes
4answers
24 views

derivative of $\ln((1+\beta)^x-1)$

How do I differentiate the term $\ln((1+\beta)^x-1)$ with respect to $x$? Is it possible to do it this way: $$\frac{1}{(1+\beta)^x-1}$$ But i get stuck if i do the normal differentiation.
2
votes
2answers
77 views

Evaluation of $\int_{0}^{\frac{\pi}{2}}\left(\frac{1+\sin 3x}{1+2\sin x}\right)dx$ and $\int_{0}^{2} \left(\sqrt{1+x^3}+\sqrt[3]{x^2+2x}\right)dx$

Evaluation of Some Integrals:: $\displaystyle (a)\;\;\int_{0}^{\frac{\pi}{2}}\left(\frac{1+\sin 3x}{1+2\sin x}\right)dx\;\;\;\;\;\;(b)\;\; \int_{0}^{2} ...
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2answers
33 views

Theorem 3.29 in Baby Rudin

Theorem 3.29 in Walter Rudin's Principles of Mathematical Analysis, 3rd ed., states that If $p>1$, then the series $$\sum_{n=2}^\infty \frac{1}{n (\log n)^p} $$ converges; if $p \leq 1$, the ...
2
votes
4answers
131 views

Find absolute maximum and minimum with domain

Find absolute maximum and minimum of the function $f(x,y)=3-x^2+y^2$ on the region $R = \{(x,y):1≥x≥0, 2≥y≥0\}$ I found that the gradient is $∇f(x,y)=(2x,2y)$ and that the critical point inside ...
2
votes
2answers
34 views

Evaluate the angle between two curves at their intersection: $y=x^2+1, x^2+y^2=1$

Evaluate the angle between two curves at their intersection: $y=x^2+1, x^2+y^2=1$ Actually don't know what the problem means
1
vote
1answer
26 views

finding the derivative of g' and h'

So I know i'm not too terribly far off of the wrong answer but i'm not sure where I went wrong so I was just looking for a little help here. and sorry ahead of time but I don't know how to use the ...
0
votes
1answer
35 views

Radius and Interval of Convergence of $\sum\limits_{n=1}^{\infty}\dfrac{x^n}{2n-1}$

This is my first time finding the radius and interval of convergence of a series, so please bear with me. I would like to find the radius and interval of convergence of ...
0
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3answers
31 views

Given f(x) =$ a/x^2 + x $

Use the second-derivative test to show that if a is positive then the graph has a local minimum, and if a is negative then the graph has a local maximum. So part 1 of this asked to find the critical ...
1
vote
1answer
44 views

A inverse Trigonometric multiple Integrals

How to calculate the closed form of the integral $$\int\limits_0^1 {\frac{{\int\limits_0^x {{{\left( {\arctan t} \right)}^2}dt} }}{{x\left( {1 + {x^2}} \right)}}} dx$$
2
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1answer
24 views

$cos(x)$ and $sin(x)$ span the solution space $V$ of $f''(x)=-f(x)$

Show that for $g(x)\in V$, $(g(x))^2+(g'(x))^2$ is constant. Show that for $g(x)\in V$ with $g(0)=g'(0)=0$, $g(x)=0$ for all $x$. Show that for $f(x)\in V$, $f(x)=f(x)-f(0)\cos(x)-f'(0)\sin(x)$. ...
3
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2answers
50 views

Tedious undefined limit without L'Hospital $\mathop {\lim }\limits_{x \to \frac{\pi }{2}} \,\,\frac{{\tan \,(x)}}{{\ln \,(2x - \pi )}}$

When I try to calculate this limit: $$\mathop {\lim }\limits_{x \to \frac{\pi }{2}} \,\,\frac{{\tan \,(x)}}{{\ln \,(2x - \pi )}}$$ I find this: $$\begin{array}{l} L = \mathop {\lim }\limits_{x \to ...
0
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1answer
21 views

Calculate the volume of intersection of $x^2+y^2+z^2=4$ and $r=2\cos \theta$ by using cylindrical coordinates.

Calculate the volume of intersection of $x^2+y^2+z^2=4$ and $r=2\cos \theta$ by using cylindrical coordinates. My try:Intersection will be a cylinder $x^2+y^2+z^2=4\implies r^2+z^2=4$ Then ...
3
votes
2answers
38 views

Integrating algebraic functions

The function $y = f(x)$, restricted on the domain $ 0 < x < 1$ and satisfying $$y^{5}+y^{4} + x = 0,$$ seems to be well-defined and smooth. So how does one integrate this thing? That is, what ...
1
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2answers
45 views

How do I approach this double integral?

Let $R$ be the region inside $$x^2+y^2 = 1$$ but outside $$x^2+y^2 = 2y$$ with $x \ge 0 $ and $y \ge 0$ Let $$u = x^2 + y^2$$ and $$v = x^2+ y^2 - 2y$$ Compute $ \iint_R xe^y dxdy$ using this change ...
1
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0answers
25 views

2 variable limit

So, I understand why these bigger limit above does not exist (I'll name it 1), but I can understand why the other (2) is $0$. It seems to me that the $y^4/(x^6+y^8)$ is a non limited function and so ...
0
votes
1answer
14 views

Mathematica Cubic Polynomial

I am interested to find the roots of a cubic polynomial f(r)=r^2 - 2 r + Q^2 - [Alpha] r^3. \Alpha is positive real number hare, Q is real. I am interested in 2 complex roots and third root to be ...
2
votes
2answers
20 views

Limits with squares

I have problem with finding such limits: a) $\displaystyle \lim_{n \to \infty}\frac{n^2}{7^{\sqrt{n}}}$ b) $\displaystyle \lim_{n \to \infty}\frac{3^{\sqrt{n}}}{2^n} $ I think the method of ...
0
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2answers
29 views

Find an equation of the straight line [on hold]

Find an equation of the straight line passes through the origin and is tangent to the curve $y=x^3+2$ I got stuck because I have no points.
1
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0answers
31 views

Closed-form of $\sum_{k=1}^{\infty }\left(\psi_1(k)\right)^n$

Inspired by answers to this question, for which $n$ values could we specify a closed-form of $$S(n)=\sum_{k=1}^{\infty }\left(\psi_1(k)\right)^n\,?$$ Here $\psi_1$ is the trigamma function, and ...
1
vote
2answers
57 views

Integration of $F(\sum_k x_k)$ over positive orthant

Problem Suppose we integrate some function $F\left(\sum\limits_{k=1}^n x_k\right)$ over the positive orthant $[0,\infty)^n$. Show that this this is proportional to the integral ...