Questions tagged [calculus]

For basic questions about limits, continuity, derivatives, integrals, and their applications, mainly of one-variable functions. For questions about convergence of sequences and series, this tag can be use with more specialized tags.

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11 views

Double integral with unknown exponents

The problem is: Compute $ I = \int \int _G \dfrac{1}{(1+3x^2+6xy+5y^2)^\alpha} dx dy$ for $\alpha > 2$ and $G =\{ (x,y): -y \leq x \leq y \}$. I have already tried plotting some examples to ...
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1answer
40 views

Calculus theorem

Let $f : [a, b] \longrightarrow \mathbb{R}$ be continuous on $[a, b]$ and differentiable on $(a, b)$. Suppose that $f(a) = a$ and $f(b) = b$. Show that there is $c \in (a, b)$ such that $f^\prime(c) = ...
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16 views

find compact operator

Let A e BI(co) and for n 􀂢 1 , define en in Co by en(n) = 1 and en(m) = 0 for m '# n. Put (Xmn = (Aen)(m) for m, n 􀂢 1 . Prove: (a) M = SUPmL:= 1 I (Xmn l < oo ; (b) for every § 2. The Banach-...
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Limit of integrals, where convergence is uniform only on certain sub-intervals.

Problem: Suppose $f_n$ and $f$ are bounded integrable functions on $[0, 1]$, and that $f_n \to f$ on $[0, 1]$, where convergence is not-uniform on $[0, 1]$, but is uniform on $[\alpha, 1]$ for all $0&...
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35 views

Newton Method in a proof question

I just find out the function of the tangent line and the relationship between $x_n$ and $x_{n+1}$ ,but how can I prove questions $(1),(3)$ and $(4)$?
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18 views

The continuity of a two-variable function and its derivative

In this question, There's a corresponding value of f at (0,1), so we can say that f has a limit at (0,1) does it correct? f is not continuous at (0,1) so we can't do derivative,so (B) & (C) are ...
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52 views

Is there a nice expansion for $\int e^{\cos(x)}dx$? [duplicate]

This indefinite integral has no closed form $$I(x)=\int_0^xe^{\cos(t)}dt$$ The usual expansions of $ I (x) $ are generally obtained by expanding $ e ^ x $ and integrating $ \int \cos ^ \alpha (x) dx $,...
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17 views

Show that a function is radial iff it is rotation invariant

Let $d\in\mathbb N$, $\Omega\subseteq\mathbb R^d$ and $f:\Omega\to\mathbb R$. Remember that $f$ is called radial if $f(x)=f(y)$ for all $x,y\in\Omega$ with $\|x\|=\|y\|$. Let $O(d)$ denote the ...
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2answers
37 views

Minimum and Maximum of $f(x,y)=\sin(y-x)-\sin(y)+\sin(x)$

I need to identify the minimum and maximum of $f(x,y)=\sin(y-x)-\sin(y)+\sin(x)$ with $0 \le x \le y \le 2\pi$. First I calculated $$ \nabla f(x,y)=(\cos(x)-\cos(y-x),\cos(y-x)-\cos(y)). $$ The ...
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1answer
34 views

Two different definitions of Gateaux differentiability

My textbook defines Gateaux derivative as the following: (Defnition) A mapping $F: D\subset \mathbb{R^n} \to \mathbb{R^m} $ is Gateaux-differentiable at an interior point $x$ of of D if there eixsts ...
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92 views

Fascinating equality

The following problem is proposed by a friend:$$4\int_0^{\pi/4}\left(\int_x^{\pi/4} (x-y)\ln(\tan (x)) \ln\left(\tan \left(y+\frac{\pi }{4}\right)\right) \textrm{d}y\right)\textrm{d}x$$ $$-\sum _{n=1}^...
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33 views

An example of a continuous function theorem

I have been looking for an example of the following theorem: If the function $f$ is continuous on real numbers and we have a sequence $a_n$ that has all the terms in the domain of $f$ and converges to ...
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1answer
42 views

Finding the degree of a differential equation [duplicate]

I am trying to find the degree of the following differential equation: $$ \sqrt{\frac{d^2y}{dx^2}}+\frac{dy}{dx}=y^3. $$ I am not $100$% sure, but I know (correct me if I'm wrong) that for a ...
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3answers
57 views

Find the area of the part of the sphere $x^2+y^2+z^2=2$ that lies inside the cylinder $y^2+z^2=1$

This is what I have tried so far. I am just not sure whether I got it right or not. We have $y^2+z^2=1$, then $r=1,\quad x=\sqrt{2-y^2-z^2}\quad\text{and}\quad x=\sqrt{2-r^2}$ $$\frac{\partial{x}}{\...
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1answer
45 views

Proof of Fermat's Theorem for a Local Minimum

Fermat's Theorem for Local Extrema states that if a function $f(x)$ has a local extremum at $c$ and $f'(c)$ exists, then $f'(c)=0$. I saw a textbook proof for the local maximum case that used the ...
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2answers
56 views

Looking for other approaches to evaluate $\lim _{x\to 0^+}\frac{\sin 3x}{\sqrt{1-\cos ^3x}}$

Here is my approach: $$\lim _{x\to 0^+}\frac{\sin 3x}{\sqrt{1-\cos ^3x}}=\lim _{x\to 0^+}\frac{1}{\sqrt{\cos^2x+\cos x+1}}\times\frac{\sin 3x}{\sqrt{1-\cos x}}$$ $$=\lim _{x\to 0^+}\frac1{\sqrt3}\...
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1answer
40 views

Is the given manipulation of the differential operator $D$ valid?

What I know till now is, the differential operator $D$ means $D=\frac{d}{dx}$ and $D^2=\frac{d^2}{dx^2}$. So, $D^2y=D(Dy)$, $$ \frac{d}{dx}\,\frac{d}{dx}=\left(\frac{d}{dx}\right)^2, $$ and $$ \frac{d}...
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17 views

Effect of Rectifying Function on Fourier Transform

Let there be some rectifying function, e.g. $$r(x) = \max (x,0).$$ What would the effect of this filter be on the Fourier transform of some generic function $f(x)$, in particular its Laplacian? I have ...
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1answer
126 views

Can the product of two functions with no inflection points have an inflection point?

If $f(x)$ and $g(x)$ are functions with no inflection points, can $h(x)=f(x)\cdot g(x)$ have an inflection point? Edit: I experimented a bit with a few functions (like $x^2\cdot x^2$) in a graphing ...
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1answer
49 views

Evaluate $f'(1)$ and $f'(6)$ if $f'(x)=5x^\frac45 -6x^\frac67$.

Suppose that $f(x)=5x^\frac45 -6x^\frac67$ . Evaluate each of the following: $f'(1)$, $f'(6).$ Should I plug in the values, use the product rule and then differentiate?... Or should I differentiate ...
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2answers
53 views

Is the proof of the connection between continuity and the limit of a function correct?

We have done the following proof: Definition: $\ f(x)\ $ is continuous at $\ x_0\ $ if $\ \vert x-x_0\vert<\delta\implies \vert f(x)-f(x_0)\vert<\varepsilon.$ The connection: $\lim_{x \...
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1answer
23 views

Show $\int \frac{-18}{3x^2-6x-9}dx=\frac{3}{2}\ln\frac{x+1}{2}-\frac{3}{2}\ln\frac{x-3}{2}$ [closed]

How do I show that $\int \frac{-18}{3x^2-6x-9}dx=\frac{3}{2}\ln\frac{x+1}{2}-\frac{3}{2}\ln\frac{x-3}{2}$
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1answer
28 views

For a bijective function show that

Let $f:A \to B$ be a bijective function such that $X \subset A$ Prove that $$f(A-X) = B -f(X) $$ Couldnt solve this question. Can someone please provide solution or some hint. Thank you
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14 views

How do I derive ui wrt qi in order to find the null [closed]

How do I derive ui with respect to qi to find the null: ui(qi,q-1)=(x-qi)*$\sum_{j=1}^n$qj According to the solution, I should obtain: (x-qi)-$\sum_{j=1}^n$qj=0
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34 views

Largest triangle in the unit circle

I need to maximize the area of a triangle which is laying in the unit circle and its area described by $f(x,y)=0.5 \cdot (sin(y-x)-sin(y)+sin(x))$ for $0\le x\le y \le 2\pi$ I calculated the gradient ...
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1answer
39 views

How to argue that $\min \left\lVert Ax-b \right\rVert_2$ on $M := \{ x \in \mathbb{R}^m \mid x \ge 0 \}$ has a solution

This is related to an old question of mine, I was not sure if I should make a new question, so please tell me if I should edit the old one and I will delete this one. Consider the following exercise: ...
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1answer
55 views

Use the chain rule to compute the derivative of an inverse function.

Given a function $f(x)$, let $g=f^{-1}(x)$. Then define the composite function $f(g(x))$. Then, by the Chain Rule, $(f(g(x)))'=f'(g(x)) g'(x)$. However, since $f(x)$ and $g(x)$ are inverses, $f(g(x))=...
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1answer
25 views

Radius of Convergence for $\sum _{n=1}^{\infty }\frac{\left(-1\right)^n}{\left(2n-2\right)!}\left(2n+1\right)x^{2n}$

Let $f(x)=\sum _{n=1}^{\infty }\frac{\left(-1\right)^n}{\left(2n-2\right)!}\left(2n+1\right)x^{2n}$ I want to find the radius of convergence. So I wrote the series with $t=-x^2$, like: $-\sum _{n=1}^{\...
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2answers
42 views

$f,g:[0,1]→R$ be $2$ continuous functions such that $f(x)g(x)≥4x^{2}$, for all $x∈[0,1]$. Prove that $|\int_{0}^1f(x)dx|≥1$ or $|\int_{0}^1g(x)dx|> 1$

Let $f,g:[0,1]\mapsto\mathbb R$ be two continuous functions such that $f(x)g(x)\ge4x^{2}$, for all $x\in[0,1]$ . Prove that $|\int_{0}^1f(x)dx|\ge1$ or $|\int_{0}^1g(x)dx|\ge1$ My approach: Since $f(x)...
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0answers
21 views

Completeness of a dual space

I am working on a problem and i have pretty much a solution to every part ( total of three parts ). I would appreciate it, if you look at the solutions and let me know if they are acceptable. If not, ...
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0answers
45 views

$(1-x)^n$ equals?

I'm struggling to find the $x$ formula in this equation, and I'm stuck in $(1-x)^{n}$ $x=\epsilon\frac{\,\alpha(n+1)\;+\;\beta(1-c))}{\alpha(n+1)\,+\,\beta}+\epsilon \frac{c\,\alpha(n+1)\,(1-x)^n}{\...
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0answers
15 views

About convergence set and radius of convergence

Find the convergence sets of each of the series: 1.$\sum_{n=1}^{\infty} 3^n sin(\frac {1}{4^nx})$. $\sum_{n=1}^{\infty} \frac {x^n-x^{-n}}{2021^n}$. 3.$\sum_{n=1}^{\infty}\frac {(1+\frac{1}{n})^n}{...
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59 views

If $6=x\sin\alpha+y\sin55^\circ$ and $x\cos\alpha=y\cos55^\circ$, find $\alpha$ for which $x$ and $y$ are as small as possible

Given equations, $$6 = x\,\sin\alpha + y\,\sin55^\circ \tag{1}$$ and $$x\,\cos\alpha = y\,\cos55^\circ \tag{2}$$ Find the value of $\alpha$ for which $x$ and $y$ are as small as possible. If the ...
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0answers
37 views

Integration with respect to a discontinuous function

I know that a function is differentiable iff it follows the first theorem of differentiability i.e. $$\lim_{h\to0}\frac{f(x+h)-f(x)}{h}$$ so if a function is discontinuous at some points it is non ...
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0answers
21 views

Let $f(x) = (x+1)\ln(x+1)−x\ln x−\ln(2x+1)$ for $x > 0$. Show that f is strictly increasing on (0, ∞). [closed]

Let $f(x) = (x+1)\ln(x+1)−x\ln x−\ln(2x+1)$ for $x > 0$. Show that f is strictly increasing on (0, ∞). Further, show that the sequence $ \frac{ (n+1)^{n+1}} {(n^n) (2n+1)} $ is strictly ...
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1answer
70 views

Let $f(x)$ be a differentiable function that satisfies the relation $f(x+y) = f(x)f(y)$. Prove $f(x) \equiv 0$ or $f(x) = e^{ax}$

My partial solution is as follows: Let $y$ be the independent variable and $x$ a constant; differentiating with respect to y, we get $$\frac {d}{dy}f(x+y)= \frac {d}{dy}f(x)f(y)=f(x)f^{'}(y) \\ \...
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0answers
29 views

How should i substitute $x$ in this integral in order to evaluate it [closed]

$${ \int_0^{\infty}((x^{\alpha}+1)^{\frac1{\alpha}}-x)\ \mathrm dx }$$
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1answer
50 views

$a_{n+1} = \frac{a^2_n}{a_n + 1}$ for each non-negative integer n. Prove that for $0 \leq n \leq 998,$ $1994 - n\leq a_n$

$a_{n+1} = \frac{a^2_n}{a_n + 1}$ for each non-negative integer $n$ and $a_0=1994$. Prove that for $0 \leq n \leq 998 $ the number $1994 - n$ is the greatest integer $\leq a_n$ Differentiating did ...
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0answers
27 views

Show that the following integral is convergent , if n>0 [closed]

I was having a trouble while solving this question, can anyone please give the solution for it?
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0answers
20 views

How to increase firs order system dynamic to second order system [closed]

So, I have system dynamic equations, and I don't know how to increase system order 1 to order second
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0answers
42 views

Is this expression for $\int e^{\cos(x)} dx$ using infinite trigonometric series correct?

TL;DR: I think I can show that $$\begin{align}\int e^{\cos(x)} dx = x\cdot \ I_{0}( 1) \ &+\sum _{n=0}^{\infty }\sum _{k=0}^{n}\frac{\sin( x( 2k-2n-1))}{k!\cdot ( 2n-k+1) !\cdot ( 2k-2n-1) \cdot ...
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1answer
103 views

Why is this integration proof incorrect?

I recently came across the fact that $\displaystyle\int_{\frac{m}{a}}^{\frac{n}{a}}\frac{f(ax)}{x}\,dx$ is independent of the value of $a$. This lead me to this integral here: $\displaystyle \int_0^\...
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0answers
19 views

A consequence of $\left(\sum_{i=1}^N f_i(t) a_i\right) \cdot b = 0$ $\forall b \in \left(\sum_{i=1}^N a_i\right)^\perp$

Let $a_i$ with $i=1, \dots, N$ be vectors in $\mathbb R^3$ and let $f_i: \mathbb R \to \mathbb R$ with $i=1, \dots, N$ (here $N\ge 2$). Assume that $$\left(\sum_{i=1}^N f_i(t) a_i\right) \cdot b = 0 \...
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0answers
17 views

Multivariable Chain Rule Dependent Variables

Why is there no multivariable chain rule for dependent variables? Also, if there is a composition of functions, say $$\psi=\phi(x(u,v),y(u,v))$$, then how can one find the partial derivative of $\psi$...
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1answer
68 views

Integrating the remainder of $\frac{1}{(1+x)}$ by hands

It can be easily shown, by simply sum everything up, that$$f(x)=\frac{1}{1+x}=\frac{(1+x)}{1+x}-\frac{(x+x^2)}{1+x}+\frac{(x^2+x^3)}{1+x}-...+\frac{[(-1)^{n-1}](x^{n-1}+x^n)}{1+x}+\frac{[(-1)^{n}]x^n}{...
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0answers
20 views

Optimitation problem of factory [closed]

A factory of a certain material can produce x tons of the material of second grade per day and y tons of top grade material per day, where the relationship between x and y is given by: y = $\frac{ 40-...
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2answers
45 views

If $u=x\cos y$ and $v=x\sin y$, then near $(x_0,y_0)$, with $x_0\neq 0$, then $(x,y)$ can be written as a differentiable function of $(u,v)$

Consider the following equations $$u = x \cos(y)\ \ \text{and}\ \ v=x \sin(y)$$ Show that near to $(x_0,y_0)$, with $x_0\neq 0$, $(x,y)$ can be written as a differentiable function of $(u,v)$. It's ...
2
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0answers
56 views

The reduction formula doesn't work for $\frac{d}{dx}\int_{2x-\frac{\pi}{2}}^{\frac{\pi}{2}+3x} \sin^{2021}{t}\,dt$

$$\frac{d}{dx}\int_{2x-\frac{\pi}{2}}^{\frac{\pi}{2}+3x} \sin^{2021}{t}\,dt$$ I used the reduction formula for this problem, but I couldn't get the result. Do you know any other ways I can use to ...
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2answers
24 views

If $f \in \mathcal{C}([a,b])$ exactly one local minimum $x^*$, then $a \le x_1 < x_2 \le x^* \implies f(x_1) > f(x_2)$

Show that if a continuous function $f: [a,b] \rightarrow \mathbb{R}$ has exactly one local minimum, then $$a \le x_1 < x_2 \le x^* \implies f(x_1) > f(x_2)$$ and $$x^* \le x_1 < x_2 \le b \...
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1answer
30 views

Stuck on a polar coordinates integral calculus question

Once every century the Klingon moon Bosok completely eclipses the moon Yan-ki. As seen from the city Mitek, the two moons appear as circular discs, with radii respectively $\sqrt{2},1$ (unit: dorks). ...

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