Mathematics Stack Exchange
Mathematics Stack Exchange

Questions tagged [calculus]

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

Calculus is the branch of mathematics studying the rate of change of quantities, which can be interpreted as slopes of curves, and the lengths, areas and volumes of objects.

Calculus is divided into differential and integral calculus, which are concerned with derivatives

$$\frac{\mathrm{d}y}{\mathrm{d}x}= \lim_{\Delta x \to 0} \frac{\Delta y}{\Delta x}$$

and integrals

$$\int_a^b f(x)\,\mathrm{d}x = \lim_{\Delta x \to 0} \sum_{k=0}^n f(x_k)\ \Delta x_k,$$

respectively.

134529 questions
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Given $y_1,y_2,\ldots,y_N \in \mathbb{R}$, Prove: $f(x)= \sum \limits_{i=1}^{N}|x-y_i|^2$ gets a minimum, and find it.

I'd love your help with this following problem: Given $y_1,y_2,\ldots,y_N \in \mathbb{R}$, I need to prove that $f(x)= \sum \limits_{i=1}^{N}|x-y_i|^2$ gets a minimum, and to find it. I don't know really how to solve it, Any suggestions? Thanks!
Jozef
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$f \geq 0$ continuous, $\lim_{x \to \infty} f(x)$ exists, $\int_{0}^{\infty}f(x)dx< \infty$, Prove: $\int_{0}^{\infty}f^2(x)dx< \infty$

Something that bothers me with the following question: $f: [0, \infty] \to \mathbb{R}$,$f \geq 0$, $\lim_{x \to \infty} f(x)$ exists and finite, and $\int_{0}^{\infty}f(x)dx$ converges, I need to show that $\int_{0}^{\infty}f^2(x)dx$ I separated …
Jozef
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Factorial identity $\left(\tfrac{1}{2}\right)!$ to get Waallis

I asked the wrong question here, my fault :( How does one see, using $n! = \prod_{k=1}^\infty \left(\frac{k+1}{k}\right)^n \frac{k}{k+n}$, that $$\left(\frac{1}{2}\right)! = \prod_{k=1}^\infty \left(\frac{k+1}{k}\right)^{\tfrac{1}{2}}\left(…
bobby
  • 689
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Logarithmic Differentiation - Always possible?

Logarithm functions in basic calculus classes are defined only for positive real numbers. But whenever we find an expression of the form $f(x)^{g(x)}$ we try to use logarithmic differentiation, we don't even care if this expression has negative…
Lages
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Euler Complexification Help.

We have$$e^{ix}=\cos x+i\sin x$$ I want $$\begin{align} I&=\large\int \underbrace{\sin x}_{\large\Im e^{ix}}\quad e^{\pi x}\,\mathrm dx \\ &=\int e^{ix}e^{\pi x}dx\\ &=\int e^{(i+\pi) x}dx\\ &=\frac{e^{(i+\pi)…
SawachikaEri
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Using the fundamental theorem of calculus when the upper limit of integration is $t^2$

I have the find the derivative of the following function: $$F(t) = \int_1^{t^2} \frac{\sqrt{1+s^2}}{s} ds$$ If the upper limit of the integral was $t$ rather than $t^2$, this would be an easy application of the fundamental theorem of calculus. As it…
Jacob
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Line through lattice points

Suppose that circles with radius r are drawn with lattice points as centres. Find the smallest value of r such that any line of form y =$\frac{2}{5}x$+c intersects some of these circles. The way I was thinking of approaching this was to find the c…
John
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Getting a wrong result when calculating $\int \frac{x^3}{\sqrt{1-x^8}}dx$

I was trying to calculate $$\int \frac{x^3}{\sqrt{1-x^8}}dx$$ Here is what I did: Let $u=x^4$ Then $du=4x^3dx$ Therefore we get: $\int \frac{\frac14du}{\sqrt{1-u^2}}$ Let $u=\sin \theta$, Then $du=\cos\theta d\theta$ Therefore we get $\frac14 \int…
pxc3110
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Evaluation of $ \lim_{x\to 0}\left\lfloor \frac{x^2}{\sin x\cdot \tan x}\right\rfloor$

Evaluation of $\displaystyle \lim_{x\to 0}\left\lfloor \frac{x^2}{\sin x\cdot \tan x}\right\rfloor$ where $\lfloor x \rfloor $ represent floor function of $x$. My Try:: Here $\displaystyle f(x) = \frac{x^2}{\sin x\cdot \tan x}$ is an even…
juantheron
  • 53,015
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How to prove $\frac{1}{2}f''(\xi) = \frac{f(a)}{(a-b)(a-c)}+\frac{f(b)}{(b-a)(b-c)}+\frac{f(c)}{(c-a)(c-b)}$

Assume $f(x)$ is continuous in $[a,b]$, and $f''$ in $(a,b)$, prove that for every $c\in (a,b)$, $\exists$ $\xi \in(a,b)$ such that $$\dfrac{1}{2}f''(\xi)=\dfrac{f(a)}{(a-b)(a-c)}+\dfrac{f(b)}{(b-a)(b-c)}+\dfrac{f(c)}{(c-a)(c-b)}.$$ I don't…
89085731
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Find all possible values of $f'(1)$

Let $f$ be a differentiable function such that $f(1)=1$ and the slope of the tangent line to the curve $y=f{[x*f(x*y)]^2 }$ at the point $A(1,1)$ is $3$. Find all possible values of $f'(1)$ . my solution $1=x^2*(f(x))^2 (f(x))^2=1/x^2…
Orkhan
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What do I have to do different when taking derivative of $2^{-x}$ opposed to $2^x$

So for $2^x$ I know the derivative would be $$2^x \ln(2)$$ What would be the thing or step I'd do different for $2^{-x}$ ? I've trying to take the derivative of $$40\over1+2^{-t}$$ & I keep getting the wrong asnwer, I think it's because I'm not…
Nolohice
  • 465
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Help me, a doubt $f(x)=\cot^{-1} \frac{1-x}{1+x}$

I have a doubt $$f(x)=\cot^{-1} \frac{1-x}{1+x}$$ $$f´(x)=\frac{1}{(\frac{1-x}{1+x})^2}\cdot\frac{(-1)(1+x)-(1-x)}{1+\frac{(1-x)^2}{(1+x)^2}}$$ mm this could to be really easy but I do not understand in the first denominator gives one, someone who…
Yobani Ordoñez
  • 496
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Studying a convergence of a series of powers

I'm studying for a calcus exam and I'm trying to solve some of the proposed exercises. I need to study the convergence for $|x+1|^5=R$ $$\sum_{n=0}^{+\infty} \frac{\log(n+12)}{(n+12)\cdot 3^n}\cdot (x+1)^{5n}.$$ The limit of $a_n/a_{n+1}$ (using…
Livio
  • 43
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How to find the maximum and minimum of $\dfrac{\sin x}{x^2+1}$?

How can we find the values of $x$ that give the maximum and minimum of $$\frac{\sin{x}}{x^2+1}$$ I took a lucky guess and found that $\dfrac\pi4$ was fairly close to giving the max, but how does one do this using calculus?
TheRealFakeNews
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