Articles of calculus

How to find the minimum value of the expression?

Let $x$, $y$, $z$ be three nonegative real numbers and $x^2 + y^2 + z^2 = 5.$ Find the minimum of the expression $$E=\dfrac{1}{2}(x^2 y^2 + y^2 z^2 + z^2 x^2) + \dfrac{96}{x + y + z + 1}.$$ What should I do find the minimum of this expression?

Integral $\int_1^2 \frac1x dx$ with a Riemann sum.

How do you find the $$ \int \dfrac{1}{x} dx$$ by using the idea of a limit of a Riemann sum on the interval [1,2]? I tried splitting the interval into a geometric progression and evaluating the Riemann sum, but i cant simplify the expression at this stage.

Is this a solution to the indefinite integral of $e^{-x^2}$?

$\int e^{-x^2} \, \mathbb{d} x$, the Gaussian integral, is notorious throughout physics and statistics. Its definite integral defined over $\mathbb{R}$ is $\sqrt{\pi}$. However, the current indefinite integral is not an elementary solution. Recently I thought about an integration technique that uses “error correcting.” If we define a function $f(x)$ such that $f^{(n)}(x)$ still contains terms […]

Show that $\limsup \frac1{x_n}\cdot \limsup x_n\geq 1$

Given a sequence $x_{n}$ and initial data that $0< a\leq x_{n}\leq b< \infty $, for $a,b\in \mathbb{R}$. I need to show that: $$\limsup \frac{1}{x_{n}}\cdot \limsup x_{n}\geq 1.$$ I think the the simplest way to do that is to show that $$\liminf \frac{1}{x_{n}}=\frac{1}{\limsup x_{n}}.$$ I started to write some things, but nothing leaded me to the […]

Range of a 1-2 function

$$f(x)=\frac x {x^2+1}$$ I want to find range of $f(x)$ and I do like below . If someone has different Idea please Hint me . Thanks in advanced . This is 1-1 function $\\f(x)=\dfrac{ax+b}{cx+d}\\$, This is 2-2 function $\\f(x)=\dfrac{ax^2+bx+c}{a’x^2+b’x+c’}\\$, This is 1-2 function $\\f(x)=\dfrac{ax+b}{a’x^2+b’x+c’}\\$

>Prove that $\frac d {dx} x^n=nx^{n-1}$ for all $n \in \mathbb R$.

Prove that $\frac d {dx} x^n=nx^{n-1}$ for all $n \in \mathbb R$. I saw some proof of $\frac d {dx} x^n=nx^{n-1}$ using binomial theorem, which is only available for $n \in\mathbb N$. Do anyone have the proof of $\frac d {dx} x^n=nx^{n-1}$ for all real $n$? Thank you.

Find all continuous functions $f(x)^2=x^2$

Find all functions $f$ which are continuous on $\mathbb R$ and which satisfy the equation $f(x)^2=x^2$ for all $x \in \mathbb R$. Clearly $f(x)=x, -x, |x|, -|x|$ all satisfy the condition. However, how can I show that these must be the only possible choices? The condition guarantees that $|f(x)|=|x|$, for all $x$ so I think […]

prove the divergence of cauchy product of convergent series $a_{n}:=b_{n}:=\dfrac{(-1)^n}{\sqrt{n+1}}$

i am given these series which converge. $a_{n}:=b_{n}:=\dfrac{(-1)^n}{\sqrt{n+1}}$ i solved this with quotient test and came to $-1$, which is obviously wrong. because it must be $0<\theta<1$ so that the series converges. my steps: $\dfrac{(-1)^{n+1}}{\sqrt{n+2}}\cdot \dfrac{\sqrt{n+1}}{(-1)^{n}} = – \dfrac{\sqrt{n+1}}{\sqrt{n+2}} = – \dfrac{\sqrt{n+1}}{\sqrt{n+2}}\cdot \dfrac{\sqrt{n+2}}{\sqrt{n+2}} = – \dfrac{(n+1)\cdot (n+2)}{(n+2)\cdot (n+2)} = – \dfrac{n^2+3n+2}{n^2+4n+4} = -1 $ did […]

If $\lim\limits_{x \to \infty} f(x)$ is a finite real number and $f''(x)$ is bounded, then $\lim\limits_{x \to \infty} f'(x) = 0$

This question already has an answer here: If $f(x)\to 0$ as $x\to\infty$ and $f''$ is bounded, show that $f'(x)\to0$ as $x\to\infty$ 3 answers

Integral of product of exponential function and two complementary error functions (erfc)

I found the following integral evaluation very interesting to me: Integral of product of two error functions (erf) and I hoped that I could use that result to evaluate the following integral: $$ \int_{-\infty}^{\infty}\exp\left(-t^{2}\right)\,\mathrm{erfc}\left(t-c\right)\,\mathrm{erfc}\left(d-t\right)\,\mathrm{d}t=\frac{4}{\pi}\int_{-\infty}^{\infty}\exp\left(-t^{2}\right)\int_{t-c}^{\infty}\int_{d-t}^{\infty}\exp\left(-u^{2}-v^{2}\right)\,\mathrm{d}u\,\mathrm{d}v\,\mathrm{d}t $$ So I note that $u\geq t-c$, $v\geq d-t$, thus $t\leq u+c$ and $t\geq d-v$, thus $d-v\leq t\leq u+c$ and $u+v\geq […]