Articles of calculus

How do I solve floor function integral equation?

This equations comes from my other question, and I thought it was ok to create another question about the same exercise. So I have to solve the equation: $$\int_0^{\lfloor x\rfloor}\lfloor t\rfloor^2\mathrm dt+\lfloor x\rfloor^2(x-\lfloor x\rfloor)=2(x-1),$$which is the same as $$\frac{(\lfloor x\rfloor^2-\lfloor x\rfloor)(2\lfloor x\rfloor-1)}{6}+\lfloor x\rfloor^2(x-\lfloor x\rfloor)=2(x-1),$$ here’s the graph of $f(x)=\int_0^{\lfloor x\rfloor}\lfloor t\rfloor^2\mathrm dt+\lfloor x\rfloor^2(x-\lfloor x\rfloor)$: One […]

Mollifiers: Approximation

Problem Given a mollifier: $\varphi\in\mathcal{L}(\mathbb{R})$ Then it acts as an approximate identity: $$f\in\mathcal{C}(\mathbb{R}):\quad\int_{-\infty}^\infty n\varphi(nx)f(x)dx\to f(0)\cdot\int_{-\infty}^\infty\varphi(x)dx$$ How to prove this under reasonable assumptions? Example As an example regard the Gaussian: $$f\in\mathcal{C}(\mathbb{R}):\quad\frac{n}{\sqrt{\pi}}\int_{-\infty}^\infty e^{-(nx)^2}f(x)\mathrm{d}x\to f(0)$$ (This is a useful technique when studying operator semigroups.)

How is $\displaystyle\lim_{n\to\infty}{(1+1/n)^n} = e$?

This question already has an answer here: Evaluating $\lim_{n \to \infty} (1 + 1/n)^{n}$ [duplicate] 4 answers

Derivative of a function is the equation of the tangent line?

So what exactly is a derivative? Is that the EQUATION of the line tangent to any point on a curve? So there are 2 equations? One for the actual curve, the other for the line tangent to some point on the curve? How can the equation of the tangent line be the same equation throughout […]

Question on using sandwich rule with trig and abs function to show that a limit exists.

Past paper Question: For the following function, determine whether $\lim_{x\to\infty}f(x)$ exists, and compute the limit if it exists. Justify your answers. $$f(x)= \dfrac{\sin(x)+1}{\left| x \right|}$$ Attempt: Consider the fact that $-1 \le \sin(x) \le 1$ (for all $x$), which implies $0 \le \sin(x) +1\le 2$. Dividing by $\left| x \right|,$ $$\color{green}{ \frac{0}{\left| x \right|}} \le […]

How to prove that no constant can bound the function f(x) = x

I know this is a trivial question, but how would one mathematically demonstrate this using a proof?

Definition of local maxima, local minima

Wikipedia says that: A real-valued function $f$ defined on a real line is said to have a local (or relative) maximum point at the point $x^*$, if there exists some $\varepsilon > 0$ such that $f(x^*) \ge f(x)$ whenever $\lvert x − x^*\rvert < \varepsilon$. The value of the function at this point is called […]

Spherical Cake and the egg slicer

Recently I baked a spherical cake (3cm radius) and invited over a few friends, 6 of them, for dinner. When done with main course, I thought of serving this spherical cake and to avoid uninvited disagreements over the size of the shares, I took my egg slicer with equally spaced wedges(and designed to cut 6 […]

Integral of $\int \frac {\sqrt {x^2 – 4}}{x} dx$

I am trying to find $$\int \frac {\sqrt {x^2 – 4}}{x} dx$$ I make $x = 2 \sec\theta$ $$\int \frac {\sqrt {4(\sec^2 \theta – 1)}}{x} dx$$ $$\int \frac {\sqrt {4\tan^2 \theta}}{x} dx$$ $$\int \frac {2\tan \theta}{x} dx$$ From here I am not too sure what to do but I know I shouldn’t have x. $$\int […]

Help on differential equation $y''-2\sin y'+3y=\cos x$

$y”-2\sin y’+3y=\cos x$ I’m trying to solve it by power series, but I just can’t find the way to get $\sin y’$. Is there any special way to find it?