calculus

I am asked to give a mini lecture about improper integrals for students of Computer Science in their second semester. So they know the classical Riemann integral on compact intervals for bounded ...

symmetric monoidal (∞,1)-category of spectra analysis (differential/integral calculus, functional analysis, topology) metric space, normed vector space open ball, open subset, neighbourhood convergence, limit of a sequence compactness, sequential compactness … … The different types of square root partial functions on the real numbers that satisfy the functional equation on some subset of the real…

Let $r>0$ and $f:\mathbb{R} \to \mathbb{R}$ be a differentiable function. Define its graph $$G_f := \{(x,f(x)) \mid x \in \mathbb{R}\},$$ the disk $$B_r := \{ p\in\mathbb{R}^2 \mid \|p\| \leq r ...

So I'm new to engineering and have studied some of the calculus but until now, I still have a hard time to understand what is exactly Differential Equations, what is it for and how can I use it in the future classes as an Engineering Physics student

This is a continuation to the previous question, there I realized that maybe the following is true: let $g(x)$ been a continuous function such: $g(x)$ is zero at $x=0$: $\quad g(0)=0$ $g(x)/x$ is ...

analysis (differential/integral calculus, functional analysis, topology) metric space, normed vector space open ball, open subset, neighbourhood convergence, limit of a sequence compactness, sequential compactness … … constructive mathematics, realizability, computability propositions as types, proofs as programs, computational trinitarianism basic constructions: strong axioms further In real ana…

In Riemann integration, one defines both lower and upper sums $ L(f,P), U(f,P), $ and declares a bounded function $f:[a,b]\to\mathbb{R} $ to be integrable if $ \sup_P L(f,P)=\inf_P U(f,P). $ On the ...

I recently wanted to show the area of an ellipse $ (x/a)^2 + (y/b)^2 = 1 $ is $ \pi \cdot a b $. There are multiple approaches, but I first decided to try direct integration in polar coordinates. ...

I was going to append this to @chwala 's thread here , but thought it deserved a new thread. For ##n \geq 1##, define $$f_n : [0,1] \to \mathbb{R} : x \mapsto \begin{cases} 1 & x = 0, \\ n(1 - nx) & x \in (0, \tfrac 1n], \\ 0 & x \in (\tfrac 1n, 1]. \end{cases}$$ (Note that ##f_n## is... Read more

I have a time dependent function as follows: $$f(x + \delta (t)) = \cos(2\pi \cdot (x - \delta(t)) + 0.05\sin(60\pi(x - \delta(t)))$$ where $\delta(t) = 0.05t$ and $ f(x,t=0) = f(x)$. Using the Taylor ...

I am studying real analysis and the book I am reading assumes $f:(a,b)\to\mathbb{R}$ when defining the derivative. It doesn't specify whether the extremes are finite or not. It just requires the value ...

I'd like to store a list of functions that are derivatives of a given function. This is a working example of what I want f = Function[{x,y},Exp[y x]] then assoc= <"Function"->f, ...

Problem: Let $C_1$ and $C_2$ be two curves passing through the origin as indicated in Figure 5.2. A curve $C$ is said to “bisect in area” the region between $C_1$ and $C_2$ if, for each point $P$ of ...

I can't help but to admire those who take on such a challenge. I have an older brother who took up mathematics in college with the help of the GI Bill. I heard that he passed such classes such as advanced Calculus and Trigonometry first in his class. But in life, it didn't appear to do him... Read more

I have this integral I am trying to evaluate as a function of parameters $a$ and $b$. As far as I understand $-1<a<0$ and $b$ is any real number. $$\int_{0}^{\infty} \frac{ x^{a} \left( ...

The limit in question: $$\lim_{x \to 0} \frac{\tan(\tan x) - \sin(\sin x)}{\tan x - \sin x} = 2$$ As stated in the TL;DR, I understand how to get to the correct answer, but I don't know that I would have figured it out without a little help/direction at the beginning because it involves adding... Read more

I was reading Gelfand&Fomin Calculus of Variations book, in which I found this Lemma 2 If $\alpha(x)$ is continuous in $[a, b]$, and if $$\int_a^b \alpha(x) h'(x)\, dx = 0$$ for every function ...

f[n_Integer] := Nest[-D[#, x]/2 &, Cos[x], n] Sum[f[n], {n, 0, Infinity}](*does not work*) The sum of that series can be manually found as $$s(x) = \frac{4}{5} \cos x + \frac{2}{5} \sin x$$ Is ...

The previous post looks at the nonlinear pendulum equation and what difference it makes to the solutions if you linearize the equation. If the initial displacement is small enough, you can simply replace sin θ with θ. If the initial displacement is larger, you can improve the accuracy quite a bit by solving the linearized […] The post Closed-form solution to the nonlinear pendulum equation first …

There’s a nice formula for the nth derivative of a product. It looks a lot like the binomial theorem. There is also a formula for the nth derivative of a quotient, but it’s more complicated and less known. We start by writing the quotient rule in an unusual way. Applying the quotient rule twice gives the following. […] The post nth derivative of a quotient first appeared on John D. Cook .