calculus

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 .

I wanted to try calculating some tangent vectors for a picture, just for fun. This requires taking partial derivatives. It will be some time before I can automate partial derivatives algebraically for ...

Fundamental Theorem of Calculus Although the notion of area is intuitive, its mathematical treatment requires a rigorous definition. This post introduces the Riemann integral, and proves the fundamental theorem of calculus—a beautiful result that connects integrals and derivatives. Riemann integral § Given a bounded1 1 Note that continuity is not required here; boundedness alone ensures the subin…

I have to solve this triple integral ∭ x y|z|³/(1+ (x²+y²)⁴)  dx  dy dz, with a domain T={(x, y, z) ∈ ℝ³: x ≤ 0, y ≥ 0, z² ≤ x²+y² ≤ 1}. Plotting with DESMOS 3D I see this: Actually I am not able to ...

I have been converting double integrals to polar coordinates and I found a wrong way to convert to polar. Can someone explain to me the error in doing it? $$\iint _Df(x,y)\text dx\text dy$$Let ...

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 Michael Tabor's "Chaos and Integrability in Nonlinear Dynamics", 1989, page 12, the nonlinear pendulum differential equation is integrated by introducing an invariant $E'$ which yields ...