calculus

The lower corner of the page of a book is folded over so as to just reach the inner edge of the page. Find the width of the part folded over, measured along the bottom of the page, when the length of ...

When trying to design asymmetric smooth-step functions $f_{\lbrack 0,1\rbrack}(x)$ that for $0<a<1$ satisfy: $f(0)=0,\ f'(0)=0,\kappa\lbrack f\rbrack(0)=0$ $\kappa\lbrack f\rbrack(a)=0$ ...

I came across this problem in Facebook. What is the limit of the definite integral $$\int\limits_{x^2+y^2\le R^2}\frac{2 x^2+1}{x^4+6 x^2 y^2+y^4+1}-\frac{y^2+1}{x^4+y^4+2} \,dx dy $$ as $R\to\infty$? ...

I wanted to take a crack at proving the shell theorem for gravity using integrals. Suppose we have a sphere of radius $R$ and mass $M$ and a point mass of mass $m$ inside the sphere, then we draw a ...

Nearly everyone who as seen partial fraction decomposition was introduced to it as a way to compute integrals. If P(x) and Q(x) are polynomials, then you can break their ratio P(x)/Q(x) into a sum of terms that can each be integrated in closed form. As with most topics in a calculus class, partial fractions go by in […] The post Partial fraction decomposition first appeared on John D. Cook .

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( ...