Mathematics Stack Exchange
Mathematics Stack Exchange

Questions tagged [calculus]

For basic questions about limits, continuity, derivatives, differentiation, integrals, and their applications, mainly of one-variable functions.

Calculus is the branch of mathematics studying the rate of change of quantities, which can be interpreted as slopes of curves, and the lengths, areas and volumes of objects.

Calculus is divided into differential and integral calculus, which are concerned with derivatives

$$\frac{\mathrm{d}y}{\mathrm{d}x}= \lim_{\Delta x \to 0} \frac{\Delta y}{\Delta x}$$

and integrals

$$\int_a^b f(x)\,\mathrm{d}x = \lim_{\Delta x \to 0} \sum_{k=0}^n f(x_k)\ \Delta x_k,$$

respectively.

134529 questions
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Simple integral help

How do I integrate $$\int_{0}^1 x \bigg\lceil \frac{1}{x} \bigg\rceil \left\{ \frac{1}{x} \right\}\, dx$$ Where $\lceil x \rceil $ is the ceiling function, and $\left\{x\right\}$ is the fractional part function
Ethan Splaver
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What's the biggest circle that can fit between 2 Gaussian curves?

What's the biggest radius possible for a circle to fit completely between the curve $y = e^{-x^2}$ and $y = -e^{-x^2}$ ? This isn't homework, I was just thinking about this randomly. I know calculus will be involved to find the radius, but I just…
Rick
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What allows us divide/multiply dx in calculus?

I've read nearly all of the threads on this topic but none seem to answer my question or lead me in the best direction. When performing U-substituion or even in it's most basic form: $y = 2x$, $dy=2\,dx$. What allows us to do this and what/where…
salman
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Why can partial derivatives be exchanged?

In the Equality of mixed partial derivatives post in this stack exchange, one of the answers to the questions of do partial derivatives commute is: Second order partial derivatives commute if f is $C^2$ (i.e. all the second partial derivatives…
Donald
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Example of a function that is not twice differentiable

Give an example of a function f that is defined in a neighborhood of a s.t. $\lim_{h\to 0}(f(a+h)+f(a-h)-2f(a))/h^2$ exists, but is not twice differentiable. Note: this follows a problem where I prove that the limit above $= f''(a)$ if $f$ is twice…
MathMathCookie
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Show bounded and convex function on $\mathbb R$ is constant

How can we show that a bounded and convex function on $\mathbb R$ is constant? Derivatives are of no use since the function does not have to differentiable. I saw an answer here I think a while ago but did not understand it at all. Since derivatives…
DatFaceTo
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Proof of $ f(x) = (e^x-1)/x = 1 \text{ as } x\to 0$ using epsilon-delta definition of a limit

I am in calc 1 and we have just learned the epsilon-delta definition of a limit and I (on my own) wanted to try and use this methodology in order to prove $(e^x-1)/x = 1$ (one of the equivalencies), along with $\displaystyle \frac {\sin(x)}{x} = 1$,…
Arthur Collé
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Vardi's Integral: $\int_{\pi/4}^{\pi/2} \ln (\ln(\tan x))dx $

Prove that: $\displaystyle\int_{\pi/4}^{\pi/2} \ln (\ln(\tan x))dx =\frac{\pi}{2}\ln \left( \frac{\sqrt{2\pi} \Gamma \left(\dfrac{3}{4} \right)}{\Gamma \left(\dfrac{1}{4} \right)}\right)$ I know that the Vardi's Integral can be evaluated in terms of…
Shobhit Bhatnagar
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avoiding calculus

Some people may carelessly say that you need calculus to find such a thing as a local maximum of $f(x) = x^3 - 20x^2 + 96x$. Certainly calculus is sufficient, but whether it's necessary is another question. There's a global maximum if you restrict…
Michael Hardy
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Show that $f^{(n)}(0)=0$ for $n=0,1,2, \dots$

Let $f: \mathbb{R} \rightarrow \mathbb{R}$ an infinitely many times differentiable function and $f(\frac{1}{n})=0$ for each $n \in \mathbb{N}$. Show that $f^{(n)}(0)=0$ for $n=0,1,2, \dots$ $$$$ Could you give me some hint what I could do?? I got…
Mary Star
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Integration and area - why is integrating over a single point zero?

This is a naive question, but I'm only starting to learn calculus so please cut me some slack. So we all know the integral from $a$ to $b$ of a function over an interval measures the area under the curve $(x, f(x))$ from $a, b$. For any point $c$ in…
user80838
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Does the series $\sum n!/n^n$ converge or diverge?

so I used the root test, but i'm not quite sure if i'm allowed to. I think im performing the operations correctly,a and i keep ending up with $(1)^{\infty}$. So really my question is am i performing the operations wrong or do i have to use a…
lawlipop
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Prove $f(x)=ax+b$

Let $f(x)$ be a continuous function in $\mathbb R$ that for all $x\in(-\infty,+\infty)$, satisfies $$ \lim_{h\rightarrow+\infty}{[f(x+h)-2f(x)+f(x-h)]}=0. $$ Prove that $f(x)=ax+b$ for some $a,b\in\mathbb R$. This is a problem from my exercise…
pxchg1200
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Euler-Maclaurin Summation Formula for Multiple Sums

The Euler-Maclaurin summation formula is \begin{eqnarray} \sum_{k = a}^{b} f(k) = \int_{a}^{b} f(t) \, dt + B_1 (f(a) + f(b)) + \sum_{n = 1}^{N} \frac{B_{2n}}{(2n)!} ( f^{(2n-1)}(b) - f^{(2n-1)}(a) ) + R_{N}, \end{eqnarray} where $B_{n}$ is the…
user02138
  • 17,064
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4 answers

How to find out the global minimum of the following expression

What is the global minimum of the expression \begin{align} |x-1| &+ |x-2|+|x-5|+|x-6|+|x-8|+|x-9|+|x- 10| \\&+ |x-11|+|x-12|+|x-17|+|x-24|+|x-31|+ |x-32|? \end{align} I've solved questions of this sort before but there were only 3 terms. I…
user619072
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