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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Angle between 2 points

Given the following image: $\hskip{1.5 in}$ Supposing $A(100, 300)$ and $B(300, 100)$, how can I find the angle $\alpha$ between A and B? On a side note, what's the main difference between a point and a vector? translating a point to a vector is…
aljndrrr
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What if we change the def of limit as following

In definition of limits why can't we have " there exist delta for all epsilon" instead of " for all epsilon there exist delta"
saqib
  • 101
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4 answers

Compute $\lim_\limits{n\to\infty}a_n$ where $a_{n+2}=\sqrt{a_n.a_{n+1}}$

I managed to show that the limit exists, but I don't know how to compute it. EDIT: There are initial terms: $a_1=1$ and $a_2=2$.
Chris
  • 1,075
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Apply function fractional times

For example, one can apply $\cos x$ to number $a$ one time to get $\cos a$, two times $\cos \cos a$, three times $\cos \cos \cos a$, and so on. Is there a way to define fractional application for $\cos$? Or for any other function? Maybe exists…
Valery Tolstov
  • 89
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Proving $\sup(A+B)=\sup(A)+\sup(B)$

Possible Duplicate: How can I prove $\sup(A+B)=\sup A+\sup B$ if $A+B= \lbrace a+b\mid a\in A, b\in B\rbrace $ here's a homework question I'm currently working on: Let $A,B \subset \mathbb{R}$ non-empty sets bounded from above and from below.…
yotamoo
  • 2,753
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2 answers

Calculus by Apostol Exercise 5.5 Problem 17

There is a function $f$, defined and continuous for all real $x$, which satisfies the equation of the form $$\int_0^x f(t)\,dt = \int_x^1 t^2f(t)dt + \frac{x^{16}}{8} + \frac{x^{18}}{9} + c,$$ I tried to differentiate both sides giving me $f(t) =…
shinobi20
  • 407
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What is a hump function?

I have been in trouble with the hump function(s) What are them? Could you give me an explicit formula for "Hump"(not bump) function. Thanks
honolululurahmi
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Equal values given by Mean Value Theorems

So I was doing some calculus homework the other day, and the following question occurred to me: what functions have the property that the value of $c$ guaranteed by the Mean Value Theorem for Derivatives is the same as the value of $c$ guaranteed by…
Samir Khan
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Question about the sign function $\operatorname{sgn}(x)$

I know I can integrate $|x|$ using the the sign function $\operatorname{sgn}(x)$ as $\int|x|dx=$$\frac{x^2}{2}\operatorname{sgn}(x)+C$ where $\operatorname{sgn}(x)=\frac{x}{|x|}=\frac{d}{dx}|x|$. But when I differentiate…
Hautdesert
  • 1,606
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Is $dy = -\frac{1}{x^2}dx$ a valid rearrangement of $\frac{dy}{dx} = -\frac{1}{x^2}$?

$$ y = \frac{1}{x}$$ $$\frac{dy}{dx} = -\frac{1}{x^2}$$ Is $$dy = -\frac{1}{x^2}dx$$ a valid rearrangement of $\frac{dy}{dx} = -\frac{1}{x^2}$? That is, is that a mathematically meaningful/legitimate operation?
Stan Shunpike
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Why and how could the integral $\int_{-1}^{1} \left(x^2 - \frac{1}{3}\right) dx$ be $0$?

How can this integral be equal to $0$? $$\int_{-1}^{1} \left(x^2 - \frac{1}{3}\right) \; dx$$ The integrand is even, and the bounds are symmetric, how could this be $0$?
gem
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SIB 2009, Problem #2

Let $f(x)$ be a continuous function, whose first and second derivatives are continuous on $[0,2π]$ and $f''(x)≥0\:\:∀\:x∈[0,2π]$. Show that$$ \int _0^{2\pi} f(x) \cos x dx \ge 0$$
Ishaan Singh
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Related rate questions and the intuition behind them

I am trying to do the following question from the Schaum Calculus book. Gas is escaping from a spherical balloon at the rate of $2$ ft$^3$/min. How fast is the surface area shrinking when the radius is $12$ ft? A sphere of radius $r$ has volume $V…
user3754366
  • 287
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Absolute maximum value of $\sin^2(x)-\sin(x)$ in $[0,\frac{3\pi}{2}]$

I thought it does not have absolute maximum, but wanted to just check and see why
Jeremy Carlos
  • 378
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Can every function be written as a total derivative?

This is motivated by a question in the Physics SE, but it's a math question not a physical one. For every function $G(x, t)$, can the function be written as a total derivative (wrt $t$) of some other function $F(x, t)$? A simple counterexample would…
John Rennie
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