Questions tagged [calculus]

For basic questions about limits, continuity, derivatives, integrals, and their applications, mainly of one-variable functions. For questions about convergence of sequences and series, this tag can be use with more specialized tags.

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 sometimes 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. The Fundamental Theorem of Calculus relates these two concepts.

While ideas related to calculus were known for some time (Archimedes' method of exhaustion was a form of calculus), it was not until the independent work of Newton and Leibniz that the modern elegant tools and ideas of calculus were developed. Riemann and Lebesgue later extended the ideas of integration. More recently, the Henstock–Kurzweil integral has led to a more satisfactory version of the second part of the Fundamental Theorem of Calculus.

Even so, many years elapsed until mathematicians such as Cauchy and Weierstrass put the subject on a mathematically rigorous footing; it was Weierstrass who formalized the definition of continuity of a function, proved the intermediate value theorem, and proved the Bolzano-Weierstrass Theorem.

Source: Wolfram Mathworld

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Why is $1^{\infty}$ considered to be an indeterminate form

From Wikipedia: In calculus and other branches of mathematical analysis, an indeterminate form is an algebraic expression obtained in the context of limits. Limits involving algebraic operations are often performed by replacing subexpressions by…
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Lesser-known integration tricks

I am currently studying for the GRE math subject test, which heavily tests calculus. I've reviewed most of the basic calculus techniques (integration by parts, trig substitutions, etc.) I am now looking for a list or reference for some lesser-known…
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Striking applications of integration by parts

What are your favorite applications of integration by parts? (The answers can be as lowbrow or highbrow as you wish. I'd just like to get a bunch of these in one place!) Thanks for your contributions, in advance!
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Is there any integral for the Golden Ratio?

I was wondering about important/famous mathematical constants, like $e$, $\pi$, $\gamma$, and obviously the golden ratio $\phi$. The first three ones are really well known, and there are lots of integrals and series whose results are simply those…
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Why does factoring eliminate a hole in the limit?

$$\lim _{x\rightarrow 5}\frac{x^2-25}{x-5} = \lim_{x\rightarrow 5} (x+5)$$ I understand that to evaluate a limit that has a zero ("hole") in the denominator we have to factor and cancel terms, and that the original limit is equal to the new and…
Emi Matro
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How to determine with certainty that a function has no elementary antiderivative?

Given an expression such as $f(x) = x^x$, is it possible to provide a thorough and rigorous proof that there is no function $F(x)$ (expressible in terms of known algebraic and transcendental functions) such that $ \frac{d}{dx}F(x) = f(x)$? In other…
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Why does the series $\sum_{n=1}^\infty\frac1n$ not converge?

Can someone give a simple explanation as to why the harmonic series $$\sum_{n=1}^\infty\frac1n=\frac 1 1 + \frac 12 + \frac 13 + \cdots $$ doesn't converge, on the other hand it grows very slowly? I'd prefer an easily comprehensible explanation…
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Identification of a curious function

During computation of some Shapley values (details below), I encountered the following function: $$ f\left(\sum_{k \geq 0} 2^{-p_k}\right) = \sum_{k \geq 0} \frac{1}{(p_k+1)\binom{p_k}{k}}, $$ where $p_0 > 0$ and $p_{k+1} > p_k$ for all $k$. In…
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Proving an alternating Euler sum: $\sum_{k=1}^{\infty} \frac{(-1)^{k+1} H_k}{k} = \frac{1}{2} \zeta(2) - \frac{1}{2} \log^2 2$

Let $$A(p,q) = \sum_{k=1}^{\infty} \frac{(-1)^{k+1}H^{(p)}_k}{k^q},$$ where $H^{(p)}_n = \sum_{i=1}^n i^{-p}$, the $n$th $p$-harmonic number. The $A(p,q)$'s are known as alternating Euler sums. Can someone provide a nice proof that $$A(1,1) =…
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How to find ${\large\int}_0^1\frac{\ln^3(1+x)\ln x}x\mathrm dx$

Please help me to find a closed form for this integral: $$I=\int_0^1\frac{\ln^3(1+x)\ln x}x\mathrm dx\tag1$$ I suspect it might exist because there are similar integrals having closed forms: $$\begin{align}\int_0^1\frac{\ln^3(1-x)\ln x}x\mathrm…
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Why is integration so much harder than differentiation?

If a function is a combination of other functions whose derivatives are known via composition, addition, etc., the derivative can be calculated using the chain rule and the like. But even the product of integrals can't be expressed in general in…
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How to prove $\int_0^1\tan^{-1}\left[\frac{\tanh^{-1}x-\tan^{-1}x}{\pi+\tanh^{-1}x-\tan^{-1}x}\right]\frac{dx}{x}=\frac{\pi}{8}\ln\frac{\pi^2}{8}?$

How can one prove that $$\int_0^1 \tan^{-1}\left[\frac{\tanh^{-1}x-\tan^{-1}x}{\pi+\tanh^{-1}x-\tan^{-1}x}\right]\frac{dx}{x}=\frac{\pi}{8}\ln\frac{\pi^2}{8}?$$
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Is there an integral that proves $\pi > 333/106$?

The following integral, $$ \int_0^1 \frac{x^4(1-x)^4}{x^2 + 1} \mathrm{d}x = \frac{22}{7} - \pi $$ is clearly positive, which proves that $\pi < 22/7$. Is there a similar integral which proves $\pi > 333/106$?
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What is the probability that a point chosen randomly from inside an equilateral triangle is closer to the center than to any of the edges?

My friend gave me this puzzle: What is the probability that a point chosen at random from the interior of an equilateral triangle is closer to the center than any of its edges? I tried to draw the picture and I drew a smaller (concentric)…
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What is the Jacobian matrix?

What is the Jacobian matrix? What are its applications? What is its physical and geometrical meaning? Can someone please explain with examples?