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

123312 questions
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Rare Integral $\int_0^1 \frac{\cosh \left( \alpha \cos ^{-1}x \right)\cos \left( \alpha \sinh ^{-1}x \right)}{\sqrt{1-x^{2}}} dx$

I need to prove if this statement is true, some ideas? $$\int_0^1 \frac{\cosh\left(\alpha \cos ^{-1}x\right)\cos \left( \alpha \sinh^{-1} x \right)}{\sqrt{1-x^2}} \, dx = \frac \pi 4 + \frac 1 {2\alpha }\cdot \sinh \frac{\alpha \pi } 2$$
whitexlotus
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Evaluation of the limit $\lim\limits_{n \to \infty } \frac1{\sqrt n}\left(1 + \frac1{\sqrt 2 }+\frac1{\sqrt 3 }+\cdots+\frac1{\sqrt n } \right)$

Evaluate the limit : $$\lim_{n \to \infty } {1 \over {\sqrt n }}\left( {1 + {1 \over {\sqrt 2 }} + {1 \over {\sqrt 3 }} + \cdots + {1 \over {\sqrt n }}} \right)$$ I can use the sandwich principle, certain convergence criteria, Cesaro mean theorem,…
Adar Hefer
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Chain rule for discrete/finite calculus

In the context of discrete calculus, or calculus of finite differences, is there a theorem like the chain rule that can express the finite forward difference of a composition $∆(f\circ g)$ in simplified or otherwise helpful terms? It's probably not…
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Maximum value of a function. I am not able to check double derivative.

Can someone explain me how $\sin^p x \cos^q x$ attains maximum at $\tan^2 x = \frac pq$. I am not able to check whether double derivative is positive or negative. Question Show that $$\sin^p\theta\cos^q\theta$$ attains a maximum when…
HoLyDeViL
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Integration by parts, the cases when it does not matter what $u$ and $dv$ we choose.

I was reviewing Integration By Parts on Brilliant.org where an example they use is $$\int x \ln x \;dx$$ Let $u=\ln x$ and $dv=x\;dx$ such that $$\begin{align} \int x \ln x\;dx&\;=\;\frac 12x^2\ln x\;-\;\frac 12\int \frac{x^2}x\;dx\\ \\ &\;=\;…
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Is the proof of $\lim_{\theta\to 0} \frac{\sin \theta}{\theta}=1$ in some high school textbooks circular?

I was taught the following proof in high school. By constructing triangles with $0<\theta<\pi/2$ and a circle with radius $r$ and by comparing the areas, we have $$\frac{1}{2}r^2\sin\theta\cos\theta \le…
velut luna
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Proving that $\sin x \ge \frac{x}{x+1}$

Prove that $$ \sin x \ge \frac{x}{x+1}, \space \space\forall x \in \left[0, \frac{\pi}{2}\right]$$
user 1591719
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Calculus of variations with two functions and inequality

I wish to extremise $$ Q = \int_0^h u \, \,dy $$ with the following constraints $$B = \int_0^h u g \, \, dy \\ M = \int_0^h u^2 +\left(\int_0^y g \, \,dy^*\right) \, \, dy$$ where $M,B$ are constant. So my idea is to set this up as a variational…
mch56
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Proving Ramanujan's Integral Formula

In a letter to Hardy, Ramanujan described a simple identity valid for $0
Crescendo
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Convexity of $x\left(1+\frac1x\right)^x,\ x\ge 0$

This may turn out to be really simple but I do not see a quick way to the proof. How would one show $\displaystyle x\Big(1+\frac1x\Big)^x,\ x\ge 0$ is convex? I derived the second derivative. It has a negative term. I suppose I could combine certain…
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Evaluating $\int \frac {\sqrt{\tan \theta}} {\sin 2\theta} \ d \theta$

I am trying to evaluate $$\int \frac {\sqrt{\tan \theta}} {\sin 2\theta} \ d \theta$$ I tried rewriting it as $$\int {\sqrt{\tan \theta}} \cdot \csc(2\theta) \ d\theta$$ Supposedly letting $u = \sqrt{\tan \theta}$ cleans up the integral to just…
Joe
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On $\int_0^1\arctan\,_6F_5\left(\frac17,\frac27,\frac37,\frac47,\frac57,\frac67;\,\frac26,\frac36,\frac46,\frac56,\frac76;\frac{n}{6^6}\,x\right)\,dx$

Reshetnikov gave the remarkable evaluation, \begin{align} I&= \int_0^1\arctan{_4F_3}\left(\frac15,\frac25,\frac35,\frac45;\frac24,\frac34,\frac54;\frac{1}{64}\,x\right)\,dx…
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Using Wallis' formula to show the limit of $a_n:=n!\left(\frac{e}{n}\right)^n n^{-1/2}.$

Let $$a_n:=n!\bigg(\frac{e}{n}\bigg)^n n^{-1/2}.$$ With the help of Wallis' formula $$\frac{\pi}{2} = \prod_{n=1}^\infty \frac{4n^2}{4n^2-1}=\lim_{m\to \infty}\frac{2^{4m}(m!)^4}{((2m)!)^2 (2m+1)}$$ show that the limit of $a_n$ is $\sqrt{2\pi}$. I…
nomadicmathematician
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How to prove $_2F_1\big(\tfrac16,\tfrac16;\tfrac23;-2^7\phi^9\big)=\large \frac{3}{5^{5/6}}\,\phi^{-1}\,$ with golden ratio $\phi$?

(Note: This is the case $a=\frac16$ of ${_2F_1\left(a ,a ;a +\tfrac12;-u\right)}=2^{a}\frac{\Gamma\big(a+\tfrac12\big)}{\sqrt\pi\,\Gamma(a)}\int_0^\infty\frac{dx}{(1+2u+\cosh x)^a}.\,$ There is also $a=\frac13$ and $a=\frac14$.) After investigating…
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Integral of a function's derivative does not equal the original function?

I am struggling with assessing the validity of this statement. $$\int ^{x}_{a}f'\left( t\right) dt \neq f\left( x\right) $$ I can understand that the left side yields a class of functions $F(x)$ whose derivative is $f(x)$, but doesn't that mean…
rb612
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