# Tag: approximation

## integration – Find the average of $\sin^{100} (x)$ in 5 minutes?

The Question : 113 people think this question is useful I read this quote attributed to VI Arnold. “Who can’t calculate the average value of the one hundredth power of the sine function within five minutes, doesn’t understand mathematics – even if he studied supermanifolds, non-standard calculus or embedding theorems.” EDIT Source is “A mathematical

## trigonometry – A 1,400 years old approximation to the sine function by Mahabhaskariya of Bhaskara I

The Question : 210 people think this question is useful The approximation $$\sin(x) \simeq \frac{16 (\pi -x) x}{5 \pi ^2-4 (\pi -x) x}\qquad (0\leq x\leq\pi)$$ was proposed by Mahabhaskariya of Bhaskara I, a seventh-century Indian mathematician. I wondered how much this could be improved using our computers and so I tried (very immodestly) to see

## alculus – How is the derivative truly, literally the “best linear approximation” near a point?

The Question : 53 people think this question is useful I’ve read many times that the derivative of a function $f(x)$ for a certain $x$ is the best linear approximation of the function for values near $x$. I always thought it was meant in a hand-waving approximate way, but I’ve recently read that: “Some people

## alculus – Is there an integral that proves $\pi > 333/106$?

The Question : 122 people think this question is useful 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 $\pi > 333/106$? The Question Comments : Hey, how do you know that it is

## alculus – Motivation for Ramanujan’s mysterious $\pi$ formula

The Question : 102 people think this question is useful The following formula for $\pi$ was discovered by Ramanujan: $$\frac1{\pi} = \frac{2\sqrt{2}}{9801} \sum_{k=0}^\infty \frac{(4k)!(1103+26390k)}{(k!)^4 396^{4k}}\!$$ Does anyone know how it works, or what the motivation for it is? The Question Comments : This article by W. Zudilin may give you some references where you may