Let's Compare 3^π and π^3

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Опубликовано:

5 окт 2024

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Комментарии : 23

@seanfraser3125 8 месяцев назад

x^(1/x) is decreasing for x>e. Since pi>3>e, 3^(1/3) > pi^(1/pi). Thus 3^pi > pi^3

@shakawtwf 7 месяцев назад

You showed that 3^1/3 is greater than π^1/π but I would like to have seen it put back to the original terms to definitively show that 3^π > π^3. I feel that the last step is missing.

@SyberMath 7 месяцев назад

I agree. Sorry

@GreenMeansGOF 8 месяцев назад

These problems have been solved many times. Since 3 is closer to e, then 3^π is greater. The interesting case is when one number is less than e and the other is greater than e. Unfortunately, the solution to that problem uses the Lambert W function which requires a calculator to evaluate. I don’t know of a better solution for that case.

@yusufdenli9363 8 месяцев назад

The same function (x^(1/x)) can also be used to compare e^pi and pi^e

@paulg444 2 месяца назад

I take the ratio of the two and then take the log. THen expand the log function in a taylor series about 3. .. the result is dispositive.

@farhansadik5423 8 месяцев назад

Educational as always! If you don't mind, would you be able to make videos covering topics on probability or geometry? That would be awesome, since I struggle in both of those. But Thanks for you videos regardless!

@SyberMath 8 месяцев назад

Great suggestion!

@dariosilva85 8 месяцев назад

Calculus is not maths gift to physics, but it is physics gift to math. It was a physicst who invented calculus, namely Isaac Newton.

@avaraportti1873 7 месяцев назад

Newton, well-known non-mathematician

@dariosilva85 7 месяцев назад

@@avaraportti1873 Newton, the well-known physicst.

@dariosilva85 7 месяцев назад

@geraldsmith6225 Physics was the inspiration for developing calculus in the first place.

@marcosrodriguez2496 3 месяца назад

Newton didn't "invent" calculus, he discovered it.

@als2cents679 7 месяцев назад

3^(1/3) is the cube root of 3 and is definitely not transcendental.

@scottleung9587 8 месяцев назад

Yay, I guessed correctly!

@gamemakingkirb667 8 месяцев назад

Really cool vid!

@SyberMath 8 месяцев назад

Glad you think so!

@Nobodyman181 8 месяцев назад

Lets find minimum of x^x please

@Qermaq 8 месяцев назад

I would like three pies. :D

@SyberMath 8 месяцев назад

🥧🥧🥧

@forcelifeforce 8 месяцев назад

@ SyberMath -- 3^(pi) vs. (pi)^3 (pi)*ln(3) vs. 3*ln(pi) (pi)/3 vs. [ln(pi)]/[ln(3)] Let 1 + a = (pi)/3, where a < 1, for convenience sake. Continuing: 1 + a vs. ln[3(1 + a)]/[ln(3)] 1 + a vs. [ln(3) + ln(1 + a)]/[ln(3)] 1 + a vs. [ln(3)]/[ln(3)] + [ln(1 + a)]/[ln(3)] 1 + a vs. 1 + [ln(1 + a)]/[ln(3)] a vs. [ln(1 + a)]/[ln(3)] e^x > x + 1 for x > 0 or, equivalently, x > ln(x + 1) for x > 0. The numerator ln(1 + a) < a, for a > 0. Also, the denominator ln(3) > 1. So, a > [ln(1 + a)]/[ln(3)]. Thus, 3^(pi) > (pi)^3.

@SyberMath 8 месяцев назад

Nice