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ThatDonGuy
ThatDonGuy
Joined: Jun 22, 2011
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February 22nd, 2024 at 4:33:03 PM permalink
Quote: Wizard

What is the limit as x approaches 0, of ln(1-x)/sin(x)?
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Apply L'Hopital's Rule:

d/dx of ln (1-x) = -1 / (1-x)
d/dx of sin x = cos x
The answer is the limit as x approaches 0 of -1 / ((1 - x) cos x) = -1

Wizard
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Wizard
Joined: Oct 14, 2009
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February 22nd, 2024 at 5:11:37 PM permalink
Quote: ThatDonGuy



Apply L'Hopital's Rule:

d/dx of ln (1-x) = -1 / (1-x)
d/dx of sin x = cos x
The answer is the limit as x approaches 0 of -1 / ((1 - x) cos x) = -1


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I agree!

L'Hopital's Rule would have been an applicable rule in the Mean Girls calculus competition.


Direct: https://www.youtube.com/watch?v=oDAKKQuBtDo
"For with much wisdom comes much sorrow." -- Ecclesiastes 1:18 (NIV)
DogHand
DogHand
Joined: Sep 24, 2011
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February 23rd, 2024 at 2:00:47 AM permalink
Quote: Wizard

Quote: ThatDonGuy



Apply L'Hopital's Rule:

d/dx of ln (1-x) = -1 / (1-x)
d/dx of sin x = cos x
The answer is the limit as x approaches 0 of -1 / ((1 - x) cos x) = -1


link to original post



I agree!

L'Hopital's Rule would have been an applicable rule in the Mean Girls calculus competition.


Direct: https://www.youtube.com/watch?v=oDAKKQuBtDo
link to original post



Wiz,

Speaking of L'Hopital, (or more correctly, L'Hôpital), how would you like to have the "correct" spelling of your name changed retroactively?

Back when I took calculus, we spelled his name "L'Hospital" (which is how he himself spelled his name) and generally referred to him jokingly as "the Hospital guy".

According to Wikipedia:

"In the 17th and 18th centuries, the name was commonly spelled "l'Hospital", and he himself spelled his name that way. Since then, French spellings have changed: the silent 's' has been removed and replaced with a circumflex over the preceding vowel."

This whole "change the spelling" business reminds me of an old Dolton Edwards science fiction short story, "Meihem in ce Klasrum": read it here.

Dog Hand
Wizard
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Wizard
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February 23rd, 2024 at 4:18:57 AM permalink
Quote: DogHand

Wiz,

Speaking of L'Hopital, (or more correctly, L'Hôpital), how would you like to have the "correct" spelling of your name changed retroactively?

Back when I took calculus, we spelled his name "L'Hospital" (which is how he himself spelled his name) and generally referred to him jokingly as "the Hospital guy".

According to Wikipedia:

"In the 17th and 18th centuries, the name was commonly spelled "l'Hospital", and he himself spelled his name that way. Since then, French spellings have changed: the silent 's' has been removed and replaced with a circumflex over the preceding vowel."

This whole "change the spelling" business reminds me of an old Dolton Edwards science fiction short story, "Meihem in ce Klasrum": read it here.

Dog Hand
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Nobody has ever asked me that before. I think I favor changing the spelling to conform with the way language changes.
"For with much wisdom comes much sorrow." -- Ecclesiastes 1:18 (NIV)
gerback123
gerback123
Joined: May 13, 2020
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February 27th, 2024 at 2:47:19 PM permalink
In the state lottery, a person selects a 3 digit number and wins $500 if that number is drawn from a box containing tickets labeled 000,···,999. If it costs $1 to play the game, what is the player's expected gain?
If 1000 people play the game each day, find the state lottery expected gain and describe its distribution?

Let S denote the state lottery total gain in 100 days, assuming that 1000 people play on each day.
Find ν:= E(S) and a value of c for which P {ν − c < S ≤ ν + c} ≈ .95
ChesterDog
ChesterDog
Joined: Jul 26, 2010
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February 27th, 2024 at 6:10:10 PM permalink

The player's expected gain on a $1 ticket is exactly -$0.50. (That's 0.001 * 499.00 + 0.999 * (-1.00).)

If 1000 people play the lottery each day, the state lottery's expected gain is 1000 * $0.50 = $500 per day. If the people select their numbers randomly (or if the lottery assigns the numbers randomly), the distribution of the lottery's gain per day is a binomial distribution with a standard deviation of 500 * sqrt(1000 * 0.001 * 0.999) ≈ $499.75.

S, the lottery's total gain in 100 days at 1000 tickets per day, would have the same distribution as selling 100,000 tickets. The distribution for S would be a binomial distribution with a standard deviation of 500 * sqrt(100,000 * 0.001 * 0.999) ≈ $4,997.50.

To find the value of c, I would use a normal distribution with mean of $50,000 and standard deviation of $4,997.50. I find that c is about 1.96 * $4,997.50 ≈ $9,795.

(I welcome any corrections to my answers.)

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