Math thread! Come in and have some pi...

[NephilimRising]

https://imgur.com/a/OBB8C

I don’t see how this integral is zero. They seem to imply that the terms in the integrand cancel out but I’m not seeing it. Some help?

Sorry for not embedding. It seems like imgur works differently now.


Edit: nvm it’s because when you integrate each term separately from 0 to 2pi you get 0.

[numberguy12]

https://imgur.com/a/OBB8C

I don’t see how this integral is zero. They seem to imply that the terms in the integrand cancel out but I’m not seeing it. Some help?

Sorry for not embedding. It seems like imgur works differently now.


Edit: nvm it’s because when you integrate each term separately from 0 to 2pi you get 0.

It won't always be 0 (say if m and n are equal).

In the case m and n are different integers, use integration by parts twice to establish the definite integral is 0.

[rubinkazan]

Yup, this does seem like a big step up from differentiation. Not too sure I understand how integration by parts works. I've performed the steps in the 'formula'. But I feel like I am missing a step.
Example: Find the integral of x^2 * (e^-x)

∫x^2 * e^-x= x^2 * -e^-x - ∫2x * -e^-x

[nordiclifter]

I have tried different ways to factorize the top, I realize I cant use polynomal division.. Im stuck

edit; sorry about the fuked pic



EDIT: I used polynomial division wrong I see, by using -2 gives the nominator = 0, so I can therefor use it

[hidingwithmusic]

calculus - absolute maximum and minimum..


I have tried for hours to solve part a. I just cant get it. Anyone please show me steps how.

[numberguy12]

calculus - absolute maximum and minimum..


I have tried for hours to solve part a. I just cant get it. Anyone please show me steps how.

To find absolute maximum in an interval, must find (1) the value at the endpoints of the interval and (2)the value at all local extrema in the interval. The greatest of these numbers = absolute maximum.

(1) So first find P(0) and P(100).

(2)Then to find extrema, find P'(t) and set P'(t) = 0. The t solutions of this equation in the interval 0<t<100 will be your extrema points. Find P(t) for each of these, if there are any.

Compare the above P(t) values, and the greatest is your maximum. Remember to convert back from the t-value to the actual year.

[hidingwithmusic]

To find absolute maximum in an interval, must find (1) the value at the endpoints of the interval and (2)the value at all local extrema in the interval. The greatest of these numbers = absolute maximum.

(1) So first find P(0) and P(100).

(2)Then to find extrema, find P'(t) and set P'(t) = 0. The t solutions of this equation in the interval 0<t<100 will be your extrema points. Find P(t) for each of these, if there are any.

Compare the above P(t) values, and the greatest is your maximum. Remember to convert back from the t-value to the actual year.

They are asking me to find p(t) and p(0) and p(100) are both incorrect.. I've tried everything and I'm unable to solve. Would you mind attempting to figure it out and I can look at the steps?

[numberguy12]

They are asking me to find p(t) and p(0) and p(100) are both incorrect.. I've tried everything and I'm unable to solve. Would you mind attempting to figure it out and I can look at the steps?

Right. It's not just between P(0) and P(100) though. If you read the above again, you have to do step (2) also and find the possible local maximums as well. This means:

-Determine P'(t). This means find the derivative of P(t).
-Solve P'(t) = 0. This will need approximation techniques/computer assistance as this is a cubic equation.

You will get one t value in 0<t<100 that solves P'(t) = 0. This is the absolute maximum of the problem. Hint, it is between t=70 and t=75. Remembering of course to convert this to the actual year, you can't leave your answer in t-value form.

[hidingwithmusic]

Right. It's not just between P(0) and P(100) though. If you read the above again, you have to do step (2) also and find the possible local maximums as well. This means:

-Determine P'(t). This means find the derivative of P(t).
-Solve P'(t) = 0. This will need approximation techniques/computer assistance as this is a cubic equation.

You will get one t value in 0<t<100 that solves P'(t) = 0. This is the absolute maximum of the problem. Hint, it is between t=70 and t=75. Remembering of course to convert this to the actual year, you can't leave your answer in t-value form.

I got t = 71 .. the part bolded is what I can't get. Will you please show me st wrongrps for reps? I'm losing plenty of confidence because it's been 2 days I've tried figuring this out... :-/

[numberguy12]

I got t = 71 .. the part bolded is what I can't get. Will you please show me st wrongrps for reps? I'm losing plenty of confidence because it's been 2 days I've tried figuring this out... :-/

Yep, t=71.8056, so you are doing it right. In terms of the bolded part, you are solving the equation P'(t) = 0.

This means solving the equation:
.0000000872 t^3 - 0.0000501 t^2 + 0.00312 t+ 0.002 == 0

Without even worrying about the exact numbers here, just focus on what you have: an equation in t with the highest power t^3. This is a cubic equation, and there is no easy solution like the quadratic formula for quadratic equations.

To solve a cubic like this, you usually just use a program like Wolfram Alpha or a Ti-89, and it'll pop out the answer. I mean there are approximation techniques you could use like Newton's method, or using Cardan's cubic formula to solve the above cubic, but I can pretty much assure you, your professor is not asking you to do this. You are mostly likely expected to use computer software to solve the cubic. Simply put something like

solve(.0000000872 t^3 - 0.0000501 t^2 + 0.00312 t+ 0.002 = 0,t)

into Wolfram Alpha, and it'll generate the solution leading to t=71.8056

[hidingwithmusic]

Yep, t=71.8056, so you are doing it right. In terms of the bolded part, you are solving the equation P'(t) = 0.

This means solving the equation:
.0000000872 t^3 - 0.0000501 t^2 + 0.00312 t+ 0.002 == 0

Without even worrying about the exact numbers here, just focus on what you have: an equation in t with the highest power t^3. This is a cubic equation, and there is no easy solution like the quadratic formula for quadratic equations.

To solve a cubic like this, you usually just use a program like Wolfram Alpha or a Ti-89, and it'll pop out the answer. I mean there are approximation techniques you could use like Newton's method, or using Cardan's cubic formula to solve the above cubic, but I can pretty much assure you, your professor is not asking you to do this. You are mostly likely expected to use computer software to solve the cubic. Simply put something like

solve(.0000000872 t^3 - 0.0000501 t^2 + 0.00312 t+ 0.002 = 0,t)

into Wolfram Alpha, and it'll generate the solution leading to t=71.8056

I was using 71 instead of 71.8056 for t. I knew I was doing it right. Thanks so much and I'll return when I am in more calculus trouble!

[IronProdigy]

How do you prove the polarization identity. I'm rusty with dot product so I get stuck at
<x,y> = (x•y) + i(x•iy)
I don't even believe my distribution up to the point I got to was correct.

[IronProdigy]

can you provide more context? i have never heard of this, and wikipedia is suggesting it is dependent on the norm, and only gives an existence statement as the definition of the identity.

http://www.wow.com/wiki/Polarization_identity
The third formula

[IronProdigy]

have you tried starting from the more complicated side? it's usually a menial task when it comes to these things. just cancel, cancel, cancel.

the derivation of course will be harder going from left to right. the denominator of 4 suggests double symmetry. so maybe starting with something such as

<x+iy, x-iy> + <x-iy, x+iy> = 2<x,y>^2

and discovering relationships which you can play with and alter the starting point.

What do you mean by complicated side? I expanded the right and just started cancelling as much terms as I can. I ended up with
(1/4)[4(x•y) + 4i(x•iy)]
= (x•y) + i(x•iy)

I assumed the 1/4 was to cancel out the 4 left over

[IronProdigy]

Im not sure what work you have but it looks okay to me. Note that semilinear gives <y,iy> = Conj(<iy,y>) = Conj(i*<y,y>) = -i * Conj(<y,y>) = -i*<y,y>, since <y,y> is real.


A = <x+y, x+y> - <x-y, x-y> = <x,x> + <x,y> + <y,x> + <y,y> - <x,x> + <x,y> + <y,x> - <y,y> = 2(<x,y> + <y,x>)

B = <x+iy, x+iy> - <x-iy, x-iy> = <x,x> + i<x,y> - i<y,x> - <y,y> - <x,x> + i<x,y> - i<y,x> + <y,y> = -2i(<x,y>-<y,x>)

A + Bi = 2(<x,y> + <y,x>) + i* (-2i(<x,y> - <y,x>)) = 4<x,y>

<x,iy> != <iy,x> ? I thought they were the same. So,
<x,iy> = -i<x,y> and
<iy,x> = i<y,x>
Or do i have it backwards? Thanks for the computation. Converting the inner product to dot products made my computation confusing.

[SCAR-H]

Approached using the hypergeometric model:


Solution is as follows:


Is my approach valid for an exam? (I understand this would be impossible to compute on a calculator)

EDIT: on second thought, if the sample size (18) were larger, the final probability would have a bigger difference. Is this the only reason the probabilities are close?

[SCAR-H]

No. N*p is not even guaranteed to be an integer. At this level (n choose k) is the number of ways to do something, although it does have an analytic continuation. N=1700 is actually irrelevent to the problem.

Can you elaborate on this?

And is this no to it being a valid approach, or no to the reason why the probabilities are close?

[numberguy12]

Approached using the hypergeometric model:


Solution is as follows:


Is my approach valid for an exam? (I understand this would be impossible to compute on a calculator)

EDIT: on second thought, if the sample size (18) were larger, the final probability would have a bigger difference. Is this the only reason the probabilities are close?

It depends on professor whether certain methods are valid, there is no way that can be answered. That being said, to answer your last question, yes the huge difference between the sample size 18 and 17,000 is why you can approximate this as "with replacement" (binomial dist). This is the main point of this problem.

I see nothing that wrong with your approach using the hypergeometric dist, as it is actually more true to the problem being without replacement. The problem itself is inexact using the phrase "almost 70%", so there is no issue really in rounding to the nearest integer for N*p (not needed in this case). If the problem was exact, and said p proportion of the N people had some characteristic, well then you know N*p must be an integer by definition.

Your method is still valid imo, it is just not the method most would use here. They would use the binomial approximation.

[SCAR-H]

It depends on professor whether certain methods are valid, there is no way that can be answered. That being said, to answer your last question, yes the huge difference between the sample size 18 and 17,000 is why you can approximate this as "with replacement" (binomial dist). This is the main point of this problem.

I see nothing that wrong with your approach using the hypergeometric dist, as it is actually more true to the problem being without replacement. The problem itself is inexact using the phrase "almost 70%", so there is no issue really in rounding to the nearest integer for N*p (not needed in this case). If the problem was exact, and said p proportion of the N people had some characteristic, well then you know N*p must be an integer by definition.

Your method is still valid imo, it is just not the method most would use here. They would use the binomial approximation.

Ah, I did not realize the phrasing was almost 70%. I understand now.

What is the cutoff for A >> B to approximate with the binomial model? (unless this also depends on the professor)

[numberguy12]

Ah, I did not realize the phrasing was almost 70%. I understand now.

What is the cutoff for A >> B to approximate with the binomial model? (unless this also depends on the professor)

Some one with more familiarity in stats than me can probably give you a better answer, but I don't think there is a set rule for when you can treat hypergeometric models as binomial models. You generally treat them in a case by case basis and use judgment to determine A>>B. When in doubt can use the hypergeometric version like you did- it is exact.

In this case 17,000, as well as 17000*.7 is clearly much larger than 18, so the binomial approximation is indicated. If the problem were picking 18 from 50, I would not use the binomial approximation.

[rubinkazan]

Okay so we're like 90% the way through our Calc I course atm. One thing I wish the lecturer told us was that the "Integration by parts" method can be derived from integrating the product rule. It took me a while to figure out what was actually going on in integration by parts, but simply integrating the product rule helped me out big time. Thought I'd share incase other people having difficulty with learning integration!

[rubinkazan]

Why does pi behave differently than e when they are both constants


I think prof left it out because we pressured for time due to protests resulting in classes being stopped for like a week, in response to above comment post.

[numberguy12]

Why does pi behave differently than e when they are both constants


I think prof left it out because we pressured for time due to protests resulting in classes being stopped for like a week, in response to above comment post.

What do you mean by behave differently? They are different numbers...defined differently.....but they are still both just real numbers (constants).

[gwem32170191]

Math boyos, how can math be so interesting and so boring at the same time? I spent a good chunk of today reading about Transcendental numbers (Pi and e), interesting as fuk.

[numberguy12]

Math boyos, how can math be so interesting and so boring at the same time? I spent a good chunk of today reading about Transcendental numbers (Pi and e), interesting as fuk.

I think you have to differentiate between what "math" truly is (example, what you are saying about the transcendence of pi and e), and what arithmetic is- the boring tedious calculations you do in grade school. Sadly when most people think math, they think of just arithmetic, as this is their only exposure to the subject. It is like comparing painting the wall of a Walmart to painting the Mona Lisa- both use paints, brushes, etc, but one is art. Just as in our case, both subjects use numbers as a medium, but the subject of mathematics is an art form.

An example of each:

1. A question of arithmetic:

Sqrt(2)^(sqrt(2)*sqrt(2)) = sqrt(2)^2= 2

2. A mathematically interesting statement:

-There exists an irrational number raised to an irrational number, that results in a rational number.

Proof: The above is not obvious at all. It can be shown non-constructively though. Consider sqrt(2)^sqrt(2). This number is either rational or irrational. If it is rational, we are done, since we have an irrational number being raised to an irrational number resulting in a rational number. If, on the other hand, sqrt(2)^sqrt(2) is irrational, then raise it to sqrt(2), an irrational power. We just showed above in (1) that this equals 2, a rational number, and we are done.

[rubinkazan]

Yeah thought I was onto something...thought e had some special properties or something. Was so used to seeing e^x integrated to e^x + c when it actually just follows the rule a^x/lna and lne is just 1

[rubinkazan]

I follow the first integration but im not sure about the second.
1) Why do we need to integrate it again?
2) Where does the 1/9 come from in the second line of the second integration?

[IronProdigy]

I follow the first integration but im not sure about the second.
1) Why do we need to integrate it again?
2) Where does the 1/9 come from in the second line of the second integration?

1) We don't want our final solution to have an integral
2) How do you integrate (1/3)(e^3x) ?

[rubinkazan]

Oh I thought we were integrating e^3x.

Also why do i get an error when integrating e^-x^2

[IronProdigy]

Oh I thought we were integrating e^3x.

Also why do i get an error when integrating e^-x^2

Why are you interested in integrating that function? It's not easy to integrate unless you have something like xe^(-x^2). Then you can use a u-substition