The Math Thread

[Arkhi]

Sure. I won't say it's my forte, but I'm sure other members here know their ways around stats.

[Spineblade]

Welp, I got another one.

The following equations:

produce the following graphs (large image warning):

The question asks "Find the area enclosed by the curves." (which I assume means the purple section)

So, basically... how?

P.S. minimum point of the blue graph is (-0.2, -0.2)

Edited March 12, 2016 by Spineblade

[Arkhi]

You'll want to use integration to figure this one out.
The first step is to determine the x-values where the two graphs intersect. If you set them equal to each other (or look at the graph), you'll end up with

x = -2

and

x= 1

If you need further explanation of this step, feel free to ask.

From there, determine which graph is "above" the other in the interval of intersection. In this case, the top curve is

This means your bottom curve must be

From there, integrate with your bounds from smaller to larger in the order of the top function minus the bottom function. This leaves you with the following:

Depending on how you're meant to solve this, use a calculator or perform it by hand. If there's an issue there, post another reply and we'll integrate by hand.

[Spineblade]

Ah, so it was just a matter of subtracting the respective integrated functions. Thanks, Ark.

[Kookies]

So I don't really know the level of mathematics that you are familiar with but here goes: I need to find asymptotic series expansion of the complete Elliptic integrals (first and second order). I tried a few things I always find something like log(ε) for the firstish terms, but I am not sure. What do you think?

[Arkhi]

I've not dealt with elliptic integrals, so I'm afraid I can't help.

[Cool Girl]

Could someone explain to me how to do trigonometric form of a complex number, please. (Also, I'm a junior in high school who's in Pre-Calculus, so try your best to explain it on that level, please. Thanks guys!

[Hiss13]

Could someone explain to me how to do trigonometric form of a complex number, please. (Also, I'm a junior in high school who's in Pre-Calculus, so try your best to explain it on that level, please. Thanks guys!

You referring to the n(cosθ+isinθ)?

[Cool Girl]

Yes, that one!

[Hiss13]

It's just the concept that any complex number can be written like that.

For example, take complex number z=a+bi

On a complex plane, you can locate z where one axis is the real axis and the other is the imaginary axis like so:


n=(a²+b²)^1/2

n is the length of this vector and is called the modulus.

and θ=tan^-1(b/a)

θ is the angle the vector makes with the real number axis.

[Cool Girl]

Ok, thanks!

[Alaris]

Sure. I won't say it's my forte, but I'm sure other members here know their ways around stats.

ok, so here I go:

Do you know of any statistical analysis to be performed on a sample that does not follow a normal distribution neither has homogeneous variance, in order to form homogeneous groups?

As an example, if the sample had normality and homoscedasticity, I'd perform an ANOVA and then just make the groups with pairwise comparison. I you want to know, the ANOVA would have 1 factor and 26 levels, which would have between 4 to 246 replicas, depending on the level.

I'm aware that this question might be quite diffivult. But anyway, thanks in advanced!

[Notus]

Ok, I have an Calculus exam coming up so I guess I´ll be using this quite a bit .

I´m trying to solve some exercises about the Maclaurin series, but I´ve got no clue how to start. Can someone help me as to how to solve them?

ex) Use a binomial series to find the Maclaurin series for the given function and determine it´s radius of convergence.

a)f(x) = (1+x)^1/2

[Hiss13]

The Maclaurin Series is basically this.



Where f(x) is whatever your function is. So, f(x)=(1+x)^1/2 in this case. Plug it into the above.
Once you do, find the recursive formula for the series and then write it as Σa_n .

With the recursive formula in hand, use the following formula:

lim |a_n+1|
_x->∞ |a_n|

To find the radius of convergence.

Edited April 1, 2016 by Hiss13

[Notus]

Thanks a ton Hiss, I managed to do that and all the following ones!

[Cool Girl]

Hey guys,

So I need help doing these problems. These problems are pre-calculus (involves Annuities, Compound Interest, and Salaries) Could someone help me?

1) A principal of $2500 is invested at 8% interest. Find the amount after 20 years if the interest is compounded a) annually, semiannually, c) quarterly, d) monthly, and e) daily

2) A deposit of $100 is made at the beginning of each month in an account that pays 6%, compounded monthly. The balance A in the account at the end of 5 years is

A = 100(1 + 0.06/12)¹+ . . . + 100(1+0.06/12)⁶⁰

Find A.

3) Consider an initial deposit of P dollars in an account earning an annual interest rate r, compounded monthly. At the end of each month, a withdrawal of W dollars will occur and the account will be depleted in t years. The amount of the initial deposit required is

P = W(1 + r/12)^-1 + W(1 + r/12)^-2 + . . . + W(1 + r/12)^-12t

Show that the initial deposit is

P = W(12/r)[1-(1 + r/12)^-12t]

4) Determine the amount required in a retirement account for an individual who retires at age 65 and wants an income of $2000 from the account each month for 20 years. Use the result of the previous problem and assume that the account earns 9% compounded monthly.

5) An investment firm has a job opening with a salary of $30,000 for the first year. Suppose that during the next 39 years, there is a 5% raise each year. Find the total compensation over the 40-year period.

Any help on these problems would be greatly appreciated. Yeah, I suck at math problems.

[Marcello]

If nobody else has given any help tomorrow when I'm less falling asleep I'll happily give more specifics on problems, but generally the most important thing you have to remember with problems like this is that compound interest is, well, compound.

So, taking the first question, part a), if it increases by 8 percent every year after the first year your total is 2500*1.08 (Or 2500*1 + 2500*0.08 as I've always preferred to break down these sortsa things). Then the second year, the new total gets the same increase. So we get (2500*1.08)*1.08. And that goes on and on and so you end up with a power of the 1.08, so for example in that first one you end up getting 2500*(1.08^20), as you've had that *1.08 factor 20 times, once every year. The latter parts of that question look to be the same gist, just working out how many times the increase is.

That's the crux of this, although the latter questions look more involved, so yeah. I'll look back and can give more specific help when needed when it's not 1am for me. Hopefully this was clear enough for you, just ask if not.

[Cool Girl]

I'm really sorry, but it's still not clear to me.

[Marcello]

Totally okay, my bad. I'll still focus on breaking down the first problem but I'll try to actually break it down this time.

So we have $2500 and it increases by 8% every year.

We start by getting 8% of $2500, which is 2500 * 0.08 (Which is 200.)

So after one year, we have 2500+200, so 2700. But this can also be written like 2500 * 1 + 2500 * 0.08, which can in turn be written as 2500 * 1.08, which as said before, is equal to $2700.

Looking from the second year, we now have $2700 in the account, and so we add on 8% of this amount now (as that's how compound interest works).

Once again, it's a similar sum, 8% of 2700 is 2700 * 0.08 = 216.

Once again, we add them up for the amount after year two, 2700 + 216 = 2916. The important part is we can rewrite this in the same way we did for the first year:

2700 * 1 + 2700 * 0.08 = 2700 * 1.08

From here though, we use the sum we used to work out the initial 2700, as follows:

2700 = 2500 * 1.08, Therefore:

2700 * 1.08 = (2500 * 1.08) * 1.08

This we can rewrite as 2500 * (1.08)^2 [Just in case this isn't familiar notation, that's raised to the power of 2, so 1.08 squared].

If you work through the third year, the pattern will continue, you'll find you get 2916 * 1.08, which using the above step, we can rewrite as 2500 * (1.08)^3.

And the pattern continues on like that, you can work out each step individually if you're not completely confident, or jump to the end if you are.

I hope that explains the general idea of how to work with compound interest. If it's still not clear, I'm really sorry for trying when I'm sleepy and potentially being confusing, and I hope somebody who's better at explaining stuff comes along!

[Cool Girl]

Ok, thank you for explaining compound interests. I still don't get annuities though.

Also, could someone explain to me combinations and permutations?

Edited April 22, 2016 by Cool Girl

[Drymus]

Combinations are the number of ways you can get a set number of items from a pool of items, when order doesn't matter.

n is the size of the pool, and r is how many items you are taking out of the pool.

The number of combinations can be determined by the formula: nCr = (n!)/(r!(n-r)!)

For example, if you have 6 items and you want to pick 3 of them.

The number of potential outcomes is 6C3 = 6!/(3!(6-3)!) = 20 possible combinations

Permutations are basically the same thing, except order does matter.

(The same elements but a new order is considered to be a separate possibility, like placings in a race.)

The formula you use for permutations is nPr = (n!)/((n-r)!)

For the same numbers above, we have 6P3 = 6!/((6-3)!) = 120 possible permutations

Hope that helps!

And sorry, no idea what annuities are...

Edited April 22, 2016 by Drymus

[Cool Girl]

Ok, thanks!