Math Is Fun Forum / Creating Functions?

therussequilibrium · 2012-12-22 19:30:37

I know that I'm capable, it's just that I'm 24 almost 25, and I haven't taken any math since I was 16 and even then in High School I never applied myself. It's a shame really, because now I'm in college and I'm taking college Algebra, which is almost sad really. I just feel like I'm really far behind, and I'm really not sure how to go about catching up quickly.

bobbym · 2012-12-22 19:55:07

Okay, we will talk about that later.

For now the third idea:

{1,5,14,30,55,91,140,204,285,385}

{4,9,16,25,36,49,64,81,100}

{5,7,9,11,13,15,17,19}

{2,2,2,2,2,2,2}

when you come to a constant difference in this case 2 you stop. Because there are 4 rows that means the answer is third degree polynomial.

There is now a simple formula for your recurrence.

therussequilibrium · 2012-12-22 20:13:34

Now that's interesting, when I was trying to work out a formula that didn't involve a recurrence, I actually worked out these first two patterns and were trying to figure a way to use them both, but ultimately couldn't.

{1,5,14,30,55,91,140,204,285,385}

{4,9,16,25,36,49,64,81,100}

And actually, I don't remember what I had done, but I had come to a pattern that went up by 4 constantly. I was working on the computer though and didn't save my notes.

Last edited by therussequilibrium (2012-12-22 20:15:03)

bob bundy · 2012-12-22 20:15:53

hi therussequilibrium

How to get a formula?

There are loads of methods. Here's one using your celcius to fahrenheit example.

There are two key points on the celcius scale: freezing point of water ... 0 and boiling point of water ... 100

The equivalent values in farhenheit are 32 and 212.

So on a graph, with C across and F up, (0,32) and (100,212) are known points for the conversion. (see picture ... this is a sketch only ... it is not accurately drawn)

So for 100 across you have to go 180 up to stay on the line. ie. gradient = 180/100 = 9/5

So if the equation is

Substitute C = 0 and F = 32 gives

Your empirical formula was very close so you did well!

Bob

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bobbym · 2012-12-22 20:21:56

Hi Bob;

How are you?

Hi therussequilibrium;

{1,5,14,30,55,91,140,204,285,385}

{4,9,16,25,36,49,64,81,100}

{5,7,9,11,13,15,17,19}

{2,2,2,2,2,2,2}

this is called a difference table. Each element in the row below is made by subtracting two elements in the row above.

5 - 1 = 4, 140 - 91 = 49.

bob bundy · 2012-12-22 20:25:10

hi bobbym,

I'm fine thanks. Throat almost back to normal. How are you ?

I think I've got another approach to the 'Pyramid numbers' example. But I need to have a long think first .....

Bob

therussequilibrium · 2012-12-22 20:27:35

I actually do this with patterns already, naturally, I just had no idea how to apply those patterns, or if they could even really be applied to do anything.

bobbym · 2012-12-22 20:36:33

Hi Bob;

Glad you are back to almost 100%. There are many approaches to his problem. We are just looking one, the recurrence he came up with.

Hi therussequilibrium;

You can use the numbers you have to come up with this formula in terms of binomials.

therussequilibrium · 2012-12-22 20:47:45

I'm not sure I know what you mean in your last post.

bobbym · 2012-12-22 20:51:04

Hi;

Have you had combinations yet?

therussequilibrium · 2012-12-22 20:53:15

No, I haven't learned of Combinations yet. /sigh so much to learn lol

bobbym · 2012-12-22 20:57:38

Okay, we do not need them. They just simplify the notation. Do you understand the difference table?

therussequilibrium · 2012-12-22 21:01:31

Yes, I completely understand the table.

bobbym · 2012-12-22 21:08:37

Let's work on that for a second. When you had 0 squares how many squares did you count?

therussequilibrium · 2012-12-22 21:11:15

0

bobbym · 2012-12-22 21:12:23

Very good so your sequence really looks like this

{0,1,5,14,30,55,91,140,204,285,385...}

therussequilibrium · 2012-12-22 21:15:27

Correct, and the second sequence really starts at 1 and not at 4, etc.

bobbym · 2012-12-22 21:20:34

Yes, the difference table now looks like this.

Take the first number in every row and say

a = 0;
b = 1
c = 3
d = 2

the rest are zero. Take those and plug into this formula.

therussequilibrium · 2012-12-22 21:36:45

Not sure how to type out a fraction on the forum, but I think this is the answer:

bobbym · 2012-12-22 21:42:42

Hi;

That is not correct. Let me show you what I meant.

I dropped the ellipses on the end which are just there to show that there might be more terms.

When you substitute a = 0, b = 1, c = 3, d = 2 into that you get,

you can leave it like that and say

or you can simplify it,

That is what we got by the other methods. So this method solves your recurrence by just making a table and plugging into a formula.

therussequilibrium · 2012-12-22 21:47:19

hmm, that's exactly the equation I entered and that's the answer it gave me; and I'm not sure how to simplify a problem with radicals in it, on my own.

bobbym · 2012-12-22 21:52:58

I am sorry, that is a factorial.

2! means 2 * 1, it equals 2

3! means 3 * 2 * 1, it equals 6

I did away with the factorials.

therussequilibrium · 2012-12-22 22:08:18

Well, I can't figure out how to simplify this problem. But question, where does this equation that we are using come from?

bobbym · 2012-12-22 22:12:18

Hi;

Offhand I do not remember the derivation of that. It is from the difference calculus. But it does work.

Try simplifying it by hand. If you want to I will help but it is really not necessary.

therussequilibrium · 2012-12-22 22:18:23

Yeah, I'm trying by hand, I'll get it eventually. You've given me enough help, but thank you. I do appreciate you taking your time to walk through this with me.

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#26 2012-12-22 19:30:37

Re: Creating Functions?

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Re: Creating Functions?

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Re: Creating Functions?

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