Need help to differentiate this

Original post by Jackabc

Btw do you know how find the horizontal asymptote of a rational function, well do you know how to sketch them in general? There is not much advice in the textbook.

For something like 7x/(3x+1)

Yes.

Write it in divided out form.

\displaystyle \frac{7x}{3x+1} = A + \frac{B}{3x+1}

Then the horizontal asymptote is y = A.

Reply 25

davros

Original post by Jackabc

Btw do you know how find the horizontal asymptote of a rational function, well do you know how to sketch them in general? There is not much advice in the textbook.

For something like 7x/(3x+1)

Polynomial division? You've basically got 7/3 + some fraction which will tend to 0 as x gets bigger.

Reply 26

Original post by Mr M

Yes.

Write it in divided out form.

\displaystyle \frac{7x}{3x+1} = A + \frac{B}{3x+1}

Then the horizontal asymptote is y = A.

Original post by davros

Polynomial division? You've basically got 7/3 + some fraction which will tend to 0 as x gets bigger.

Why exactly does that work?

Reply 27

Original post by Jackabc

Why exactly does that work?

Davros explained it really. The fraction approaches zero for large and small x so can (almost) be ignored.

Edit: Small here means large negative.

(edited 11 years ago)

Reply 28

davros

Original post by Jackabc

Why exactly does that work?

Not sure what you're asking? If the numerator and the denominator are both linear (Ax + B) then you can always divide the bottom into the top and get an expression in the form MrM wrote.

As x gets bigger then the remainder with the linear factor in the denominator gets smaller and smaller, so y gets closer and closer to the constant term which is therefore the horizontal asymptote.

Reply 29

Original post by Mr M

Yes.

Write it in divided out form.

\displaystyle \frac{7x}{3x+1} = A + \frac{B}{3x+1}

Then the horizontal asymptote is y = A.

Original post by davros

Polynomial division? You've basically got 7/3 + some fraction which will tend to 0 as x gets bigger.

Why exactly does that work?

Original post by Mr M

Yes.

Write it in divided out form.

\displaystyle \frac{7x}{3x+1} = A + \frac{B}{3x+1}

Then the horizontal asymptote is y = A.

Original post by davros

Polynomial division? You've basically got 7/3 + some fraction which will tend to 0 as x gets bigger.

I think I get why actually now. Would this work if the polynomial on top had degree 2 and the denominator degree 3?

Edit: Just saw what you said but does that only work for certain cases? Doesn't matter if it takes too long to explain as probably don't need to know too much just interesting.

(edited 11 years ago)

Reply 30

Original post by Jackabc

I think I get why actually now. Would this work if the polynomial on top had degree 2 and the denominator degree 3?

Other way up. If the numerator has degree 3 and the denominator 2 then you get an oblique (wonky) asymptote of the form y = Ax + B.

Reply 31

Original post by Mr M

Other way up. If the numerator has degree 3 and the denominator 2 then you get an oblique (wonky) asymptote of the form y = Ax + B.

How an earth do you know all this? You must have a good textbook because mine hardly explains anything, either that or I am just stupid.

Use quotient rule

Original post by Jackabc

How an earth do you know all this?

I'm a mathematics teacher.

Reply 34

Original post by Mr M

I'm a mathematics teacher.

Oh right, do you enjoy it as I was hoping one day to be a teacher but if the work outside of school was too much then might not be worth it. Do you think textbooks are crap these days as they hardly explain any knowledge?

Reply 35

Original post by Jackabc

It's ok. I do lots of work outside school but I could do less if I wanted. I think most textbooks are perfectly fine but it is a good idea to refer to a range of sources on obscure topics.

Reply 36

Original post by Mr M

It's ok. I do lots of work outside school but I could do less if I wanted. I think most textbooks are perfectly fine but it is a good idea to refer to a range of sources on obscure topics.

I suppose, its just sometimes I wonder why is that the case and I think the textbooks bad but it could be out of syllabus. For example in s2 when finding E(x) for a p.d.f it just says replace the sum sign by an integral sign but I don't get how it gives the sum of the probabilities multiplied by x as f(x) is not a probability if you get what I am saying. Is that something I will find out at uni or am I missing something? Last question I promise :smile:

Reply 37

davros

Original post by Jackabc

you would really need to study the fundamentals at uni - it's a principle that crops up in different areas. Basically, the integral is the limiting form of a sum so when you move from a discrete distribution to a continuous one the simple formula you have that involves a sum becomes a more complicated formula involving an integral and a probability density.

Reply 38

Original post by davros

Thanks a lot for all your help. This is why I don't like a-level maths, its almost empty of meaning but like you said, at uni it isn't. Is it safe to assume that anything that the textbook doesn't explain is not meant to be explained?

Reply 39