Turn on thread page Beta
    • Thread Starter
    Offline

    3
    ReputationRep:
    Hi all, I just want some clarification as to whether using FP2 methods of solving C4 Binomial expansion is valid. I get how the C4 binomial expansion works and I know the formula without looking at the sheet, but when being asked binomial questions I prefer using the FP2 trick of differentiating the term again and again and sticking in f(0), f'(0)...etc. I get same answers for the questions I have done, but is it true for all functions being asked at C4?

    So the Binomial formula for is (1+ nx+ n(n-1)(x^2)/2! + n(n-1)(n-2)(x^3)/3!........)

    Which is similar to the Taylor series f(0)+f'(0)x+f''(0)(x^2)/2!+f'''(0)(x^3)/3!

    Say we have function, (9+8x)^0.5 or sqrt (9+8x),

    The binomial gives (3) (1+ 4x/9 - 8x^2/81) = 3+4x/3-8x^2/27.

    Taylor series gives

    f(x)=(9+8x)^0.5
    f'(x)=4(9+8x)^-0.5
    f''(x)=-16(9+8x)^-1.5

    And subbing 0s into the derivatives with the appropriate factorial division and x^n terms gives 3+4x/3-8x^2/27 as well.

    Is this type of approach valid for all of these question types and will I get credit if I use this method, and is the binomial expansion just a special case of a taylor/maclaurin series?
    • Study Helper
    Offline

    9
    ReputationRep:
    Study Helper
    (Original post by lecafe88)
    Hi all, I just want some clarification as to whether using FP2 methods of solving C4 Binomial expansion is valid. I get how the C4 binomial expansion works and I know the formula without looking at the sheet, but when being asked binomial questions I prefer using the FP2 trick of differentiating the term again and again and sticking in f(0), f'(0)...etc. I get same answers for the questions I have done, but is it true for all functions being asked at C4?

    So the Binomial formula for is (1+ nx+ n(n-1)(x^2)/2! + n(n-1)(n-2)(x^3)/3!........)

    Which is similar to the Taylor series f(0)+f'(0)x+f''(0)(x^2)/2!+f'''(0)(x^3)/3!

    Say we have function, (9+8x)^0.5 or sqrt (9+8x),

    The binomial gives (3) (1+ 4x/9 - 8x^2/81) = 3+4x/3-8x^2/27.

    Taylor series gives

    f(x)=(9+8x)^0.5
    f'(x)=4(9+8x)^-0.5
    f''(x)=-16(9+8x)^-1.5

    And subbing 0s into the derivatives with the appropriate factorial division and x^n terms gives 3+4x/3-8x^2/27 as well.

    Is this type of approach valid for all of these question types and will I get credit if I use this method, and is the binomial expansion just a special case of a taylor/maclaurin series?
    I THINK YOU WOULD GET FULL CREDSIT FOR IT AND YES THE bINOMIAL IS JUST AN EXAMPLE OF A TAYLOR SERIES, HOWEVER I WOULD THINK THAT IT IS A LOT QUICKER TO SIMPLY QUOTE THE STANDARD SERIES RATHER THAN USING REPEATED DIFFERENTIATION. I know that is certainly how I would do them.
    • Thread Starter
    Offline

    3
    ReputationRep:
    (Original post by brianeverit)
    I THINK YOU WOULD GET FULL CREDSIT FOR IT AND YES THE bINOMIAL IS JUST AN EXAMPLE OF A TAYLOR SERIES, HOWEVER I WOULD THINK THAT IT IS A LOT QUICKER TO SIMPLY QUOTE THE STANDARD SERIES RATHER THAN USING REPEATED DIFFERENTIATION. I know that is certainly how I would do them.
    Thanks for the reply. I have trouble using a calculator especially fractions as I don't know how to input fractions properly, to do the square of (9/8) for example I usually do 9x9/8/8. That's plenty of places to trip up on for a binomial question, so I'd rather do differentiation and sticking 0s in.
    Offline

    20
    ReputationRep:
    (Original post by lecafe88)
    ...
    Yes, this approach works for all binomial expansion questions. You can easily prove it by setting f(x) = (a+x)^n.

    Binomial expansion is a special case of Maclaurin's series. It is quicker to just use binomial expansion though rather than differentiating.

    You will get credit if you provide enough explanation of what you are dong to the examiner.
    • Study Helper
    Offline

    16
    ReputationRep:
    Study Helper
    (Original post by lecafe88)
    Thanks for the reply. I have trouble using a calculator especially fractions as I don't know how to input fractions properly, to do the square of (9/8) for example I usually do 9x9/8/8. That's plenty of places to trip up on for a binomial question, so I'd rather do differentiation and sticking 0s in.
    You shouldn't need a calculator to work out (9/8)^2 and in fact unless a fraction should have been simplified first you'll typically just be using (a/b)^2 = a^2/b^2 all the time anyway

    But the answer to your original question is yes - you'll get the same answer, basically because if you represent a function by a power series then that series is unique in the interval of convergence.
 
 
 
Reply
Submit reply
Turn on thread page Beta
Updated: January 20, 2015
Poll
Do you think parents should charge rent?
Useful resources

Make your revision easier

Maths

Maths Forum posting guidelines

Not sure where to post? Read the updated guidelines here

Equations

How to use LaTex

Writing equations the easy way

Student revising

Study habits of A* students

Top tips from students who have already aced their exams

Study Planner

Create your own Study Planner

Never miss a deadline again

Polling station sign

Thinking about a maths degree?

Chat with other maths applicants

Can you help? Study help unanswered threads

Groups associated with this forum:

View associated groups

The Student Room, Get Revising and Marked by Teachers are trading names of The Student Room Group Ltd.

Register Number: 04666380 (England and Wales), VAT No. 806 8067 22 Registered Office: International House, Queens Road, Brighton, BN1 3XE

Write a reply...
Reply
Hide
Reputation gems: You get these gems as you gain rep from other members for making good contributions and giving helpful advice.