FP2: Taylor Series confusion

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    Hi,

    The question I need help with is:

    Use Taylor's expansion to express each of the following as a series in ascending powers of x as far as the term x^4.

    A) cos(x+(pi/4))

    For this one, I got the correct answer, it being:
    1/root2 (1 - x - x^2/2 + x^3/6 + x^4/24...)

    By using f(pi/4) and its respective deriviates.

    B) Ln(x+5)

    For this one, the answers say the answer is:
    Ln5 + (1/5)x...

    My issue is, this answer appears to be using the results of f(0) and further derivatives where x = 0.
    Why did they not use f(5), as this contradicts part a)?
    Why is the answer not:
    Ln10 + (1/10)x...

    Thank you. I hope I managed to make myself clear

    This is taken from the Edexcel textbook.
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    (Original post by JLegion)
    Hi,

    The question I need help with is:

    Use Taylor's expansion to express each of the following as a series in ascending powers of x as far as the term x^4.

    A) cos(x+(pi/4))

    For this one, I got the correct answer, it being:
    1/root2 (1 - x - x^2/2 + x^3/6 + x^4/24...)

    By using f(pi/4) and its respective deriviates.

    B) Ln(x+5)

    For this one, the answers say the answer is:
    Ln5 + (1/5)x...

    My issue is, this answer appears to be using the results of f(0) and further derivatives where x = 0.
    Why did they not use f(5), as this contradicts part a)?
    Why is the answer not:
    Ln10 + (1/10)x...

    Thank you. I hope I managed to make myself clear

    This is taken from the Edexcel textbook.
    For the first question, you have expanded about x=0. What you've written as f(\frac{\pi}{4}) is in fact f(0) as the function contains a translation that leads to f(0)=\cos(\frac{\pi}{4}).

    As an aside, you could, if you wished, expand the function about the point x=5, though what you've written there is not quite correct, the second term would be \frac{1}{10}(x-5), but expanding about x=0 is often easier.
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    Oh I see!

    That completely makes sense now.

    Thank you for all your help
 
 
 
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