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    MEI NM Jan 09

    For part i) abs error = 0.00033648912, rel error = 0.00035222053

    I don't understand part ii), i looked in mark scheme and they equal kx^4 to the error from part i) and then work out k. But I don't understand the logic behind it whatsoever, why do they equal kx^4 to the error?
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    when x= 0.3, the absolute error (or the key point - the "missing bit in the more accurate approximation") is goint to equal the addition of the term kx^{4}

    (the first 2 terms are the start of the Maclaurin series for cos(x), and 2 terms isn`t very accurate).

    you discover in 1) that there`s an error in the approximation, and in 2) this error is reduced by introducing a certain multiple of the next term in the series - (k)x^4 - which, if you want a better approximation, has to be equal to the error.

    so, Cos(0.3)=0.955336 (straightforward evaluation)

    and in 2), result above in 2) = (i.e. error = added term so that error in new calculation is as close to zero as we can get it): \displaystyle 0.00336489= 0.3^{4}k^4 (=> k = 0.0415419 (which = 0.997005/24 where 24 is the numerator in the 3rd term of the mac series)

    (EDITED)

    below, the purple graph is cos(x), the other the approximmation for small x with the value of k - seems to be accurate up to just before 1.5708 = Pi/2 rads - the first approximation is only accurate to about 0.5 rads)
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    (Original post by Hasufel)
    when x= 0.3, the absolute error (or the key point - the "missing bit in the more accurate approximation") is goint to equal the addition of the term kx^{4}

    (the first 2 terms are the start of the Maclaurin series for cos(x), and 2 terms isn`t very accurate).

    you discover in 1) that there`s an error in the approximation, and in 2) this error is reduced by introducing a certain multiple of the next term in the series - (k)x^4 - which, if you want a better approximation, has to be equal to the error.

    so, Cos(0.3)=0.955336 (straightforward evaluation)

    and in 2), result above in 2) = (i.e. error = added term so that error in new calculation = 0): \displaystyle 0.00336489= 0.3^{4}k^4 (=> k = 0.0415419 (which = 0.997005/24 where 24 is the numerator in the 3rd term of the mac series)

    (EDITED)

    below, the purple graph is cos(x), the other the approximmation for small x with the value of k - seems to be accurate up to about 1.5708 = Pi/2 rads)
    Thank you for taking time to write a long informational post! That clears it up, thank you very much!
 
 
 
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