Date | November 2020 | Marks available | 2 | Reference code | 20N.3.AHL.TZ0.Hca_4 |
Level | Additional Higher Level | Paper | Paper 3 | Time zone | Time zone 0 |
Command term | Write down | Question number | Hca_4 | Adapted from | N/A |
Question
The function is defined by where .
The seventh derivative of is given by .
Use the Maclaurin series for to write down the first three non-zero terms of the Maclaurin series for .
Hence find the first three non-zero terms of the Maclaurin series for .
Use your answer to part (a)(i) to write down an estimate for .
Use the Lagrange form of the error term to find an upper bound for the absolute value of the error in calculating , using the first three non-zero terms of the Maclaurin series for .
With reference to the Lagrange form of the error term, explain whether your answer to part (b) is an overestimate or an underestimate for .
Markscheme
substitution of in (M1)
A1
[2 marks]
(M1)
Note: Award (M1) if this is seen in part (a)(i).
attempt to differentiate their answer in part (a) (M1)
M1
Note: Award M1 for equating their derivatives.
A1
[4 marks]
A1
Note: Accept an answer that rounds correct to or better.
[1 mark]
attempt to find the maximum of for (M1)
maximum of occurs at (A1)
(for all ) (A1)
use of (M1)
substitution of and and their value of and their value of into Lagrange error term (M1)
Note: Award (M1) for substitution of and and their value of and their value of into Lagrange error term.
upper bound A1
Note: Accept an answer that rounds correct to or better.
[6 marks]
(for all ) R1
Note: Accept or “the error term is negative”.
the answer in (b) is an overestimate A1
Note: The A1 is dependent on the R1.
[2 marks]