Date | November 2010 | Marks available | 9 | Reference code | 10N.3ca.hl.TZ0.3 |
Level | HL only | Paper | Paper 3 Calculus | Time zone | TZ0 |
Command term | Find, Hence, and Write down | Question number | 3 | Adapted from | N/A |
Question
(a) Using the Maclaurin series for the function ex, write down the first four terms of the Maclaurin series for e−x22.
(b) Hence find the first four terms of the series for ∫x0e−u22du.
(c) Use the result from part (b) to find an approximate value for 1√2π∫10e−x22dx.
Markscheme
(a) ex=1+x+x22!+x33!+x44!+…
putting x=−x22 (M1)
e−x22≈1−x22+x422×2!−x623×3!≈(1−x22+x48−x648) A2
[3 marks]
(b) ∫x0e−u22du≈[u−u33×2+u55×22×2!−u77×23×3!]x0 M1(A1)
=x−x33×2+x55×22×2!−x77×23×3! A1
(=x−x36+x540−x7336)
[3 marks]
(c) putting x = 1 in part (b) gives ∫10e−x22dx≈0.85535… (M1)(A1)
1√2π∫10e−x22dx≈0.341 A1
[3 marks]
Total [9 marks]
Examiners report
This was one of the most successfully answered questions. Some candidates however failed to use the data booklet for the expansion of the series, thereby wasting valuable time.