Date | November 2009 | Marks available | 10 | Reference code | 09N.3ca.hl.TZ0.2 |
Level | HL only | Paper | Paper 3 Calculus | Time zone | TZ0 |
Command term | Find, Show that, and Hence or otherwise | Question number | 2 | Adapted from | N/A |
Question
The function f is defined by f(x)=e(ex−1) .
(a) Assuming the Maclaurin series for ex , show that the Maclaurin series for f(x)
is 1+x+x2+56x3+… .
(b) Hence or otherwise find the value of limx→0f(x)−1f′(x)−1 .
Markscheme
(a) ex−1=x+x22+x26+… A1
eex−1=1+(x+x22+x36)+(x+x22+x36)22+(x+x22+x36)36+… M1A1
=1+x+x22+x36+x22+x32+x36+… M1A1
=1+x+x2+56x3+… AG
[5 marks]
(b) EITHER
f′(x)=1+2x+5x22+… A1
f(x)−1f′(x)−1=x+x2+5x3/6+…2x+5x2/2+… M1A1
=1+x+…2+5x/2+… A1
→12 as x→0 A1
[5 marks]
OR
using l’Hopital’s rule, M1
limx→0e(ex−1)−1e(ex−1)−1′−1=limx→0e(ex−1)−1e(ex+x−1)−1 M1A1
=limx→0e(ex+x−1)e(ex+x−1)×(ex+1) A1
=12 A1
[5 marks]
Total [10 marks]
Examiners report
Many candidates obtained the required series by finding the values of successive derivatives at x = 0 , failing to realise that the intention was to start with the exponential series and replace x by the series for ex−1. Candidates who did this were given partial credit for using this method. Part (b) was reasonably well answered using a variety of methods.