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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(ex1) .

(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 limx0f(x)1f(x)1 .

Markscheme

(a)     ex1=x+x22+x26+     A1

eex1=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 x0     A1

[5 marks]

OR

using l’Hopital’s rule,     M1

limx0e(ex1)1e(ex1)11=limx0e(ex1)1e(ex+x1)1     M1A1

=limx0e(ex+x1)e(ex+x1)×(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 ex1. Candidates who did this were given partial credit for using this method. Part (b) was reasonably well answered using a variety of methods.

Syllabus sections

Topic 9 - Option: Calculus » 9.7 » The evaluation of limits of the form limxaf(x)g(x) and limxf(x)g(x) .

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