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Date May 2008 Marks available 9 Reference code 08M.3ca.hl.TZ1.2
Level HL only Paper Paper 3 Calculus Time zone TZ1
Command term Show that, Determine, and Find Question number 2 Adapted from N/A

Question

(a)     Using l’Hopital’s Rule, show that limxxex=0 .

(b)     Determine 0axexdx .

(c)     Show that the integral 0xexdx is convergent and find its value.

Markscheme

(a)     limxxex=limx1ex     M1A1

= 0     AG

[2 marks]

 

(b)     Using integration by parts     M1

0axexdx=[xex]0a+0aexdx     A1A1

=aea[ex]0a     A1

=1aeaea     A1

[5 marks]

 

(c)     Since ea and aea are both convergent (to zero), the integral is convergent.     R1

Its value is 1.     A1

[2 marks]

Total [9 marks]

Examiners report

Most candidates made a reasonable attempt at (a). In (b), however, it was disappointing to note that some candidates were unable to use integration by parts to perform the integration. In (c), while many candidates obtained the correct value of the integral, proof of its convergence was often unconvincing.

Syllabus sections

Topic 9 - Option: Calculus » 9.7 » Using l’Hôpital’s rule or the Taylor series.

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