Date | May 2019 | Marks available | 9 | Reference code | 19M.3.AHL.TZ0.Hca_4 |
Level | Additional Higher Level | Paper | Paper 3 | Time zone | Time zone 0 |
Command term | Find | Question number | Hca_4 | Adapted from | N/A |
Question
Using L’Hôpital’s rule, find limx→0(tan3x−3tanxsin3x−3sinx).
Markscheme
* This question is from an exam for a previous syllabus, and may contain minor differences in marking or structure.
limx→0(tan3x−3tanxsin3x−3sinx)
limx→0(3sec23x−3sec2x3cos3x−3cosx)(=limx→0(sec23x−sec2xcos3x−cosx)) M1A1A1
Note: Award M1 for attempt at differentiation using l'Hopital's rule, A1 for numerator, A1 for denominator.
METHOD 1
using l’Hopital’s rule again
=limx→0(18sec23xtan3x−6sec2xtanx−9sin3x+3sinx)(=limx→0(6sec23xtan3x−2sec2xtanx−3sin3x+sinx)) A1A1
EITHER
=limx→0(108sec23xtan23x+54sec43x−12sec2xtan2x−6sec4x - 27cos3x+3cosx) A1A1
Note: Not all terms in numerator need to be written in final fraction. Award A1 for 54sec43x+…−6sec4x…−. However, if the terms are written, they
must be correct to award A1.
attempt to substitute x=0 M1
=48−24
OR
ddx(18sec23xtan3x−6sec2xtanx)|x=0=48 (M1)A1
ddx(−9sin3x+3sinx)|x=0=−24 A1
THEN
(limx→0(tan3x−3tanxsin3x−3sinx))=−2 A1
METHOD 2
=limx→0(3cos23x−3cos2x3cos3x−3cosx) M1
=limx→0(cos2x−cos23xcos23xcos2x(cos3x−cosx)) A1
=limx→0(cosx+cos3x−cos23xcos2x) M1A1
attempt to substitute x=0 M1
=2−1
=−2 A1
[9 marks]