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Date November 2020 Marks available 4 Reference code 20N.2.AHL.TZ0.F_5
Level Additional Higher Level Paper Paper 2 Time zone Time zone 0
Command term Show that Question number F_5 Adapted from N/A

Question

Assuming the Maclaurin series for cosx and ln(1+x), show that the Maclaurin series for cos(ln(1+x)) is

1-12x2+12x3-512x4+

[4]
a.

By differentiating the series in part (a), show that the Maclaurin series for sin(ln(1+x)) is x-12x2+16x3+ .

[4]
b.

Hence determine the Maclaurin series for tan(ln(1+x)) as far as the term in x3.

[5]
c.

Markscheme

* This question is from an exam for a previous syllabus, and may contain minor differences in marking or structure.

METHOD 1 

attempts to substitute lnx+1=x-12x2+13x3-14x4+ into

cosx=1-12x2+124x4-        M1

cosx=ln1+x=1-12x-12x2+13x3+2+124x+4+        A1

attempts to expand the RHS up to and including the x4 term        M1

=1-12x2-x3+14x4+23x4+124x4+        A1

=1-12x2+12x3-512x4+        AG

 

METHOD 2

attempts to substitute lnx+1 into cosx=1-12x2+124x4-        M1

cosln1+x=1-12ln1+x2+124ln1+x4-

attempts to find the Maclaurin series for ln1+x2 up to and including the x4 term        M1

ln1+x2=x2-x3+1112x4-        A1

ln1+x2=x4-

=1-12x2-x3+1112x4++124x4+        A1

=1-12x2+12x3-512x4+        AG


[4 marks]

a.

-sin(ln(1+x))×11+x=-x+32x2-53x3+        A1A1

sin(ln(1+x))=-1+x-x+32x2-53x3+

attempts to expand the RHS up to and including the x3 term         M1

=x-32x2+53x3+x2-32x3+        A1

=x-12x2+16x3+        AG


[4 marks]

b.

METHOD 1

let tan(ln(1+x))=a0+a1x+a2x2+a3x3+

uses sin(ln(1+x))=cos(ln(1+x))×tan(ln(1+x)) to form         M1

x-12x2+16x3+=1-12x2+12x3+a0+a1x+a2x2+a3x3+        A1

=a0+a1x+a2-12a0x2+a3-12a1+12a0x3+         (A1)

attempts to equate coefficients,

a0=0,  a1=1,  a2-12a0=-12,  a3-12a1+12a0=16         M1

a0=0,  a1=1,  a2=-12,  a3=23        A1

so tan(ln(1+x)) =x-12x2+23x3+

 

METHOD 2

uses tan(ln(1+x))=sin(ln(1+x))cos(ln(1+x)) to form         M1

=x-12x2+16x3+1-12x2+12x3+-1        A1

=1-12x2+12x3+-1=1+12x2-12x3+         (A1)

attempts to expand the RHS up to and including the x3 term         M1

=x-12x2+16x3+1+12x2-12x3+

=x+12x3-12x2+16x3+

=x-12x2+23x3+        A1


Note: Accept use of long division.


[5 marks]

c.

Examiners report

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Syllabus sections

Topic 5 —Calculus » AHL 5.19—Maclaurin series
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