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Date May 2016 Marks available 3 Reference code 16M.3ca.hl.TZ0.2
Level HL only Paper Paper 3 Calculus Time zone TZ0
Command term Obtain Question number 2 Adapted from N/A

Question

A function ff is given by f(x)=x0ln(2+sint)dtf(x)=x0ln(2+sint)dt.

Write down f(x).

[1]
a.

By differentiating f(x2), obtain an expression for the derivative of x20ln(2+sint)dt with respect to x.

[3]
b.

Hence obtain an expression for the derivative of x2xln(2+sint)dt with respect to x.

[3]
c.

Markscheme

ln(2+sinx)    A1

Note:     Do not accept ln(2+sint).

[1 mark]

a.

attempt to use chain rule     (M1)

ddx(f(x2))=2xf(x2)    (A1)

=2xln(2+sin(x2))    A1

[3 marks]

b.

x2xln(2+sint)dt=x20ln(2+sint)dtx0ln(2+sint)dt    (M1)(A1)

ddx(x2xln(2+sint)dt)=2xln(2+sin(x2))ln(2+sinx)    A1

[3 marks]

c.

Examiners report

Many candidates answered this question well. Many others showed no knowledge of this part of the option; candidates that recognized the Fundamental Theorem of Calculus answered this question well. In general the scores were either very low or full marks.

a.

Many candidates answered this question well. Many others showed no knowledge of this part of the option; candidates that recognized the Fundamental Theorem of Calculus answered this question well. In general the scores were either very low or full marks.

b.

Many candidates answered this question well. Many others showed no knowledge of this part of the option; candidates that recognized the Fundamental Theorem of Calculus answered this question well. In general the scores were either very low or full marks.

c.

Syllabus sections

Topic 9 - Option: Calculus
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