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Date None Specimen Marks available 7 Reference code SPNone.3ca.hl.TZ0.4
Level HL only Paper Paper 3 Calculus Time zone TZ0
Command term Prove Question number 4 Adapted from N/A

Question

Let f(x)=2x+|x| , xR .

Prove that f is continuous but not differentiable at the point (0, 0) .

[7]
a.

Determine the value of aaf(x)dx where a>0 .

[3]
b.

Markscheme

we note that f(0)=0, f(x)=3x for x>0 and f(x)=x for x<0

limx0f(x)=limx0x=0     M1A1

limx0f(x)=limx03x=0     A1

since f(0)=0 , the function is continuous when x = 0     AG

limx0f(0+h)f(0)h=limx0hh=1     M1A1

limx0+f(0+h)f(0)h=limx0+3hh=3     A1

these limits are unequal     R1

so f is not differentiable when x = 0     AG

[7 marks]

a.

aaf(x)dx=0axdx+a03xdx     M1

=[x22]0a+[3x22]a0     A1

=a2     A1

[3 marks]

b.

Examiners report

[N/A]
a.
[N/A]
b.

Syllabus sections

Topic 9 - Option: Calculus » 9.3 » Continuity and differentiability of a function at a point.

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