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| , x∈R .
Prove that f is continuous but not differentiable at the point (0, 0) .
[7]
a.
Determine the value of ∫a−af(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
limx→0−f(x)=limx→0−x=0 M1A1
limx→0−f(x)=limx→0−3x=0 A1
since f(0)=0 , the function is continuous when x = 0 AG
limx→0−f(0+h)−f(0)h=limx→0−hh=1 M1A1
limx→0+f(0+h)−f(0)h=limx→0+3hh=3 A1
these limits are unequal R1
so f is not differentiable when x = 0 AG
[7 marks]
a.
∫a−af(x)dx=∫0−axdx+∫a03xdx M1
=[x22]0−a+[3x22]a0 A1
=a2 A1
[3 marks]
b.
Examiners report
[N/A]
a.
[N/A]
b.