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Date May 2013 Marks available 4 Reference code 13M.2.hl.TZ1.13
Level HL only Paper 2 Time zone TZ1
Command term Verify Question number 13 Adapted from N/A

Question

The function f has inverse f1 and derivative f(x) for all xR. For all functions with these properties you are given the result that for aR with b=f(a) and f(a)0

(f1)(b)=1f(a).

Verify that this is true for f(x)=x3+1 at x = 2.

[6]
a.

Given that g(x)=xex2, show that g(x)>0 for all values of x.

[3]
b.

Using the result given at the start of the question, find the value of the gradient function of y=g1(x) at x = 2.

[4]
c.

(i)     With f and g as defined in parts (a) and (b), solve gf(x)=2.

(ii)     Let h(x)=(gf)1(x). Find h(2).

[6]
d.

Markscheme

f(2)=9     (A1)

f1(x)=(x1)13     A1

(f1)(x)=13(x1)23     (M1)

(f1)(9)=112     A1

f(x)=3x2     (M1)

1f(2)=13×4=112     A1

Note: The last M1 and A1 are independent of previous marks.

 

[6 marks]

a.

g(x)=ex2+2x2ex2     M1A1

g(x)>0 as each part is positive     R1

[3 marks]

b.

to find the x-coordinate on y=g(x) solve

2=xex2     (M1)

x=0.89605022078     (A1)

gradient =(g1)(2)=1g(0.896)     (M1)

=1e(0.896)2(1+2×(0.896)2)=0.172 to 3sf     A1

(using the dydx function on gdc g(0.896)=5.7716028

1g(0.896)=0.173

[4 marks]

c.

(i)     (x3+1)e(x3+1)2=2     A1

x=0.470191     A1

 

(ii)     METHOD 1

(gf)(x)=3x2e(x3+1)2(2(x3+1)2+1)     (M1)(A1)

(gf)(0.470191)=3.85755     (A1)

h(2)=13.85755=0.259 (232)     A1

Note: The solution can be found without the student obtaining the explicit form of the composite function.

 

METHOD 2

h(x)=(f1g1)(x)     A1

h(x)=(f1)(g1(x))×(g1)(x)     M1

=13(g1(x)1)23×(g1)(x)     M1

h(2)=13(g1(2)1)23×(g1)(2)

=13(0.896051)23×0.171933

=0.259 (232)     A1     N4

[6 marks]

d.

Examiners report

There were many good attempts at parts (a) and (b), although in (b) many were unable to give a thorough justification.

a.

There were many good attempts at parts (a) and (b), although in (b) many were unable to give a thorough justification.

b.

Few good solutions to parts (c) and (d)(ii) were seen although many were able to answer (d)(i) correctly.

c.

Few good solutions to parts (c) and (d)(ii) were seen although many were able to answer (d)(i) correctly.

d.

Syllabus sections

Topic 6 - Core: Calculus » 6.2 » Derivatives of xn , sinx , cosx , tanx , ex and \lnx .
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