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Date May 2014 Marks available 4 Reference code 14M.2.hl.TZ1.10
Level HL only Paper 2 Time zone TZ1
Command term Find Question number 10 Adapted from N/A

Question

Let f(x)=e2x+1ex2.

The line L2 is parallel to L1 and tangent to the curve y=f(x).

Find the equations of the horizontal and vertical asymptotes of the curve y=f(x).

[4]
a.

(i)     Find f(x).

(ii)     Show that the curve has exactly one point where its tangent is horizontal.

(iii)     Find the coordinates of this point.

 

[8]
b.

Find the equation of L1, the normal to the curve at the point where it crosses the y-axis.

[4]
c.

Find the equation of the line L2.

[5]
d.

Markscheme

xy12 so y=12 is an asymptote     (M1)A1

ex2=0x=ln2 so x=ln2 (=0.693) is an asymptote     (M1)A1

[4 marks]

a.

(i)     f(x)=2(ex2)e2x(e2x+1)ex(ex2)2     M1A1

          =e3x4e2xex(ex2)2

(ii)     f(x)=0 when e3x4e2xex=0     M1

          ex(e2x4ex1)=0

          ex=0, ex=0.236, ex=4.24 (or ex=2±5)     A1A1

 

Note:     Award A1 for zero, A1 for other two solutions.

     Accept any answers which show a zero, a negative and a positive.

 

          as ex>0 exactly one solution     R1

 

Note:     Do not award marks for purely graphical solution.

 

(iii)     (1.44, 8.47)     A1A1

[8 marks]

b.

f(0)=4     (A1)

so gradient of normal is 14     (M1)

f(0)=2     (A1)

so equation of L1 is y=14x2     A1

[4 marks]

c.

f(x)=14     M1

so x=1.46     (M1)A1

f(1.46)=8.47     (A1)

equation of L2 is y8.47=14(x1.46)     A1

(or y=14x+8.11)

[5 marks]

d.

Examiners report

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

Syllabus sections

Topic 2 - Core: Functions and equations » 2.2 » Investigation of key features of graphs, such as maximum and minimum values, intercepts, horizontal and vertical asymptotes and symmetry, and consideration of domain and range.
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