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Date May 2019 Marks available 3 Reference code 19M.1.SL.TZ2.S_10
Level Standard Level Paper Paper 1 Time zone Time zone 2
Command term Find Question number S_10 Adapted from N/A

Question

Let y=(x3+x)32.

Consider the functions f(x)=x3+x and g(x)=63x2x3+x, for x ≥ 0.

The graphs of f and g are shown in the following diagram.

The shaded region R is enclosed by the graphs of f, g, the y-axis and x=1.

Find dydx.

[3]
a.

Hence find (3x2+1)x3+xdx.

[3]
b.

Write down an expression for the area of R.

[2]
c.

Hence find the exact area of R.

[6]
d.

Markscheme

* This question is from an exam for a previous syllabus, and may contain minor differences in marking or structure.

evidence of choosing chain rule       (M1)

eg   dydx=dydu×dudx,  u=x3+x,  u=3x2+1

dydx=32(x3+x)12(3x2+1)(=32x3+x(3x2+1))        A2  N3

[3 marks]

a.

integrating by inspection from (a) or by substitution       (M1)

eg   2332(3x2+1)x3+xdxu=x3+xdudx=3x2+1, u12u321.5

correct integrated expression in terms of x       A2 N3

eg   23(x3+x)32+C,  (x3+x)1.51.5+C

[3 marks]

 

 

b.

integrating and subtracting functions (in any order)        (M1)

eg   gf,  fg

correct integral (including limits, accept absence of dx)       A1 N2

eg   10(gf)dx,  1063x2x3+xx3+xdx,  10g(x)10f(x)

[2 marks]

c.

recognizing x3+x is a common factor (seen anywhere, may be seen in part (c))       (M1)

eg   (3x21)x3+x6(3x2+1)x3+x,   (3x21)x3+x

correct integration      (A1)(A1)

eg   6x23(x3+x)32

Note: Award A1 for 6x and award A1 for 23(x3+x)32.

substituting limits into their integrated function and subtracting (in any order)       (M1)

eg   623(13+1)32,  0[623(13+1)32]

correct working       (A1)

eg   623×22,  623×4×2

area of R=6423(=6238,623×232,18423)       A1  N3

[6 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 5—Calculus » AHL 5.11—Indefinite integration, reverse chain, by substitution
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