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Unformatted text preview: ation. (ii) Find the solution to the equation g(x) = f(x)
(4)
(Total 10 marks) 703. Let h(x) = (x – 2)sin(x – 1) for –5 ≤ x ≤ 5. The curve of h(x) is shown below. There is a
minimum point at R and a maximum point at S. The curve intersects the xaxis at the points
(a, 0) (1, 0) (2, 0) and (b, 0).
y
4
3
2 –5 –4 –3 –2 S 1 (a, 0)
–1 (b, 0)
1 –1 R 2 3 4 5 x –2
–3
–4
–5
–6
–7 (a) Find the exact value of
(i) a; (ii) b.
(2) 473 The regions between the curve and the xaxis are shaded for a ≤ x ≤ 2 as shown.
(b) (i) Write down an expression which represents the total area of the shaded regions. (ii) Calculate this total area.
(5) (c) (i) The ycoordinate of R is –0.240. Find the ycoordinate of S. (ii) Hence or otherwise, find the range of values of k for which the equation
(x – 2)sin(x – 1) = k has four distinct solutions.
(4)
(Total 11 marks) 704. Reaction times of human beings are normally distributed with a mean of 0.76 seconds and a
standard deviation of 0.06 seconds.
(a) The graph below is that of the standard normal curve. The shaded area represents the
probability that the reaction time of a person chosen at random is between 0.70 and 0.79
seconds. a 0b (i) Write down the value of a and of b. (ii) Calculate the probability that the reaction time of a person chosen at random is
(a) greater than 0.70 seconds; (b) between 0.70 and 0.79 seconds.
(6) 474 Three percent (3 %) of the population have a reaction time less than c seconds.
(b) (i) Represent this information on a diagram similar to the one above. Indicate clearly
the area representing 3 %. (ii) Find c.
(4)
(Total 10 marks) 705. Let f(x) =
(a) 1
.
1+ x2 Write down the equation of the horizontal asymptote of the graph of f.
(1) (b) Find f′(x).
(3) (c) The second derivative is given by f″(x) = 6x 2 − 2
.
(1 + x 2 ) 3 Let A be the point on the curve of f where the gradient of the tangent is a maximum. Find
the xcoordinate of A.
(4) 475 (d) Let R be the region under the graph of f, between x = – 1
1
and x = ,
2
2 as shaded in the diagram below
y
2 1 R
–1 –1
2 x 1 1
2 –1 Write down the definite integral which represents the area of R.
(2)
(Total 10 marks) 706. Let y = g(x) be a function of x for 1 ≤ x ≤ 7. The graph of g has an inflexion point at P, and a
minimum point at M.
Partial sketches of the curves of g′ and g″ are shown below.
g’ ( x ) g’ ’ (x) 6 6 5 5 4 4 3 3 2 2 1 1 0 1 2 3 4 –1 5 6 7 8 x 0 1 2 3 4 –1 –2 8 x –4 –5 7 –3 –4 6 –2 –3 5 –5 –6 –6 y = g’ (x) y = g’’ (x) 476 Use the above information to answer the following.
(a) Write down the xcoordinate of P, and justify your answer.
(2) (b) Write down the xcoordinate of M, and justify your answer.
(2) (c) Given that g (4) = 0, sketch the graph of g. On the sketch, mark the points P and M.
(4)
(Total 8 marks) 707. Consider f(x) = x3 – 2x2 – 5x + k. Find the value of k if(x + 2) is a factor of f(x).
Working: Answer:
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 Fall '13
 Apple
 The Land

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