(Solved) CC1452A Semester 1 2016 Page 10 of 23 Extended Answer Section

(Solved) CC1452A Semester 1 2016 Page 10 of 23 Extended Answer Section

CC1452A Semester 1 2016 Page 10 of 23 Extended Answer Section

CC1452A Semester 1 2016 Page 10 of 23
Extended Answer Section
There are three questions in this section, each with a number of parts. Write your
answers in the space provided below each part. If you need more space there are extra pages
at the end of the examination paper.

  1. (a) The data points (x, y) below obey an approximately exponential law y ≈ Aekx:
    x 0 2 4 6 8
    y 7.389 33.12 54.60 90.02 403.4
    (i) Perform an appropriate scaling to the experimental data points (x, y) to obtain
    new scaled data points (x, Y ). Round the scaled data to one decimal place.
    (ii) Obtain the equation of the ‘line of best fit’ using the ‘method of least squares’.
    From the data in the table, find Xx,
    Xx
    2
    ,
    XY and XxY . Hence, find
    the equation of the ‘line of best fit’ in the form Y = mx + b.
    CC1452A Semester 1 2016 Page 11 of 23
    (iii) Write y as an approximating function of x, i.e. find A and k such that y ≈
    Aekx
    .
    (b) Consider the function h(x) = x − ln x.
    (i) Find and classify all critical points (x, h(x)) of the function h.
    (ii) Explain why h(x) has no inflection points.
    Question 1 continues on the next page.
    CC1452A Semester 1 2016 Page 12 of 23
    (iii) Find the global maximum and global minimum values of h(x) on the interval
    0.5 ≤ x ≤ 2.
    (c) Evaluate the following sums:
    (i)
    X
    77
    k=7
    9
    k
    (leave your answer in exact form).
    (ii)
    X
    49
    k=1

    1
    (k + 1)2

    1
    k
    2

    .
    This is the end of Question 1.
    CC1452A Semester 1 2016 Page 13 of 23
  2. (a) Let f(x) = cos (πx) − sin (πx).
    (i) Find r, and α (in radians), such that f(x) can be written in the form
    f(x) = r sin (πx + α).
    Note: you may simply quote and use any main results and formulae from the
    lectures; you need not derive them.
    Question 2 continues on the next page.
    CC1452A Semester 1 2016 Page 14 of 23
    (ii) Hence sketch one period of the graph of y = f(x), clearly identifying the
    amplitude, period, mean value, and intercepts with the coordinate axes.
    Question 2 continues on the next page.
    CC1452A Semester 1 2016 Page 15 of 23
    (b) The rate of increase of the water level in a dam, at time t, is given by
    3t
    2
    e
    −t
    3
    where t ≥ 0,
    and t is measured in hours and the height in metres. Calculate the increase in the
    level of the dam in the first 30 minutes.
    (c) Consider the function defined by
    f(x) = e
    x
    cos(x) − x
    2
    sin(x) for x ≥ 0.
    How many critical points, of the type where f
    0
    (x) is undefined, are there? You must
    explain your answer using words.
    This is the end of Question 2.
    CC1452A Semester 1 2016 Page 16 of 23
  3. Let T(x, y) = x
    2 − 6xy + 2x − 2y
    2 + 5y + 1.
    (a) Find ∂T
    ∂x ,
    ∂T
    ∂y ,

    2T
    ∂x2
    ,

    2T
    ∂y2
    , and ∂
    2T
    ∂x∂y .
    (b) Find the critical point of T and determine its nature.
    Question 3 continues on the next page.
    CC1452A Semester 1 2016 Page 17 of 23
    (c) Calculate the equation of the tangent plane of T(x, y) at the critical point you found
    in part (b).
    (d) Suppose a particular material in the shape of a flat right-angled triangle is heated
    in a non-uniform way. The verticies of the triangle are located at (0, 0),(1, 0),(0, 1)
    in the (x, y)-plane. The temperature T at any point (x, y) on the triangle is given
    by T(x, y).
    Let B1, B2, B3 denote the boundary edges of the triangle. B1 is the edge along the
    x-axis, B2 is the edge along the y-axis, and B3 is the edge along the hypotenuse.
    The longest edge of the boundary, B3, can be written mathematically as
    B3 := {(x, y)| y = 1 − x, 0 ≤ x ≤ 1} .
    You are given that the only critical points of T(x, y) on the other two edges occur
    at the corners of the triangle.
    (i) Find all critical points of T along B3.
    Question 3 continues on the next page.
    CC1452A Semester 1 2016 Page 18 of 23
    (ii) Taking all critical points into consideration, find the global maximum and
    minimum temperature on the triangle-shaped material.
    This is the end of Question 3.

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