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TMA MST121 03 - Two sections of advanced calculus





TMA MST121 03 Cut-off date 21 March 2012


********SEE ATTACHED PDFS FOR QUESTIONS AND EQUATIONS THAT ARE PROPERLY FORMATTED *********

This assignment covers Block C. It has six questions.
Question 1 – 20 marks

You should be able to answer this question after studying Chapter C1.
This question concerns the function
f(x) = -2x3 -9x2 + 24x + 40.
(a) Find the stationary points of this function. [6]
(b) (i) Using the strategy to apply the First Derivative Test, classify the
left-hand stationary point found in part (a). [4]
(ii) Using the Second Derivative Test, classify the right-hand
stationary point found in part (a). [3]
(c) Find the y-coordinate of each of the stationary points on the graph of
the function f(x), and also evaluate f(0). [3]
(d) Hence draw a rough sketch of the graph of the function f(x). [4]
Question 2 – 15 marks
You should be able to answer this question after studying Chapter C1.
In each of the following parts, you should simplify your answers where it is
appropriate to do so.
(a) (i) Write down the derivative of each of the functions
f(x) = x4 and g(x) = sin(5x). [2]
(ii) Hence, by using the Product Rule, differentiate the function
k(x) = x4 sin(5x). [2]
(b) (i) Write down the derivative of each of the functions
f(t) = e6t and g(t) = t3 + 8. [2]
(ii) Hence, by using the Quotient Rule, differentiate the function

(ii) Hence, by using the Composite Rule, differentiate the function


Question 3 – 10 marks
You should be able to answer this question after studying Chapter C1.
In the quadrilateral PQRS shown below, the side QR has length 1 metre,
the side RS has length 2 metres, and the angle at R is a right angle. The
diagonal PR has length 2 metres. The point P is a perpendicular distance
x metres from QR. The value of x is between 0 and 2. (The quadrilateral
described cannot exist for other values of x.)
(You are not asked to derive this formula.)
For parts (a) and (b) (and for part (c), if you use Mathcad there) you
should provide a printout annotated with enough explanation to make it
clear what you have done.
Note: If you define x to be a range variable in part (a) and wish to use x
in a symbolic calculation in part (b), then you will need to insert the
definition x := x between the two parts in your worksheet. (For more
details, see the bottom of page 49 in A Guide to Mathcad.)
(a) Use Mathcad to obtain the graph of the function A(x). [2]
(b) This part of the question requires the use of Mathcad in each sub-part.
(i) By using the differentiation facility, and if you wish the symbolic
keyword ‘simplify’, find an expression for the derivative A!(x). [2]
(ii) By either applying a solve block or solving symbolically, find a
value of x for which A!(x) = 0. [2]
(iii) Verify, by the Second Derivative Test, that this value of x
corresponds to a local maximum of A(x). (It should be apparent
from the graph obtained in part (a) that this is also an overall
maximum within the domain of A(x).) [2]
(c) Using Mathcad, or otherwise, calculate the maximum possible area of
the quadrilateral, according to the model. [2]

Question 4 – 25 marks
You should be able to answer this question after studying Chapter C2.
(a) Find the indefinite integrals of the following functions.
(i) f(t) = 8 cos(16t) - 21e7t [3]


Question 5 – 10 marks
You should be able to answer this question after studying Chapter C2.
A rocket is modelled by a particle that moves along a vertical line. From
launch, the rocket rises until its motor cuts out after 17 seconds. At this
time it has reached a height of 580 metres above the launch pad and
attained an upward velocity of 120ms-1. From this time on, the rocket
has constant upward acceleration -10ms-2 (due to the effect of gravity
alone).
Choose the s-axis (for the position of the particle that represents the
rocket) to point upwards, with origin at the launch pad. Take t = 0 to be
the time when the rocket motor cuts out.
(a) What is the maximum height (above the launch pad) reached by the
rocket? [4]
(b) How long (from launch) does the rocket take to reach this maximum
height? [2]
(c) After how long (from launch) does the rocket crash onto the launch
pad? Give your answer in seconds correct to one decimal place. [4]


Question 6 – 20 marks
You should be able to answer this question after studying Chapter C3.
(a) Solve the initial-value problem
(iii) Find the corresponding particular solution (in implicit form) that
satisfies the initial condition y = 1 when x = 0. [3]
(iv) Find the explicit form of this particular solution. [2]
(v) What is the value of y given by this particular solution when
x = 2? Give your answer to four significant figures. [2]


===========================

TMA MST121 04 Cut-off date 23 May 2012
This assignment covers the whole module.
You should answer ALL FOUR questions. One of the requirements for
passing the module is that you must obtain a score of at least 30 out
of 100 for this assignment.
Question 1 – 25 marks
In this question, positions are given with reference to a Cartesian
coordinate system whose x- and y-axes point due East and due North,
respectively. Distances are measured in kilometres.
A light aircraft travels at a constant speed of 480 kilometres per hour
(kph) in a straight line from airport A, located at position
(xA, yA) = (-200, 100), to airport B at (xB, yB) = (400,-100).
(a) (i) Find the equation of the line of travel of the aircraft. [3]
(ii) Find the direction of travel of the aircraft, as a bearing, with the
angle in degrees correct to one decimal place. [2]
(iii) Calculate the distance between airports A and B, in kilometres,
correct to the nearest kilometre. [3]
(iv) How long does the aircraft take to travel from A to B? Give your
answer in hours and minutes, correct to the nearest minute. [3]
(v) Find parametric equations for the line of travel of the aircraft.
Your equations should be in terms of a parameter t, and should
be such that the aircraft is at airport A when t = 0 and at airport
B when t = 1. [2]
(b) During its journey, the aircraft passes a landmark, L, located at
position (90,-30).
(i) Let d kilometres be the distance between the location of the
aircraft at parameter value t and the landmark L. Find an
expression for d2 in terms of t. Simplify your result as far as
possible. [4]
(ii) Using your answer to part (b)(i) and the method of completing
the square, determine the shortest distance, to the nearest
kilometre, between the aircraft and the landmark L. [3]
(iii) Landmark L is visible from a distance of at most 100 kilometres.
Calculate the parameter values t1 and t2 when the landmark L
can first and last be seen from the aircraft. For how many
minutes is the landmark visible from the aircraft? [5]


Question 2 – 25 marks
This question requires the use of Mathcad throughout, and no marks will
be awarded to answers obtained by other means. For each of parts (a)–(c)
you should provide an appropriate printout, though a printout on one page
may cover your answers to several parts. Annotate your printouts with
Mathcad text or handwriting, or reference them from a separate sheet, in
order to explain clearly what you have done and what your conclusions are.
You may find it useful in all parts of this question to refer to A Guide to
Mathcad.

A flexible wire PQRS, of total length 16 metres, is bent into a three-edged
planar shape, and its ends P, S placed at the edge of a disc of radius
9 metres with centre O, as shown in the diagram below. (The arc PS is
not part of the wire.) The end-segments PQ and RS of the wire lie along
straight lines through O, while the arc QR forms part of a circle with
centre O and subtends an angle x (in radians) at O.
This question concerns the area A enclosed between the wire and the edge
of the disc, which is shown shaded below. This area can be expressed by
A= f(x), where
(You are NOT asked to show this.)

In part (a), you may either use the Mathcad graph plotter file (121A3-04)
or plot the graph in a new worksheet of your own.
(a) (i) Plot the graph of the function f(x). Your graph should cover the
interval [0,1.78] in the x-direction and [0, 60] in the y-direction. [4]
(ii) By using the ‘Trace’ facility (and also, if you wish, the ‘Zoom’
facility), estimate to two decimal places the coordinates of the
point on this graph at which y = f(x) takes its maximum value. [2]
(iii) On the same graph, plot the line y = 36. Using the ‘Trace’
facility, estimate to two decimal places both solutions of the
equation f(x) = 36. (These solutions give the values of x for
which the shaded area is 36m2.) [3]
page 5 of 21
(b) Each of the following recurrence relations has the property that if a
sequence generated by the recurrence relation converges to a limit in
the interval

(i) For each of these recurrence relations, generate the sequence with
starting value x0 = 1.7, and tabulate your results to six decimal
places. Which sequence converges more rapidly? (That is, which
sequence gives an estimate with specified accuracy for its limit
with the smaller value of n?) [5]
(ii) Use your tabulated results to write down the solutions of the
equation f(x) = 36 to six decimal places. [3]
(c) This part of the question concerns finding the maximum value of the
function f(x), as estimated in part (a)(ii), and hence the maximum
possible value of the shaded area A.
You may need to put x := x in your worksheet before answering
part (c)(i).
(i) Use symbolic differentiation and the ‘simplify’ keyword to find an
expression for the derivative f!(x). [3]
(ii) The maximum value of f(x) occurs where f!(x) = 0. Use a solve
block to solve the equation f!(x) = 0 for x, giving your answer to
six decimal places. [3]
(iii) Calculate the corresponding maximum possible value of the
area A, giving your answer to four decimal places. [2]
page 6 of 21
Question 3 – 25 marks
As for other questions, remember to show your working explicitly
throughout your answer to this question.
(a) (i) Use the Composite Rule to differentiate the function
f(x) = e(-3x+2 sin x)/6. [5]
(ii) Use the Quotient Rule and your answer to part (a)(i) to show
that the function
g(x) =
e(-3x+2 sin x)/6
3 + 2 cos x
(0 = x = 2p)
has derivative
6(3 + 2 cos x)2 . [4]
(iii) Find any stationary points of the function g(x) defined in
part (a)(ii), and use the First Derivative Test to classify each
stationary point as a local maximum or a local minimum of g(x).
(Note that the domain of the function g is the interval [0,2p], and
recall that | sin x| = 1 for all values of x.) [5]
(iv) Using your answers to parts (a)(ii) and (a)(iii), find the area
below the graph of
y =
20(5 - 2 sin x)(2 sin x - 1) e(-3x+2 sin x)/6
(3 + 2cos x)2 (0 = x = 2p)
and above the x-axis. Give your answer to five significant figures. [4]
(b) (i) Using your answer to part (a)(i), find the general solution of the
differential equation
dy
dx
= (2 cos x - 3) e(-3x+2 sin x)/6 y5/6 (y > 0),
giving the solution in implicit form. [4]
(ii) Find the particular solution of the differential equation in
part (b)(i) for which y = 1 when x = 0, and then give this
particular solution in explicit form. [3]


Question 4 – 25 marks
This question requires the use of OUStats.
(a) The file AUSTEN.OUS contains data on the lengths of 60 sentences in
Jane Austen’s Sense and Sensibility. A page was chosen at random
and a sentence selected by one of two different methods: rolling a die
to choose the sentence number and choosing blindly, with a pin.
Thirty sentence lengths were obtained using each method of sentence
selection. ‘Die’ contains the lengths of the sentences selected by
rolling a die; ‘Pin’ contains the lengths of the sentences selected with
a pin. [Data collected by C. E. Graham, 1996.]
(i) Use OUStats to produce boxplots of the two samples of data on a
single diagram. [2]
(ii) Comment on what the boxplots produced in part (a)(i) tell you
about the lengths of sentences obtained using each method. [2]
(iii) Obtain a printout of a histogram of the data for ‘Pin’, using the
OUStats default values for the starting value of the first interval
and the interval width. Add the fitted normal curve to this
histogram before printing. Write down the mean and the
standard deviation of the fitted normal distribution. Is this
normal distribution a good fit for the data? Explain your answer. [4]
(iv) Use the two-sample z-test to investigate whether there is a
difference between the mean length of sentences obtained by the
two methods. Your answer should include the following:
• A statement of the null and alternative hypotheses, including
an explanation of the meanings of any symbols that you use
(other than H0 and H1).
• The means of the lengths of sentences obtained by the two
methods, and the value of the test statistic. (You should use
OUStats to find these values.)
• Your conclusion. [8]
(b) The file BEARS.OUS contains the results of aerial surveys carried out
on each of 20 days in a particular part of Alaska. The average wind
speed on each day is in ‘WindSpeed’, and the number of black bears
sighted is in ‘Bears’. The source does not report the units in which
the wind speed was measured. The data are paired. [Lindgren, B.W.
and Berry, D.A. (1981) Elementary Statistics, p. 137, Macmillan.]
(i) Obtain a printout of a scatterplot of the data with ‘WindSpeed’
as the explanatory variable (which must be on the x-axis). Add
the least squares fit line to the plot. [3]
(ii) Describe any pattern that you observe in the plot. [1]
(iii) Obtain the equation of the least squares fit line of ‘Bears’ on
‘WindSpeed’. If you use labels other than ‘Bears’ and
‘WindSpeed’, specify which variable represents ‘Bears’ and which
represents ‘WindSpeed’. [2]
(iv) Making use of the model from part (b)(iii), find the number of
bears for wind speed 40. Briefly explain why this is an invalid
prediction. [3]



About the Solutions
Solutions will be downloadable in word and pdf formats!

Question Files
Section_1.pdf
Section_2.pdf

Other Details about the Project/Assignment
Subjects: Mathematics -> Calculus
Topic: University Level Calculus
Level: College / University
Tags:

Calculus, MST121, TMA, OU


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Project Details
Question Files
Section_1.pdf
Section_2.pdf

Subjects: Mathematics -> Calculus
Topic: University Level Calculus
Level: College / University
Tags:

Calculus, MST121, TMA, OU


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