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MST121 Using Mathematics - TMA MST121 03 and CMA MST121 41 Solutions





Mathematics, Computing and Technology
MST121 Using Mathematics
MST121 Assignment Booklet I 2011B
Contents Cut-off date
13 TMA MST121 03
(covering Block C) 06 July 2011
17 CMA MST121 41
(covering Block D) 31 August 2011


===> The following was pasted from the PDF so may not look right - Please view the pdf for full questions / solutions<====



Instructions for submitting the TMAs
Send your answers to each tutor-marked assignment (TMA), together with
a completed TMA form (PT3), to reach your tutor on or before the cut-off
date shown above. However, you should not send a TMA form with Part 2
of TMA 01 – see the submission instructions for TMA 01 on page 3.
There are instructions on how to fill in the TMA form in the Preparatory
Assignment Booklet. Remember to fill in the correct assignment number.
Remember also to allow sufficient time in the post for the TMA to reach
your tutor on or before the cut-off date, and to pay the correct postage
charge to cover the weight and size of the TMA and its envelope. Do not
use recorded delivery. You are advised to keep a copy of your answers in
case of loss in the mail, to obtain proof of posting, and to keep all your
marked TMAs so you can refer to them later.
Your tutor will inform you about the address to use for submitting your
TMAs. Please don’t submit your TMAs directly to the University.
Regrettably, the University is unable to accept TMAs submitted
electronically on this module. If you have any questions about how best to
prepare and submit your TMAs, please contact your tutor.
Copyright c 2011 The Open University WEB 02083 8
21.1
 
TMA MST121 03 Cut-off date 06 July 2011
This assignment covers Block C. It has six questions.
Question 1 – 20marks
You should be able to answer this question after studying Chapter C1.
This question concerns the function
f(x) = x3 - 15x2 + 48x+ 29.
(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 – 15marks
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) = e 3x and g(x) = ln(6x) (x > 0). [2]
(ii) Hence, by using the Product Rule, differentiate the function
k(x) = e
-3x ln(6x) (x > 0). [2]
(b) (i) Write down the derivative of each of the functions
f(t) = 2 - t2 and g(t) = cos(4t). [2]
(ii) Hence, by using the Quotient Rule, differentiate the function
2 - t2
k(t) = (-1p < t < 1p). [3] cos(4t) 8 8
(c) (i) Write down the derivative of the function
5
f(x) = (x > 0). [1]
x
(ii) Hence, by using the Composite Rule, differentiate the function
5
k(x) = sin (x > 0). [5]
x
13

Question 3 – 10marks
You should be able to answer this question after studying Chapter C1.
In the quadrilateral PQRS shown below, the sides QR and PS each have
length 1 metre, the side RS has length 2 metres, and the angle at R is a
right angle. The point P is a perpendicular distance x metres from QR.
The value of x is between 1 and 3. (The quadrilateral described cannot
exist for other values of x.)
Q
2
x
A(x)
P
1 1
R S
The total area A(x)m2 of the quadrilateral is given by
A(x) = 1x+ (x - 1)(3 - x) (1< x < 3). 2
(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 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]
14
 

Question 4 – 25marks
You should be able to answer this question after studying Chapter C2.
(a) Find the indefinite integrals of the following functions.
(i) f(t) = 14 sin(7t) + 5e-15t [3]
8 + 39x3
(ii) g(x) = (x > 0) [4]
x
(iii) h(u) = sin2 1u [6] 8
6
(b) Evaluate x(8 - 5x2) dx. [6]
2
(c) (i) Write down a definite integral that will give the value of the area
under the curve y = x3 cos(6x) between x = 1p and x = 1p. [2] 4 3
(The expression x3 cos(6x) takes no negative values for
1p = x = 1p. You are not asked to evaluate the integral by 4 3
hand.)
Provide a printout of your working for part (c)(ii).
(ii) Use Mathcad to find the area described in part (c)(i), giving your
answer correct to four significant figures. [4]
Question 5 – 10marks
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 9 seconds. At this
time it has reached a height of 380 metres above the launch pad and
attained an upward velocity of 40ms-1. From this time on, the rocket has
a constant upward acceleration of -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 on to the launch
pad? Give your answer in seconds correct to one decimal place. [4]
15

 
Question 6 – 20marks
You should be able to answer this question after studying Chapter C3.
(a) Solve the initial-value problem
dy cos(3x)
= , y= 2 when x = 0.
dx 2 - sin(3x)
(You may find equation (2.4) in Chapter C2 helpful when integrating.) [6]
(b) (i) Using equation (2.3) in Chapter C2, show that
ex + e-2x
dx = -(2ex - e
-2x)-1/2 + c,
(2ex - e-2x)3/2
where c is an arbitrary constant. [2]
(ii) Hence find, in implicit form, the general solution of the
differential equation
5/4 -2x dy 2y ex + e
= (y > 0). [5]
dx (2ex - e-2x)3/2
(iii) Find the corresponding particular solution (in implicit form) that
satisfies the initial condition y = 16 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 = 1? Give your answer to four significant figures. [2] 2
16
CMA MST121 41 Cut-off date 31 August 2011
This assignment covers Block D. It has 24 questions.
Questions 1 to 9 are on Chapter D1.
Questions 1 to 4
Two fair dice, one six-sided and one tetrahedral, are rolled.
1 Choose the option that gives the probability that the numbers
obtained on the two dice add up to 7.
2 Choose the option that gives the probability that the numbers
obtained on the two dice add up to 6 or more.
Options for Questions 1 and 2
A 1 B 1 C 1 D 5 7 6 3 12
E 7 F 15 G 3 H 5 12 24 4 6
3 Choose the option that gives the probability that the number on the
six-sided die is 5 or 6 and the number on the tetrahedral die is odd.
4 Choose the option that gives the probability that the number 3 is
obtained on at least one of the dice.
Options for Questions 3 and 4
A 1 B 1 C 5 D 1 8 6 24 4
E 1 F 3 G 5 H 11 3 8 12 24
Questions 5 and 6
During one day in a particular hospital, the probability that a birth was
that of a boy was 26 . 41
5 Choose the option that is closest to the probability that the first boy
born on that day was the fifth child born.
6 Choose the option that is closest to the probability that there were at
least four girls born before the first boy was born.
Options for Questions 5 and 6
A 0.0114 B 0.0179 C 0.0592 D 0.1025
E 0.1853 F 0.6341 G 0.7 H 0.9615
17
Question 7
A popular pastime in a particular club is flipping beer mats balanced on
the edge of a table and trying to catch them before they land back on the
table. Ann is successful two times out of five in the long run. Choose the
option that is closest to the number of times, on average, that she has to
flip a beer mat in order to catch it.
Options
A 0.4 B 0.5 C 1 D 1.2 E 2 F 2.5
Question 8
A game uses a pack of five cards numbered 1, 2, 3, 4, 5. In each round the
cards are shuffled, the top card is turned face up, and the number is
recorded. This card is then returned to the pack. Choose the option that
gives, on average, the number of rounds to two decimal places that would
be required to obtain four different numbers.
Options
A 5.42 B 6.42 C 7.42
D 8 E 8.42 F 10.42
Question 9
Assuming that a birth is equally likely to occur on any day of the week,
choose the option that is closest to the probability that, in a family of four
children, at least two of the children were born on the same day of the
week.
Options
A 0.02 B 0.10 C 0.35
D 0.45 E 0.55 F 0.65
Questions 10 to 13 are on Chapter D2.
Question 10
A sample of concentrations of transferrin receptor for six women with
laboratory evidence of overt iron-deficiency anaemia yielded the following
data, in ppm.
12.1 5.3 7.7 5.4 5.9 13.2
Choose the option that is closest to the sample standard deviation.
Options
A 3.19 B 3.52 C 8.30
D 12.23 E 24.96 F 27.34
18
Questions 11 to 13
You should use OUStats for these questions.
The weights (in kg) of people in a sample selected at random from a
particular population are normally distributed with mean 73.15 and
standard deviation 6.6.
11 Choose the option that is closest to the weight above which
approximately 25% of the weights of people from the population will
lie.
Options for Question 11
A 55.20 B 55.87 C 56.63
D 60.24 E 68.70 F 77.60
12 Choose the option that is closest to a range of values, symmetric
about the mean, within which approximately 95% of the weights of
people will lie.
Options for Question 12
A (60.2, 86.1) B (60.6, 85.7) C (62.3, 84.0)
D (66.9, 88.4) E (62.6, 83.7) F (61.6, 84.7)
13 Choose the option that is closest to the percentage of people with
weights below 90 kg.
Options for Question 13
A 1% B 10% C 19%
D 81% E 90% F 99%
Questions 14 to 17 are on Chapter D3.
Questions 14 to 16
The distribution of the weights of the contents of boxes of a certain
breakfast cereal, labelled as containing 450 g, has mean 452.1 g and
standard deviation 6.5 g.
14 Choose the option that is closest to the standard error of the mean
contents (in grams) for samples of 30 cereal boxes.
Options for Question 14
A 0.22 B 0.32 C 1.19
D 1.21 E 1.30 F 2.5
19
You should use OUStats for Questions 15 and 16.
The sampling distribution of the mean weight of contents, for samples of
35 cereal boxes, has mean 452.1 g and standard deviation 1.1 g.
15 Choose the option that is closest to the probability that the mean
weight of the contents of a sample of 35 cereal boxes will be less
than 450 g.
Options for Question 15
A 0.03 B 0.17 C 0.31
D 0.53 E 0.79 F 0.97
16 Choose the option that gives a range of values, symmetric about the
mean, within which the mean weight of contents of approximately
90% of samples of 35 cereal boxes will lie.
Options for Question 16
A (449.2, 455.0) B (449.4, 454.8) C (449.9, 454.3)
D (450.3, 453.9) E (451.0, 453.2) F (451.4, 452.8)
Question 17
The mean monthly expenditure on petrol per household in Milton Keynes
is estimated by selecting a random sample of 36 households. The sample
mean is £186.25, and the sample standard deviation is £47.40.
Choose the option that gives an approximate 95% confidence interval for
the mean monthly expenditure on petrol per household in Milton Keynes.
Options
A (171.24, 201.26) B (173.29, 199.21) C (175.19, 197.31)
D (170.77, 201.73) E (173.61, 198.89) F (174.80, 197.70)
20
Questions 18 to 24 are on Chapter D4.
Questions 18 to 20
The salaries of employees (in £) in a certain small company are as follows.
9829 12 554 30 802 23 664 17 582
41 541 29 263 27 486 45 859 14 291
18 Choose the option that is the median salary in £ in the company.
Options for Question 18
A 9829 B 23 664 C 25 287
D 25 575 E 27 486 F 45 859
19 Choose the TWO options that give the lower and upper quartiles for
the salaries in the company.
Options for Question 19
A 9829 B 14 291 C 17 582
D 29 263 E 30 802 F 45 859
20 Choose the option that gives the range of the salaries in the company.
Options for Question 20
A 9829 B 18 015 C 25 287
D 25 575 E 36 030 F 45 859
21
Questions 21 and 22
A small supermarket wishes to test the effectiveness of two types of
coupon for a day. The amount (in £) spent by each customer using a
coupon is recorded.
A summary of the results is given in the following table.
Amount (in £) spent during the day
Coupon Sample Sample Sample standard
size mean deviation
Type I 38 26.27 4.37
Type II 32 24.05 9.4
21 Choose the option that is closest to the estimated standard error
(ESE) of the difference between the sample means for the first and
second types of coupon.
Options for Question 21
A 0.11 B 0.64 C 1.81
D 3.26 E 6.02 F 10.37
22 Choose the option that is closest to the magnitude of the test
statistic z that would be used to carry out a two-sample z-test to
determine whether there is a difference between the mean amount
spent with the two types of coupon.
Options for Question 22
A 0 B 0.06 C 1.11
D 1.23 E 2.22 F 27.80
22
Questions 23 and 24
A production manager in a certain engineering company uses a regression
line to model the relationship between production volume and production
cost. The regression line obtained using the recorded data has the equation
y = 4.4 + 3.2x,
where y is the production cost (in £000s), and x is the units produced
(in 000s).
23 Choose the option that gives the estimated production cost (in £000s)
for producing 5500 units.
Options for Question 23
A 17.60 B 22.00 C 27.40
D 5500 E 17 604.40 F 17 600 004.40
24 Choose the option that gives the estimated increase in the production
cost (in £000s) due to a production increase from 4000 to 5000 units.
Options for Question 24
A 3.2 B 4.4 C 10
D 1000 E 3200 F 4400
23


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Topic: Advanced Calculus
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