ENM1600 Engineering Mathematics, S1–2020

Assignment 2

Value: 10%. Due Date: Tuesday 28 April 2020.

· Submit your assignment electronically as one (1) PDF file via the link (Assignment 2 submis¬sion) available on study desk before the deadline. You may edit/change your assignment whilst in Draft status and it is before the deadline.

Please note: Once you have chosen “Submit assignment”, you will not be able to make any changes to your submission.

The submitted version will be marked so please check your assignment carefully before submit¬ting.

· Hand-written work is more than welcome, provided you are neat and legible. Do not waste time type-setting and struggling with symbols. Rather show that you can use correct notation by hand and submit a scanned copy of your assignment. You may also type-set your answers if your software offers quality notation if you wish.

· Offer careful hand-copies of plots if you have no printer. Do not neglect the scale and labels: they may be added by hand.

· Any requests for extensions should be made prior to the due date by contacting the examiner. Requests for an extension should be supported by documentary evidence.

An Assignment submitted after the deadline without an approved extension of time will be penalised. The penalty for late submission is a reduction by 5% of the maximum Assignment Mark, for each University Business Day or part day that the Assignment is late. An Assignment submitted more than ten University Business Days after the deadline will have a Mark of zero recorded for that Assignment.

· We expect a high standard of communication. Look at the worked examples in your texts, and the sample solutions in Tutorial Worked Examples to see the level you should aim at. Up to 15% of the marks may be deducted for poor language and notation.

· Please note: No Assignments for Assessment purposes will be accepted after as-signment solutions have been released.

QUESTION 1 (20 marks)

The currents I1, I2 and I3, measured in Amperes, in the three branches of a parallel circuit are determined by the following system of linear equations.

7I1 - 3I2 - I3 = -48 -I2 + 4I3 = 15 -4I1 + 8I2 = 13

(a) Write the system in matrix form Ax = b; (2 marks)

(b) Solve the system of linear equations using Cramer’s Rule. (18 marks)

Check your answer by substituting into the system of linear equations.

QUESTION 2 (20 marks)

The steady-state concentrations of chemicals in three vats C1, C2 and C3, measured in milligrams per cubic metre (mg/ m3), are determined by the following system of linear equations.

-3C1 + 4C2 + 2C3 = 24 -9C1 - 5C2 + 8C3 = 18 6C1 - 9C2 + C3 = -17

(a) Write the system of linear equations in augmented matrix form. (3 marks)

(b) Use Gaussian elimination on the augmented matrix to find the three concentrations, and

write the row operation used next to each new row. (17 marks)

Check your answer by substituting into the system of linear equations.

QUESTION 3 (12 marks)

1

Let f(x) = 2x - 1 and g(x) = sin4x.

(a) Find the composite function f(g(x)) and give its domain. (6 marks)

(b) Find the composite function g(f(x)) and give its domain. (6 marks)

QUESTION 4 (16 marks)

The following rational functions were found in analyzing the motion of a spring-damper system. Express these rational functions as partial fractions:

8s2 + 26s + 15; (8 marks)

s - 20 + 7s2

(b) s3 + 6s2 + 10s. (8 marks)

QUESTION 5 (16 marks)

On a fun park ride the position of the carriage at time t = 0 (in minutes) is given by the parametric function

x=4cos5pt+~10+2cos4pt~cospt

y=4sin5pt+ ~10 + 2cos4pt~ sinpt

where both x and y are measured from the axis of rotation and are given in metres.

(a) Show the squared distance from the axis of rotation, i.e. d2 = x2 + y2, can be rewritten as

d2 = 20 cos2 4pt + 120 cos 4pt + 116. (7 marks)

(b) Using (a) find all times, t = 0, (in exact form i.e. in terms of fractions) when the distance d is

4 m i.e. d=4ord2 = 16. (7 marks)

(c) What is the first time that the distance d of the carriage is 4 m?

Give your answer exactly (as a fraction) and approximately as a decimal. (2 marks)

QUESTION 6 (16 marks)

The speed, V (km/h), of a car at time t (seconds) is given by

where 0 = t = 100. V = f(t) = ~ pt ~

70 - 10 cos 100

~ pt ~

2 + cos 100

(a) We want to know when the car will reach a certain speed. Solve for the time t in terms of the

speed V , and hence find the formula for the inverse function f_1(t). (6 marks)

(b) Write down the domain of f_1(t). (2 marks)

(c) Plot f(t) from t = 0 to t = 100 and then plot f_1(t) on the same axes but only for the values

of t in the domain of the inverse function found in (b). (6 marks)

(d) Describe the geometric relationship that you can see when you consider both f(t) and f_1(t). (2 marks)

Note: You are encouraged to use MATLAB or Scilab for your plots.

Note that the MATLAB commands for cost and cos_1 t are cos(t) and acos(t). First define values of t from 0 to 100 for your plot of f.

t1=[0:0.01:100];

gives values from 0 to 100, in steps of 0.01. Then find the V-values for these t-values using

y1=(70-10.*cos(pi*t1/100))./(2+cos(pi*t1/100)); Then plot f using

plot(t1,y1).

Use the command

axis equal

to get equal scales on the axes.

To add the plot of f-1, define new t’s for its domain. That is with

t2=[...]

Then calculate the y-values.

/t + 1

For instance for the function 7 sin-1 + 5 we would use the command

t - 2

y2 = 7*asin((t2+1)./(t2-2)) + 5;

We can then plot both functions on the same graph using the command plot(t1,y1,t2,y2).

Note the sin-1 x and cos-1 x functions are evaluated in MATLAB using asin(x) and acos(x) respectively.

Include a plot style to distinguish the graphs, if you wish.

To add an additional plot of y = t to show the geometry, use the same process: define the t’s you want for the plot first (t3=[...]), calculate the y’s, then plot by including t3,y3 after y2 to the previous command

plot(t1,y1,t2,y2,t3,y3).

End of Assignment 2 (100 marks Total)

Assignment 2

Value: 10%. Due Date: Tuesday 28 April 2020.

· Submit your assignment electronically as one (1) PDF file via the link (Assignment 2 submis¬sion) available on study desk before the deadline. You may edit/change your assignment whilst in Draft status and it is before the deadline.

Please note: Once you have chosen “Submit assignment”, you will not be able to make any changes to your submission.

The submitted version will be marked so please check your assignment carefully before submit¬ting.

· Hand-written work is more than welcome, provided you are neat and legible. Do not waste time type-setting and struggling with symbols. Rather show that you can use correct notation by hand and submit a scanned copy of your assignment. You may also type-set your answers if your software offers quality notation if you wish.

· Offer careful hand-copies of plots if you have no printer. Do not neglect the scale and labels: they may be added by hand.

· Any requests for extensions should be made prior to the due date by contacting the examiner. Requests for an extension should be supported by documentary evidence.

An Assignment submitted after the deadline without an approved extension of time will be penalised. The penalty for late submission is a reduction by 5% of the maximum Assignment Mark, for each University Business Day or part day that the Assignment is late. An Assignment submitted more than ten University Business Days after the deadline will have a Mark of zero recorded for that Assignment.

· We expect a high standard of communication. Look at the worked examples in your texts, and the sample solutions in Tutorial Worked Examples to see the level you should aim at. Up to 15% of the marks may be deducted for poor language and notation.

· Please note: No Assignments for Assessment purposes will be accepted after as-signment solutions have been released.

QUESTION 1 (20 marks)

The currents I1, I2 and I3, measured in Amperes, in the three branches of a parallel circuit are determined by the following system of linear equations.

7I1 - 3I2 - I3 = -48 -I2 + 4I3 = 15 -4I1 + 8I2 = 13

(a) Write the system in matrix form Ax = b; (2 marks)

(b) Solve the system of linear equations using Cramer’s Rule. (18 marks)

Check your answer by substituting into the system of linear equations.

QUESTION 2 (20 marks)

The steady-state concentrations of chemicals in three vats C1, C2 and C3, measured in milligrams per cubic metre (mg/ m3), are determined by the following system of linear equations.

-3C1 + 4C2 + 2C3 = 24 -9C1 - 5C2 + 8C3 = 18 6C1 - 9C2 + C3 = -17

(a) Write the system of linear equations in augmented matrix form. (3 marks)

(b) Use Gaussian elimination on the augmented matrix to find the three concentrations, and

write the row operation used next to each new row. (17 marks)

Check your answer by substituting into the system of linear equations.

QUESTION 3 (12 marks)

1

Let f(x) = 2x - 1 and g(x) = sin4x.

(a) Find the composite function f(g(x)) and give its domain. (6 marks)

(b) Find the composite function g(f(x)) and give its domain. (6 marks)

QUESTION 4 (16 marks)

The following rational functions were found in analyzing the motion of a spring-damper system. Express these rational functions as partial fractions:

8s2 + 26s + 15; (8 marks)

s - 20 + 7s2

(b) s3 + 6s2 + 10s. (8 marks)

QUESTION 5 (16 marks)

On a fun park ride the position of the carriage at time t = 0 (in minutes) is given by the parametric function

x=4cos5pt+~10+2cos4pt~cospt

y=4sin5pt+ ~10 + 2cos4pt~ sinpt

where both x and y are measured from the axis of rotation and are given in metres.

(a) Show the squared distance from the axis of rotation, i.e. d2 = x2 + y2, can be rewritten as

d2 = 20 cos2 4pt + 120 cos 4pt + 116. (7 marks)

(b) Using (a) find all times, t = 0, (in exact form i.e. in terms of fractions) when the distance d is

4 m i.e. d=4ord2 = 16. (7 marks)

(c) What is the first time that the distance d of the carriage is 4 m?

Give your answer exactly (as a fraction) and approximately as a decimal. (2 marks)

QUESTION 6 (16 marks)

The speed, V (km/h), of a car at time t (seconds) is given by

where 0 = t = 100. V = f(t) = ~ pt ~

70 - 10 cos 100

~ pt ~

2 + cos 100

(a) We want to know when the car will reach a certain speed. Solve for the time t in terms of the

speed V , and hence find the formula for the inverse function f_1(t). (6 marks)

(b) Write down the domain of f_1(t). (2 marks)

(c) Plot f(t) from t = 0 to t = 100 and then plot f_1(t) on the same axes but only for the values

of t in the domain of the inverse function found in (b). (6 marks)

(d) Describe the geometric relationship that you can see when you consider both f(t) and f_1(t). (2 marks)

Note: You are encouraged to use MATLAB or Scilab for your plots.

Note that the MATLAB commands for cost and cos_1 t are cos(t) and acos(t). First define values of t from 0 to 100 for your plot of f.

t1=[0:0.01:100];

gives values from 0 to 100, in steps of 0.01. Then find the V-values for these t-values using

y1=(70-10.*cos(pi*t1/100))./(2+cos(pi*t1/100)); Then plot f using

plot(t1,y1).

Use the command

axis equal

to get equal scales on the axes.

To add the plot of f-1, define new t’s for its domain. That is with

t2=[...]

Then calculate the y-values.

/t + 1

For instance for the function 7 sin-1 + 5 we would use the command

t - 2

y2 = 7*asin((t2+1)./(t2-2)) + 5;

We can then plot both functions on the same graph using the command plot(t1,y1,t2,y2).

Note the sin-1 x and cos-1 x functions are evaluated in MATLAB using asin(x) and acos(x) respectively.

Include a plot style to distinguish the graphs, if you wish.

To add an additional plot of y = t to show the geometry, use the same process: define the t’s you want for the plot first (t3=[...]), calculate the y’s, then plot by including t3,y3 after y2 to the previous command

plot(t1,y1,t2,y2,t3,y3).

End of Assignment 2 (100 marks Total)

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