ENM1600 Engineering Mathematics, S1–2020 Assignment 3

Value: 10%. Due Date: Tuesday 26 May 2020.

• Submit your assignment electronically as one (1) PDF file via the link (Assignment 3 submission) 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 submitting.

• 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.

• 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 assignment solutions have been released.

QUESTION 1 (16 marks)

Find each of the following limits:

(a) ; (8 marks)

(b) . (8 marks)

QUESTION 2 (14 marks)

A rocket is travelling in a straight line for a short time. The distance in metres covered by the rocket during this time is described by the function

where t = 0 and time is given in seconds.

(a) Find a function that describes the speed of the rocket. (3 marks)

(b) What is the speed of the rocket at the time t = 10 seconds? (1 mark)

(c) Find all values of time t (if any) when the speed of the rocket is 50ms-1. (3 marks) (d) Find a function that describes the acceleration of the rocket. (3 marks) (e) Find the acceleration of the rocket at t = 18 seconds. (1 mark)

(f) Find all values of time t (if any) when the rocket’s acceleration is zero i.e. 0ms-2. (3 marks)

QUESTION 3 (14 marks)

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

function

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

(a) Find an expression for in terms of t. (11 marks)

(b) Using part (a) evaluate the derivative when t = 0. (3 marks) QUESTION 4 (20 marks)

Two ropes are attached to the ceiling at points 20 cm apart. The rope on the left is 7 cm long and has a pulley at its end. The rope on the right, of length is 42 cm (shown in blue), passes through the pulley and has a weight attached to its end as shown below (not drawn to scale). At rest the ropes and pulley arrange themselves such that the vertical distance from the ceiling to the weight is maximized.

(a) Express the vertical distance from the ceiling to the weight as a function of x. (4 marks)

(b) Using Calculus find the value of x (in metres) that gives the maximum vertical distance from the ceiling to the weight.

What is the maximum vertical distance?

Check your value of x by substitution into the derivative. (10 marks)

(c) Confirm that you have found the maximum amount using an appropriate test. (6 marks)

QUESTION 5 (16 marks)

The speed of a car at time t (in seconds) is given by a piecewise function V (t) (in m/s) as shown below. Determine the total distance, d, travelled by the car from t = 0 seconds to t = 24 seconds.

Note: the exponential function, ex, can be written as exp(x).

QUESTION 6 (20 marks)

To help find the velocity of particles requires the evaluation of the indefinite integral of the acceleration function, a(t), i.e. Z v = a(t)dt.

Evaluate the following indefinite integrals.

Check your value for each integral by differentiating your answer.

(a) ; (8 marks)

(b) . (12 marks)

End of Assignment 3 (100 marks Total)

Value: 10%. Due Date: Tuesday 26 May 2020.

• Submit your assignment electronically as one (1) PDF file via the link (Assignment 3 submission) 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 submitting.

• 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.

• 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 assignment solutions have been released.

QUESTION 1 (16 marks)

Find each of the following limits:

(a) ; (8 marks)

(b) . (8 marks)

QUESTION 2 (14 marks)

A rocket is travelling in a straight line for a short time. The distance in metres covered by the rocket during this time is described by the function

where t = 0 and time is given in seconds.

(a) Find a function that describes the speed of the rocket. (3 marks)

(b) What is the speed of the rocket at the time t = 10 seconds? (1 mark)

(c) Find all values of time t (if any) when the speed of the rocket is 50ms-1. (3 marks) (d) Find a function that describes the acceleration of the rocket. (3 marks) (e) Find the acceleration of the rocket at t = 18 seconds. (1 mark)

(f) Find all values of time t (if any) when the rocket’s acceleration is zero i.e. 0ms-2. (3 marks)

QUESTION 3 (14 marks)

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

function

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

(a) Find an expression for in terms of t. (11 marks)

(b) Using part (a) evaluate the derivative when t = 0. (3 marks) QUESTION 4 (20 marks)

Two ropes are attached to the ceiling at points 20 cm apart. The rope on the left is 7 cm long and has a pulley at its end. The rope on the right, of length is 42 cm (shown in blue), passes through the pulley and has a weight attached to its end as shown below (not drawn to scale). At rest the ropes and pulley arrange themselves such that the vertical distance from the ceiling to the weight is maximized.

(a) Express the vertical distance from the ceiling to the weight as a function of x. (4 marks)

(b) Using Calculus find the value of x (in metres) that gives the maximum vertical distance from the ceiling to the weight.

What is the maximum vertical distance?

Check your value of x by substitution into the derivative. (10 marks)

(c) Confirm that you have found the maximum amount using an appropriate test. (6 marks)

QUESTION 5 (16 marks)

The speed of a car at time t (in seconds) is given by a piecewise function V (t) (in m/s) as shown below. Determine the total distance, d, travelled by the car from t = 0 seconds to t = 24 seconds.

Note: the exponential function, ex, can be written as exp(x).

QUESTION 6 (20 marks)

To help find the velocity of particles requires the evaluation of the indefinite integral of the acceleration function, a(t), i.e. Z v = a(t)dt.

Evaluate the following indefinite integrals.

Check your value for each integral by differentiating your answer.

(a) ; (8 marks)

(b) . (12 marks)

End of Assignment 3 (100 marks Total)

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