1“Qualification Pearson BTEC Level 3 National Extended
1“Qualification Pearson BTEC Level 3 National Extended Diploma in Engineering
Pearson BTEC Level 3 National Extended Diploma in Electrical/Electronic Engineering
Pearson BTEC Level 3 National Extended Diploma in Mechanical Engineering
Unit or Component number and title
Unit 7: Calculus to solve engineering problems
Learning aim(s) (For NQF/RQF only) A: Examine how differential calculus can be used to solve engineering problems
Assignment title Solving engineering problems that involve differentiation
Assessor Helen Christison
Hand out date 6th December 2021
Hand in deadline 6th January 2022
Vocational Scenario or Context
You are working as an apprentice engineer at a company involved in the research, design production and maintenance of bespoke engineering solutions for larger customers.
Part of your apprenticeship is to spend time working in all departments, however a certain level of understanding needs to be shown before the managing director allows apprentices into the design team and so she has developed a series of questions on differentiation to determine if you are suitable.
Task 1
Produce a report that contains written descriptions, analysis and mathematics that shows how calculus can be used to solve engineering problems as set out below.
1 The equation for a distance, s(m), travelled in time t(s) by an object starting with an initial velocity u(ms-1) and uniform acceleration a(ms-2) is:
s=ut+1/2 at^2
The tasks are to:
Plot a graph of distance (s) vs time (t) for the first 10s of motion if u=10ms^(-1) and a=5ms^(-2).
Determine the gradient of the graph at t=2s and t=6s.
Differentiate the equation to find the functions for
Velocity (v=ds/dt)
Acceleration (a=dv/dt=(d^2 s)/?dt?^2 )
Use your result from part c to calculate the velocity at t=2s and t=6s.
Compare your results for part b and part d.
2 The displacement of a mass is given by the function
y=sin 3t .
The tasks are to:
Draw a graph of the displacement y(m) against time t(s) for the time t=0s to t=2s.
Identify the position of any turning points and whether they are maxima, minima or points of inflexion.
Calculate the turning points of the function using differential calculus and show which are maxima, minima or points of inflexion by using the second derivative.
Compare your results from parts b and c.
3 The equation for the instantaneous voltage across a discharging capacitor is given by v=V_O e^(-t/t), where V_O is the initial voltage and t is the time constant of the circuit.
The tasks are to:
Draw a graph of voltage against time for V_O=12V and t=2s, between t=0s and t=10s.
Calculate the gradient at t=2s and t=4s.
Differentiate v=12e^(-t/2) and calculate the value of dv/dt at t=2s and t=4s.
Compare your answers for part b and part c.
Calculate the second derivative of the instantaneous voltage ((d^2 v)/?dt?^2 ).
4 The same capacitor circuit is now charged up to 12V and the instantaneous voltage is v=12(1-e^(-t/2) ).
The tasks are to:
Differentiate v with respect to t to give an equation for dv/dt.
Calculate the value of dv/dt at t=2s and t=4s.
Find the second derivative ((d^2 v)/?dt?^2 ).
5 The gain of an amplifier is found to be G=20 log?(10V_out ),:
The tasks are to find equations for:
dG/(dV_Out )
(d^2 G)/?dV_Out?^2
6 The displacement, y(m), of a body in damped oscillation is y=2e^(-t) sin?3t.
The task is to:
Use the Product Rule to find an equation for the velocity of the object if v=dy/dt.
7 The velocity of a moving vehicle is given by the equation v=(2t+3)^4
The task is to:
Use the Chain Rule to determine an equation for the acceleration when a=dv/dt.
8 A communication signal is given by the function y=sin?t/t
The task is to:
Derive an equation for dy/dt using the Quotient Rule.
9 A company is required to fence off a square/rectangular area around a robot arm to comply with health and safety law. They have 750m of fencing available.
The task is to:
Find the maximum square/rectangular area they can fence off?
10 You plan to make a simple, open topped box from a piece of sheet metal by cutting a square – of equal size – from each corner and folding up the sides as shown in the diagram:
If l=200mm and w=150mm calculate:
The value of x which will give the maximum volume
The maximum volume of the box
Comment of the value obtained in part b.
Checklist of evidence required Your informal report should contain:
analysis
worked solutions to the problems
Each worked solution should be laid out clearly and contain brief explanations of the stages of the calculation to indicate your understanding of how calculus can be used to solve an engineering problem. Your explanation should be detailed in response to questions 9 and 10 to show how the variables are optimised in each case. Graphs should be well presented and clearly labelled and comparisons between methods should be accurate and well presented.
Criteria covered by this task:
Unit/Criteria reference To achieve the criteria you must show that you are able to:
7/A.D1 Evaluate, using technically correct language and a logical structure, the correct graphical and analytical differential calculus solutions for each type of given routine and non-routine function, explaining how the variables could be optimised in at least two functions.
7/A.M1 Find accurately the graphical and analytical differential calculus solutions and, where appropriate, turning points for each type of given routine and non-routine function and compare the results.
7/A.P1 Find the first and second derivatives for each type of given routine function.
7/A.P2 Find, graphically and analytically, at least two gradients for each type of given routine function.
7/A.P3 Find the turning points for given routine polynomial and trigonometric functions.
Sources of information to support you with this Assignment Books:
Pearson BTEC National Engineering. Author: A Buckenham, G. Thomas, N. Grifiths, S. Singleton, A. Serplus, M. Ryan. ISBN 978 1 292 14100 8
Websites:
http://www.mathsisfun.com/index.htm
http://www.mathcentre.ac.uk/students/topics
Other assessment materials attached to this Assignment Brief None
1“Qualification BTEC Level 3 National Extended 1″Qualification Pearson BTEC Level 3 National Extended 1″Qualification BTEC Level 3 National Extended Diploma in Engineering from Pearson
Pearson’s National Extended Diploma in Electrical/Electronic Engineering (BTEC Level 3)
Unit or Component number and title for Pearson BTEC Level 3 National Extended Diploma in Mechanical Engineering
Calculus to solve engineering issues (Unit 7)
Learning objectives (only for NQF/RQF) A: Investigate the use of differential calculus to engineering problems.
Solving engineering problems that require differentiation is the title of the assignment.
Assessor Helen Christison is a British actress.
The deadline for submissions is December 6, 2021.
The deadline for submission is January 6, 2022.
Context or Workplace Scenario
You’re an apprentice engineer at a firm that specializes in the development, manufacture, and maintenance of customised engineering solutions for larger clients.
Working in all departments is part of your apprenticeship, but the managing director requires a particular degree of expertise before allowing apprentices into the design team, so she has prepared a series of differentiating questions to assess if you are fit.
1st task
Produce a report that includes written explanations, analysis, and mathematics that demonstrates how calculus can be utilized to solve the following engineering challenges.
1 The equation for a distance travelled in time t(s) by an object having an initial velocity u(ms-1) and uniform acceleration a(ms-2) is s=ut+1/2 at. 2
The duties are as follows:
If u=10ms(-1) and a=5ms, plot a graph of distance (s) vs. time (t) during the first 10s of motion (-2).
Calculate the graph’s gradient for t=2s and t=6s.
To obtain the functions for the equation, differentiate it.
Acceleration (a=dv/dt=(d2 s)/?dt?2) Velocity (v=ds/dt)
Calculate the velocity at t=2s and t=6s using the answer from component c.
Compare and contrast your results from parts b and d.
2 The function y=sin 3t gives the displacement of a mass.
The duties are as follows:
For the time t=0s to t=2s, graph the displacement y(m) versus time t(s).
Determine the location of any turning points, as well as whether they are maxima, minima, or inflexion points.
Calculate the function’s turning points using differential calculus and use the second derivative to illustrate which are maxima, minima, or points of inflexion.
Parts b and c’s results should be compared.
3 v=V O e(-t/t) is the equation for the instantaneous voltage across a discharging capacitor, where V O is the starting voltage and t is the circuit’s time constant.
Draw a graph of voltage against time for V O=12V and t=2s, between t=0s and t=10s, for V O=12V and t=2s.
Calculate the gradient for time intervals of 2s and 4s.
Calculate the value of dv/dt at t=2s and t=4s by differentiating v=12e(-t/2).
Compare and contrast your responses for parts b and c.
Calculate the second derivative of the instantaneous voltage ((d2 v)/?dt?2). 4 The identical capacitor circuit has now been charged to 12V, and the instantaneous voltage is v=12(1-e(-t/2)
The duties are as follows:
To get an equation for dv/dt, differentiate v with regard to t.
At t=2s and t=4s, find the value of dv/dt.
(d2 v)/?dt?2), find the second derivative.
5 What is the gain of an amplifier, G=20 log?
(10V out),,,,,,,,,,,,,,,
The tasks are to solve the following equations: dG/(dV Out) (d2 G)/?dV Out?
2
6 In damped oscillation, a body’s displacement, y(m), is y=2e(-t) sin?3t.
If v=dy/dt, the aim is to find an equation for the object’s velocity using the Product Rule.
7 The equation v=(2t+3) gives the velocity of a moving vehicle. 4
The aim is to: Determine an equation for the acceleration when a=dv/dt using the Chain Rule.
8 The function y=sin?t/t produces a communication signal.
The goal is to use the Quotient Rule to derive an equation for dy/dt.
9 To meet with health and safety regulations, a corporation must fence off a square/rectangular space around a robot arm. They have 750 meters of fencing at their disposal.
The aim is to determine the largest square/rectangular area that they can encircle.
10 You want to create a simple, open-topped box out of sheet metal by cutting a square – of equal size – from each corner and folding up the sides like in the diagram:
Calculate: The value of x that will produce the greatest volume if l=200mm and w=150mm
The box’s maximum capacity
Part b’s value is discussed in detail.
Evidence checklist is necessary. The following items should be included in your informal report:
Problems were analyzed and solutions were devised.
Each completed solution should be well-organized and include brief explanations of the various stages of the calculation to demonstrate your understanding of how calculus may be applied to an engineering problem. In response to questions 9 and 10, provide a detailed description of how the variables are optimized in each situation. Graphs should be well-designed and labeled, and technique comparisons should be accurate and well-presented.
This task includes the following criteria:
Reference to the unit/criteria To meet the requirements, you must demonstrate that you can:
7/A.D1 Evaluate, using technically correct language and a logical structure, the correct graphical and analytical differential calculus solutions for each type of given routine and non-routine function, explaining how the variables could be optimised in at least two functions.
7/A.M1 Find accurately the graphical and analytical differential calculus solutions and, where appropriate, turning points for each type of given routine and non-routine function and compare the results.
7/A.P1 For each type of routine function, find the first and second derivatives.
7/A.P2 Find, graphically and analytically, at least two gradients for each type of given routine function.
7/A.P3 Find the turning points for given routine polynomial and trigonometric functions.
Information sources to help you with this Pearson BTEC National Engineering Assignment Books Author: A Buckenham, G. Thomas, N. Grifiths, S. Singleton, A. Serplus, M. Ryan. ISBN 978 1 292 14100 8
Websites: shttp://www.mathsisfun.com/index.htm
http://www.mathcentre.ac.uk/students/topics
Other assessment materials attached to this Assignment Brief None
