UNIT 7: CALCULUS TO SOLVE ENGINEERING PROBLEMS
Unit 7: Calculus to Solve Engineering
Problems
Delivery guidance
This unit builds on mathematical techniques covered in the mandatory unit Engineering
Principles, and hence it is suggested that it is delivered after or at the same time as Unit 1:
Mechanical Principles and/or Unit 57: Electrical and Electronic Principles. Learners will
investigate the rules and manipulation techniques of calculus and apply them to the
solution of engineering problems, making them aware of the importance of understanding
advanced mathematical techniques. You should encourage learners to develop their pure
mathematical skills by using them in an applied environment.
You should enable learners to appreciate why correctly modelling of an engineering system
using differential and/or integral calculus techniques is important: working with numbers is
much more cost effective than building hardware that does not perform to specification.
Your learners’ algebraic manipulation, numerical accuracy and presentation skills will develop
so that assignments can be presented to an agreed standard.
A large part of this unit involves the teaching and learning of mathematical techniques that
are transferable and not just specific to engineering. For the first two learning aims, the unit
content is structured so that your learners will initially investigate generic calculus
techniques and then develop them to produce the solutions to engineering problems. The
third learning aim focuses on the solution of a complex, defined, specialist engineering
problem by applying thinking skills (eg reductionism) and modelling techniques (analytical
and numerical).
To complete this unit, your learners will need access to a spreadsheet package. You can
use a range of delivery methods in this unit, such as:
● formal teaching
● individual and small-group investigation
● structured worksheets
● case studies.
Learners will benefit from access to web-based mathematics support.
This unit, like the other mandatory units, could be delivered in a specialist context such as aeronautical,
manufacturing or electrical and electronic engineering. For example, a centre wanting to deliver the
mandatory units in an electrical/electronic context could explore a range of problems linked to electrical
applications. However, care must be taken to ensure learners are prepared before they attempt the internal
assessment activities.
You can involve local employers in the delivery of this unit if there are local opportunities to do so.
Pearson BTEC International Level 3 Qualifications in Engineering – Delivery Guide Issue
1
1 – October 2019 © Pearson Education Limited 2019
, UNIT 7: CALCULUS TO SOLVE ENGINEERING PROBLEMS
Approaching the unit
You should make your learners aware that there are two strands to the teaching and
assessment of this unit. The first strand is to teach learners how to apply the principle of
calculus to a range of polynomial, trigonometric, logarithmic and exponential functions. To
do this will require formal classroom delivery supported by significant amounts of learner
practice in the form of graded worksheets. You should work with the universally accepted
variables ‘x’ and ‘y’ but, as the unit develops, replace them with those used by engineers, for
example ‘s’ (displacement) and ‘t’ (time). This first strand is, in effect, pure mathematics with
an engineering ‘flavour’.
The second strand is the use of calculus to solve real engineering problems and much of
your teaching will be in form of supporting learners as they carry out self-directed study
done on an individual or small-group basis. You should guide learners to use calculus
techniques when studying other units within their programme, for example, to determine
the maximum bending moment in a beam or the energy discharge from an electronic
capacitor. Throughout your teaching of this unit, do reinforce the notion that mathematics
is a modelling process that is much more interesting and valuable than simply a set of
routines to be learnt by rote. Guide your learners to correctly present what they set out on
paper, be it hand-written or computer-generated; mathematics is a communicative
language.
Delivering the learning aims
For learning aim A, introduce the topic by showing your learners a displacement– time graph
for an object that is accelerating in a straight line with constant acceleration. Discuss how you
might determine the object’s instantaneous velocity and acceleration at a given time point
on the graph. Remind them of the concept of gradient, which they will have already
investigated when they studied Unit 1: Mechanical Principles. Through discussion, develop
the concept of rate of change and think about other engineering situations where there are
time dependent processes happening, for example heat transfer, flow of electrical charge,
mechanical energy and power.
Explain to your learners that, in the initial stages of delivering this learning aim, they will learn about
differential calculus manipulation techniques, the starting point being different types of mathematical
function (building on what they learnt in Unit 1: Mechanical Principles). Follow this up with a graphical
explanation of gradient, small change and limiting value (derivative) of a simple power function such
as y = x2. Once your learners understand the concepts of variables and rates of change, you can go on
to work through the various rules of differential calculus as set out in the unit content. This will have to
be done in a formal way – with initial classroom delivery followed by worksheets. Until learners are
confident in applying the various rules of differentiation, for example the product and quotient rules,
your learners will not be able to move on.
To make the learning aim more accessible, it is important to incorporate spreadsheet
mathematics, particularly when investigating the turning points and second derivative. You
will need to explain to learners how to find the maximum and minimum points of a function
(eg finding the maximum and minimum bending moments for a beam) by using differential
calculus (analytical method) and plotting (graphical, spreadsheet method) and then use the
second derivative to confirm that the point is a maximum or minimum.
Learners can find the concept of variables tending to zero (limiting value/Leibniz notation)
very difficult to accept; for a lot of the content in learning aim A (and also B) they can just
mechanically apply the rules of differentiation and work with the table of standard
differential coefficients. They need to be convinced of what (dy/dx or ds/dt) means – this is
easily done by setting out functions in spreadsheets, plotting, and following up with ‘what if’
repetitive calculation. You should use animations sourced from on-line mathematics support
packages to reinforce your teaching of
Pearson BTEC International Level 3 Qualifications in Engineering – Delivery Guide Issue
2
1 – October 2019 © Pearson Education Limited 2019
Unit 7: Calculus to Solve Engineering
Problems
Delivery guidance
This unit builds on mathematical techniques covered in the mandatory unit Engineering
Principles, and hence it is suggested that it is delivered after or at the same time as Unit 1:
Mechanical Principles and/or Unit 57: Electrical and Electronic Principles. Learners will
investigate the rules and manipulation techniques of calculus and apply them to the
solution of engineering problems, making them aware of the importance of understanding
advanced mathematical techniques. You should encourage learners to develop their pure
mathematical skills by using them in an applied environment.
You should enable learners to appreciate why correctly modelling of an engineering system
using differential and/or integral calculus techniques is important: working with numbers is
much more cost effective than building hardware that does not perform to specification.
Your learners’ algebraic manipulation, numerical accuracy and presentation skills will develop
so that assignments can be presented to an agreed standard.
A large part of this unit involves the teaching and learning of mathematical techniques that
are transferable and not just specific to engineering. For the first two learning aims, the unit
content is structured so that your learners will initially investigate generic calculus
techniques and then develop them to produce the solutions to engineering problems. The
third learning aim focuses on the solution of a complex, defined, specialist engineering
problem by applying thinking skills (eg reductionism) and modelling techniques (analytical
and numerical).
To complete this unit, your learners will need access to a spreadsheet package. You can
use a range of delivery methods in this unit, such as:
● formal teaching
● individual and small-group investigation
● structured worksheets
● case studies.
Learners will benefit from access to web-based mathematics support.
This unit, like the other mandatory units, could be delivered in a specialist context such as aeronautical,
manufacturing or electrical and electronic engineering. For example, a centre wanting to deliver the
mandatory units in an electrical/electronic context could explore a range of problems linked to electrical
applications. However, care must be taken to ensure learners are prepared before they attempt the internal
assessment activities.
You can involve local employers in the delivery of this unit if there are local opportunities to do so.
Pearson BTEC International Level 3 Qualifications in Engineering – Delivery Guide Issue
1
1 – October 2019 © Pearson Education Limited 2019
, UNIT 7: CALCULUS TO SOLVE ENGINEERING PROBLEMS
Approaching the unit
You should make your learners aware that there are two strands to the teaching and
assessment of this unit. The first strand is to teach learners how to apply the principle of
calculus to a range of polynomial, trigonometric, logarithmic and exponential functions. To
do this will require formal classroom delivery supported by significant amounts of learner
practice in the form of graded worksheets. You should work with the universally accepted
variables ‘x’ and ‘y’ but, as the unit develops, replace them with those used by engineers, for
example ‘s’ (displacement) and ‘t’ (time). This first strand is, in effect, pure mathematics with
an engineering ‘flavour’.
The second strand is the use of calculus to solve real engineering problems and much of
your teaching will be in form of supporting learners as they carry out self-directed study
done on an individual or small-group basis. You should guide learners to use calculus
techniques when studying other units within their programme, for example, to determine
the maximum bending moment in a beam or the energy discharge from an electronic
capacitor. Throughout your teaching of this unit, do reinforce the notion that mathematics
is a modelling process that is much more interesting and valuable than simply a set of
routines to be learnt by rote. Guide your learners to correctly present what they set out on
paper, be it hand-written or computer-generated; mathematics is a communicative
language.
Delivering the learning aims
For learning aim A, introduce the topic by showing your learners a displacement– time graph
for an object that is accelerating in a straight line with constant acceleration. Discuss how you
might determine the object’s instantaneous velocity and acceleration at a given time point
on the graph. Remind them of the concept of gradient, which they will have already
investigated when they studied Unit 1: Mechanical Principles. Through discussion, develop
the concept of rate of change and think about other engineering situations where there are
time dependent processes happening, for example heat transfer, flow of electrical charge,
mechanical energy and power.
Explain to your learners that, in the initial stages of delivering this learning aim, they will learn about
differential calculus manipulation techniques, the starting point being different types of mathematical
function (building on what they learnt in Unit 1: Mechanical Principles). Follow this up with a graphical
explanation of gradient, small change and limiting value (derivative) of a simple power function such
as y = x2. Once your learners understand the concepts of variables and rates of change, you can go on
to work through the various rules of differential calculus as set out in the unit content. This will have to
be done in a formal way – with initial classroom delivery followed by worksheets. Until learners are
confident in applying the various rules of differentiation, for example the product and quotient rules,
your learners will not be able to move on.
To make the learning aim more accessible, it is important to incorporate spreadsheet
mathematics, particularly when investigating the turning points and second derivative. You
will need to explain to learners how to find the maximum and minimum points of a function
(eg finding the maximum and minimum bending moments for a beam) by using differential
calculus (analytical method) and plotting (graphical, spreadsheet method) and then use the
second derivative to confirm that the point is a maximum or minimum.
Learners can find the concept of variables tending to zero (limiting value/Leibniz notation)
very difficult to accept; for a lot of the content in learning aim A (and also B) they can just
mechanically apply the rules of differentiation and work with the table of standard
differential coefficients. They need to be convinced of what (dy/dx or ds/dt) means – this is
easily done by setting out functions in spreadsheets, plotting, and following up with ‘what if’
repetitive calculation. You should use animations sourced from on-line mathematics support
packages to reinforce your teaching of
Pearson BTEC International Level 3 Qualifications in Engineering – Delivery Guide Issue
2
1 – October 2019 © Pearson Education Limited 2019
