
Unit 1: 
Analytical Methods for Engineers 


Unit code:

A/601/1401



QCF level:

4



Credit value:

15






• Aim
This unit will provide the analytical knowledge and techniques needed to
carry out a range of engineering tasks and will provide a base for further
study of engineering mathematics.
• Unit abstract
This unit enables learners to develop previous mathematical knowledge
obtained at school or college and use fundamental algebra, trigonometry,
calculus, statistics and probability for the analysis, modelling and solution
of realistic engineering problems.
Learning outcome 1 looks at algebraic methods, including polynomial
division, exponential, trigonometric and hyperbolic functions, arithmetic and
geometric progressions in an engineering context and expressing variables as
power series.
The second learning outcome will develop learners’ understanding of
sinusoidal functions in an engineering concept such as AC waveforms, together
with the use of trigonometric identities.
The calculus is introduced in learning outcome 3, both differentiation
and integration with rules and various applications.
Finally, learning outcome 4 should extend learners’ knowledge of
statistics and probability by looking at tabular and graphical representation
of data; measures of mean, median, mode and standard deviation; the use of
linear regression in engineering situations, probability and the Normal
distribution.
• Learning outcomes
On successful completion of this unit a
learner will:
1
Be able to
analyse and model engineering situations and solve problems using algebraic
methods
2
Be able to
analyse and model engineering situations and solve problems using trigonometric
methods
3 Be able to analyse and model engineering
situations and solve problems using calculus
4
Be able to
analyse and model engineering situations and solve problems using statistics
and probability.
Unit content
1
Be able
to analyse and model engineering situations and solve problems using algebraic
methods
Algebraic methods: polynomial division; quotients and remainders; use of factor and remainder
theorem; rules of order for partial fractions (including linear, repeated and
quadratic factors); reduction of algebraic fractions to partial fractions
Exponential, trigonometric and hyperbolic functions:
the nature of algebraic functions; relationship between exponential and
logarithmic functions; reduction of exponential laws to linear form; solution
of equations involving exponential and logarithmic expressions; relationship
between trigonometric and hyperbolic identities; solution of equations
involving hyperbolic functions
Arithmetic
and geometric: notation
for sequences; arithmetic and geometric progressions; the limit of a
sequence; sigma notation; the sum of a series; arithmetic and geometric series;
Pascal’s triangle and the binomial theorem
Power series: expressing variables as power series functions and use series to find
approximate values eg exponential series, Maclaurin’s series, binomial
series
2
Be able
to analyse and model engineering situations and solve problems using
trigonometric methods
Sinusoidal functions: review of the trigonometric ratios; Cartesian and polar
coordinate systems; properties of the circle; radian measure;
sinusoidal functions
Applications: angular velocity, angular acceleration,
centripetal force, frequency, amplitude, phase, the production of
complex waveforms using sinusoidal graphical synthesis, AC waveforms and phase
shift
Trigonometric
identities: relationship between
trigonometric and hyperbolic identities; double angle and compound angle
formulae and the conversion of products to sums and differences; use of
trigonometric identities to solve trigonometric equations and simplify
trigonometric expressions
3
Be able
to analyse and model engineering situations and solve problems using calculus
Calculus: the concept of the limit and continuity;
definition of the derivative; derivatives of standard functions; notion
of the derivative and rates of change; differentiation of functions using the
product, quotient and function of a function rules; integral calculus as the
calculation of area and the inverse of differentiation; the indefinite integral
and the constant of integration; standard integrals and the application of
algebraic and trigonometric functions for their solution; the definite integral
and area under curves
Further differentiation: second order and higher derivatives; logarithmic
differentiation; differentiation of inverse trigonometric functions;
differential coefficients of inverse hyperbolic functions
Further
integration:
integration by parts; integration by substitution; integration using partial
fractions
Applications
of the calculus: eg maxima
and minima, points of inflexion, rates of change of temperature,
distance and time, electrical capacitance, rms values, electrical circuit
analysis, AC theory, electromagnetic fields, velocity and acceleration problems,
complex stress and strain, engineering structures, simple harmonic motion,
centroids, volumes of solids of revolution, second moments of area, moments of
inertia, rules of Pappus, radius of gyration, thermodynamic work and heat
energy
Engineering problems: eg stress and strain, torsion, motion,
dynamic systems, oscillating systems, force systems, heat energy and
thermodynamic systems, fluid flow, AC theory, electrical signals, information
systems, transmission systems, electrical machines, electronics
4
Be able
to analyse and model engineering situations and solve problems using statistics
and probability
Tabular and graphical form: data collection methods; histograms; bar
charts; line diagrams; cumulative frequency diagrams; scatter plots
Central tendency and dispersion: the concept of central tendency and
variance measurement; mean; median; mode; standard deviation; variance
and interquartile range; application to engineering production
Regression, linear correlation: determine linear correlation coefficients
and regression lines and apply linear regression and product moment
correlation to a variety of engineering situations
Probability: interpretation of probability;
probabilistic models; empirical variability; events and sets; mutually
exclusive events; independent events; conditional probability; sample space and
probability; addition law; product law; Bayes’ theorem
Probability distributions: discrete and continuous distributions,
introduction to the binomial, Poisson and normal distributions; use of
the normal distribution to estimate confidence intervals and use of these
confidence intervals to estimate the reliability and quality of appropriate
engineering components and systems
Learning outcomes and assessment criteria


Learning outcomes

Assessment criteria for pass






On successful completion of

The learner can:






this unit a learner will:
















LO1 Be able to analyse and model


1.1

determine the quotient and remainder for
algebraic





engineering
situations and



fractions and reduce algebraic fractions to
partial





solve
problems using



fractions





algebraic
methods


1.2

solve engineering problems that involve the
use and















solution of exponential, trigonometric and
hyperbolic








functions and equations







1.3

solve scientific problems that involve
arithmetic and








geometric series







1.4

use power series methods to determine
estimates of








engineering variables expressed in power
series








form













LO2 Be able to analyse and model


2.1

use trigonometric functions to solve
engineering





engineering
situations and



problems





solve
problems using


2.2

use sinusoidal functions and radian measure
to solve





trigonometric
methods









engineering problems
















2.3

use trigonometric and hyperbolic identities
to solve








trigonometric equations and to simplify








trigonometric expressions













LO3 Be able to analyse and model


3.1

differentiate algebraic and trigonometric
functions





engineering
situations and



using the product, quotient and function of
function





solve problems using calculus



rules







3.2

determine higher order derivatives for
algebraic,








logarithmic, inverse trigonometric and
inverse








hyperbolic functions







3.3

integrate functions using the rules, by
parts, by








substitution and partial fractions







3.4

analyse engineering situations and solve
engineering








problems using calculus













LO4 Be able to analyse and model


4.1

represent engineering data in tabular and
graphical





engineering
situations and



form





solve
problems using


4.2

determine measures of central tendency and





statistics
and probability









dispersion
















4.3

apply linear regression and product moment








correlation to a variety of engineering
situations







4.4

use the normal distribution and confidence
intervals








for estimating reliability and quality of
engineering








components and systems.



















Guidance
Links
This unit
can be linked with the core units and other principles and applications units
within the programme. It will also form the underpinning knowledge for the
study of further mathematical units such as Unit 35: Further Analytical
Methods for Engineers, Unit 59: Advanced Mathematics for Engineering.
Entry
requirements for this unit are at the discretion of the centre. However, it is
strongly advised that learners should have completed the BTEC National unit Mathematics
for Engineering Technicians or equivalent. Learners who have not
attained this standard will require appropriate bridging studies.
Essential requirements
There are no essential resources for this unit.
Employer engagement and vocational contexts
The delivery
of this unit will benefit from centres establishing strong links with employers
willing to contribute to the delivery of teaching, workbased placements and/or
detailed case study materials.
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