# Mechanical Engineering BEng/MEng

## Engineering Mathematics 1 ENG1063

**Academic Session:**2023-24**School:**School of Engineering**Credits:**40**Level:**Level 1 (SCQF level 7)**Typically Offered:**Runs Throughout Semesters 1 and 2**Available to Visiting Students:**No

### Short Description

This course provides the fundamental mathematics needed throughout all engineering disciplines. Topics covered include: numbers, algebra and geometry; functions; complex numbers; vector algebra; matrix algebra; sequences, series and limits; differential calculus and applications; integral calculus and applications; data handling and probability theory.

### Timetable

4 lectures per week

1 group tutorial per week

### Excluded Courses

None

### Co-requisites

None

### Assessment

76% Written Exam: 38% Paper 1 (Blocks 1-3) December Diet, 38% Paper 2 (Blocks 4-6) April/May Diet

24% Set Exercise: Six online End-of-Block tests

**Main Assessment In:** December and April/May

### Course Aims

The aim of this course is to ensure that students are competent in the fundamental mathematics required for engineering degree programmes.

### Intended Learning Outcomes of Course

By the end of this course students will be able to:

BLOCK 1 - Numbers, Algebra and Geometry (Chapter 1) and Functions (Chapter 2)

■ use reliably the basic rules of arithmetic and algebra;

■ solve quadratic equations and apply the results in engineering applications;

■ apply notation for sums, products, combinations and permutations;

■ explain the effect of rounding in arithmetic calculations and quote results to an appropriate number of significant figures;

■ estimate errors in sums, differences, products and quotients;

■ explain concept of a function, inverse function and zeros;

■ form composite functions, classify functions as odd, even and periodic;

■ decompose rational functions into partial fractions;

■ sketch and be able to explain fundamental properties of common circular, exponential, logarithmic and hyperbolic functions, giving special values;

■ use theorems to simplify trigonometric expressions.

BLOCK 2 - Complex Numbers (Chapter 3) and Vector Algebra (Chapter 4)

■ show the relationship between complex numbers and vectors;

■ plot complex numbers on an Argand diagram;

■ perform basic arithmetic operations on complex numbers and demonstrate the result on an Argand diagram;

■ find modulus, phase and real and imaginary parts; convert between rectangular and polar form;

■ use Euler's formula and de Moivre's theorem for simplifying trigonometric expressions and powers of complex numbers;

■ apply complex numbers to solving engineering problems, e.g. simple a.c. circuits;

■ distinguish between scalars and vectors;

■ perform addition and subtraction of vectors, showing triangle law and effect on components;

■ evaluate scalar and vector products, giving geometrical interpretations;

■ apply vectors to solving engineering problems.

BLOCK 3 - Matrix Algebra (Chapter 5) and Sequences, Series and Limits (Chapter 7)

■ reduce engineering problems to equations involving matrices;

■ add, subtract and multiply matrices;

■ evaluate determinant of a 3 x 3 matrix and other determinants, exploiting properties of determinant;

■ find inverse of a 3 x 3 matrix and other matrices, and state conditions under which an inverse exists;

■ solve simultaneous equations;

■ determine the eigenvalues and eigenvectors of a matrix;

■ apply matrix algebra to solving engineering problems;

■ state definition of sequence and series;

■ evaluate arithmetic, geometric and other simple series;

■ evaluate limit of a sequence and series; apply comparison, and ratio tests for convergence;

■ recognise power series of common functions;

■ apply concept of a limit of a function; distinguish between continuous and discontinuous functions.

BLOCK 4 - Differential calculus and applications (Sections 8.1 to 8.6, and 9.4)

■ understand differentiation as a rate of change, and as the slope of a tangent to a curve;

■ differentiate functions of a single variable from first principles;

■ find the derivative of powers, polynomial, exponential, logarithmic and trigonometric functions;

■ apply the rules of differentiation: chain rule, product and quotient rules, and implicit differentiation;

■ find higher-order derivatives;

■ find stationary and turning points, and solve problems on extrema;

■ apply Taylor's theorem;

■ apply L'HÃ´pital's rule to determine a limiting value of a function;

■ apply the Newton-Raphson method to determine numerically the zero of a function;

■ sketch the graphs of rational and other simple functions, with an appreciation of asymptotic behaviour;

■ apply differentiation to solving engineering problems.

BLOCK 5 - Integral calculus and applications (Sections 8.7 to 8.10, 9.2 and 9.3)

■ know the connection between definite integrals and area, and the Fundamental Theorem of Calculus;

■ understand indefinite integrals as anti-derivatives and hence solve indefinite integrals;

■ be able to write down the standard indefinite integrals of powers, polynomial, exponential and trigonometric functions;

■ apply the standard techniques of integration: variable substitution, integration by parts, partial fractions;

■ apply techniques for numerically evaluating definite integrals;

■ apply integration to solving engineering problems.

BLOCK 6 - Data Handling and Probability Theory (Chapter 13)

■ apply the concept of sampling, and graphically represent data samples;

■ interpret probabilities of random events;

■ extract location and dispersion values from data sets;

■ identify and use the properties of important practical distributions: the Binomial Distribution, the Poisson Distribution, and the Normal Distribution;

■ apply data handling and probability theory to engineering problems.

### Minimum Requirement for Award of Credits

Students must attend the degree examinations and submit at least 75% by weight of the other components of the course's summative assessment.

Students must complete all 5 100% tests.