### Module Details

 The information contained in this module specification was correct at the time of publication but may be subject to change, either during the session because of unforeseen circumstances, or following review of the module at the end of the session. Queries about the module should be directed to the member of staff with responsibility for the module.
 Title ENGINEERING MATHEMATICS Code MATH198 Coordinator Dr TJ Alanne Mathematical Sciences Tommi.Alanne@liverpool.ac.uk Year CATS Level Semester CATS Value Session 2021-22 Level 4 FHEQ Whole Session 22.5

### Aims

To provide a basic level of mathematicsincluding calculus and extend the student's knowledge to include an elementaryintroduction to complex variables and functions of two variables.

### Learning Outcomes

(LO1) • differentiate using the chain, product and quotient rules;

(LO2)  sketch the graphs of elementary and rational functions;

(LO3) integrate using list integrals, substitution and integration by parts with applications to simple geometrical problems;

(LO4)  understand the basic properties of three dimensional vectors and apply them to elementary geometrical problems;

(LO5)  understand the algebra of complex numbers in Cartesian and polar forms and their application to multiplication, division and roots.

(LO6)  solve elementary first and second order differential equations with and without initial conditions and make simple mechanical applications;

(LO7)  evaluate simple Laplace transforms and their inverses using tables with application to initial value problems;

(LO8)  understand the graphical representation of functions of two variables;

(LO9) find partial derivatives and use to locate and classify the stationary points of a function of two variables

### Syllabus

Topic 1 Differential Calculus and Applications Preliminary revision (exponential number and function e x ; natural logarithms and function ln x; hyperbolic functions). Definition of derivative, derivative as gradient; differentiation rules. Graph sketching (finding and classifying stationary points); revision of quadratics, cubics; sketching y = (x n – a) m ; rational functions (vertical asymptotes; asymptotic behaviour at infinity, sketching functions involving  exponentials. Topic 2  Vector Algebra Revision of basic properties. Consolidation of scalar and vector products with further applications to calculation of angles between vectors and to simple areas. 3-dim. geometry of  lines  including line vectors, position vectors, parallel lines, angle of intersection of two lines, co-linearity, vector equation of the line through two given points. Topic 3 Integration and Applications Definition of the indefinite integral, relation to differenti ation. Integration by parts and by change of variable; standard integrals; other simple methods. Definite integral and applications to simple averages, areas, centroids, centres of gravity and volumes of revolution. Topic 4 Complex numbers and Differential Equations Algebra of complex numbers in Cartesian form; transition to polar and exponential form; applications to multiplication, powers and roots. Simple first order differential equations (variable separable, and linear equations) with applications to exponential growth/decay. Second order linear differential equations with constant coefficients with simple forcing functions, applications to simple harmonic motion and, e.g,. damped vibrations. Topic 5  Laplace Transform and Applications Definition and basic properties; tables of standard transforms and their use; transform of the Heaviside function. Applications of solving differential equations with constant coefficients (second-order and simultaneous first-order equatio ns). Topic 6  Functions of Two Variables Representation by contour plots (level curves) and by surfaces; partial derivatives; finding and classifying stationary points. Revision sessions

### Assessment

EXAM Duration Timing
(Semester)
% of
final
mark
Resit/resubmission
opportunity
Penalty for late
submission
Notes
Final Assessment (possibly remote) in S2 There is a resit opportunity. Standard UoL penalty applies for late submission. This is an anonymous assessment. Assessment Schedule (When) :Second se  1 hour time on task    50
CONTINUOUS Duration Timing
(Semester)
% of
final
mark
Resit/resubmission
opportunity
Penalty for late
submission
Notes
Homework 1 Standard UoL penalty applies for late submission. This is not an anonymous assessment.  equivalent to 2-5 si    10
Homework 2 Standard UoL penalty applies for late submission. This is not an anonymous assessment.  equivalent to 2-5 si    10
Homework 3 Standard UoL penalty applies for late submission. This is not an anonymous assessment.  equivalent to 2-5 si    10
Homework 4 Standard UoL penalty applies for late submission. This is not an anonymous assessment.  equivalent to 2-5 si    10
Homework 5 Standard UoL penalty applies for late submission. This is not an anonymous assessment.  equivalent to 2-5 si    10