Programme

International year One

Module

Engineering Mathematics - Calculus and MATLAB

Subject Leader

Dr. Ravjeet Kour, ONCAMPUS Coventry rkour@oncampus.global ONCAMPUS Coventry

Number of classroom hours per week

5

Number of advised sessions per week

5

Upon successful completion of this pathway, students will:

LO1 Use and Apply the Geometric applications to engineering problems

LO2 Use the differentiation and higher derivatives

LO3 Apply the kinematics using differentiation.

LO4 Apply the integration by parts and substitutions.

LO5 Understand partial fractions.

LO6 Understand trapezium rule and Simpsons Rule

LO7 Use engineering software like Excel to engineering problems.

Upon successful completion of this pathway, students will have demonstrated that they can:

Subject assessment outcomes (knowledge and skills):

1. The assessments have been set in contexts appropriate to, and reflect the demand of, the course content. These assessments allow students to use their knowledge and understanding of course in planning, carrying out, analysing and evaluating their work. The details of these assessments are as below. This is continuous assessment throughout the whole term and is based on the learning outcome from each topic in the SOW.

2. The methods of teaching, learning and assessment are constructed so that the learning activities and assessment tasks are aligned with the learning outcomes that are intended in the programme. The assessment will cover a series of mathematical and research based questions in the form of unseen examination. In addition to this, there will be some skill-based coursework.

This module aims to further develop the students' competence in mathematical methods relevant to engineering whilst at the same time emphasising fundamental concepts which the engineer needs to understand in order to produce a mathematical formulation of a problem. Students will learn to analyse problems using appropriate mathematical techniques carrying out the manipulation themselves and also using modern mathematical software. This module encourages students to develop skills for learning and skills for work.  It supports the development of analytical and critical thinking skills while encouraging students to develop their problem solving skills.  This module also allows the students to apply Engineering real life context to their work.  This module also introduces MATLAB software which is widely used in Academia & Industry. Students will learn to analyse problems using appropriate mathematical techniques carrying out the manipulation themselves and also using modern mathematical software. Some areas of application includes: Signal, Image & Video processing.

Term

Assessment Number

Assessment Type

Assessment Name

Overall module weighting

Pass Mark

1

1

Exam

Exam

75%

65%

2

Coursework

Report

25%

65%

Term 1

Unit resources & reading list: Engineering mathematics through applications by K. Singh.

1

Complex Numbers  1

· In the first part of this module, the main focus is introducing a number of mathematical techniques and concepts, algebraic in nature, that are very useful in engineering applications, and show how problems can be analysed using the mathematical techniques and, in addition, how mathematical software can be used to help in finding solutions to the problems.

Power point presentations,  worksheets

Lecture 1

1. Complex numbers and complex functions are used in many engineering fields such as alternating current (a.c.) theory, vibration problems, fluid flow, elasticity, etc.

2. Rules of complex numbers

Lecture 2

· Using the previous lecture finding the solutions to the following equations

·

· A complex number is of the form a + jb, where a is called the real part and b is called the imaginary part.

3^rd session – Revision, Practice Problems related to Engineering application

Weekly Tests, Online Quizzes on Moodle

Exercises 10(a): 1-12. & 10(b): 1-3, 10(c): 1-3,5,8,9

Book: Engineering mathematics through applications by K. Singh: • 1st Edition: Chap. 10, pp. 488-501 & pp. 508-516 • 2nd Edition: Chap. 10, pp. 538-553 & pp. 561-581

2

Complex Numbers  2

· Powers and roots of complex numbers

· De Moivre’s theorem

· The exponential form of complex numbers

Power point presentations,  worksheets

Lecture 1:

· plot on an Argand diagram.

· De Moivre’s theorem can be applied to the problem of finding roots of complex numbers

· Roots of a complex number

Lecture 2:

· Exponential form of the complex number

3^rd session – Revision, Practice Problems related to Engineering application

Weekly Tests, Online Quizzes

• Exercises 10(d):1,2; 10(e):1-6, Misc 10:1-4,8,16,18, 11(a):1,5; 11(b):1-8

From book: • 1st Edition: Chap. 11, pp 534-548 & pp. 586 - 597 • 2nd Edition: Chap. 11, pp. 522-534 & 598-614

3

Calculus 1

· Gradient of a curve

· Differentiation

· Standard results for differentiation

· The chain rule

· Product rule for differentiation

· Quotient rule Higher derivatives

Power Point Presentations, worksheets

1^st Lecture

1. The gradient of a straight line y = mx + c, where m and c are constants, is defined to be the value of m

2. From the graph of the straight line, the gradient can be calculated by considering an interval of x (the change in x) and the resulting interval of y (the change in y).

3. The gradient of the straight line is determined by calculating the change in y divided by the change in x:.

2^nd Lecture

· To add (or The gradient of a curve, at some point on the curve, is defined to be the gradient of the tangent to the curve at that point

· The gradient of a curve, at some point on the curve, is defined to be the gradient of the tangent to the curve at that point. How do we calculate the gradient of a curve at some point on the curve?

· Consider a general function y = f ( x ) and its graph, i.e. a plot of y (the dependent variable ) against x (the independent variable).

· General formula for derivatives

· The chain rule (sometimes known as the ‘function of a function rule’) states that if y is a function of u, and u is a function of x, then

· More standard differentiation results

· The product rule

· The quotient rule

· Higher derivatives

· 3^rd session – Revision, Practice Problems related to Engineering application

Weekly Tests, Online Quizzes

Exercise 6(d): 1, 3a-g, 4 ; Ex 6(e): 1, 2, 8, 9 • Exercise 6(f): 1-3, 5 ; 6(g): 1, 2; Misc Ex 6: 16

Book:

: Engineering mathematics through applications by K. Singh:

4

Calculus 2

· Parametric differentiation;

· Implicit differentiation

· Logarithmic differentiation

· Stationary points;

· Local maxima and minima

· Points of inflexion

· Curve sketching

Power Point Presentations, worksheets

Ist Lecture:

· Parametric Equations: Sometimes it is more convenient to express a relationship between x and y using two equations called parametric equations and containing a third variable called a parameter

· Suppose the equation of a curve is given parametrically. If the gradient of the curve is required, this can be obtained using parametric differentiation

· Higher derivatives

· Higher derivatives can be found using parametric differentiation.

2^nd Lecture:

· If y is given directly in terms of x, it is an explicit function

· If y is given indirectly in terms of x, it is an implicit function

· If y consists of a product/division/powers of a number of terms involving x, the differentiation process can be made easier by taking the natural logarithm of y, before differentiating.

· A stationary point is a point on the curve where the gradient is zero.

· The point A is known as a local maximum point and B is known as a local minimum point. The point C is known as a point of inflexion.

· The function f has a local maximum at x = a if f(a) is the largest value of f(x) in the neighbourhood of x = a. The function f has a local minimum at x = a if f(a) is the smallest value of f(x) in the neighbourhood of x = a.

· If x = a gives a local minimum or a local maximum of f(x), then a is said to be a turning point of f(x).

· A stationary point that is not a turning point is a point of inflexion

3^rd session – Revision, Practice Problems related to Engineering application

Weekly Tests, Online Quizzes

Tutorial Exercises: • Exercise 7(a): 1,4-5,10; Ex 7(c): 1-2

Engineering mathematics through applications by K. Singh: • 1st Edition: Chap. 7, pp. 320-324, 330-332, 360 - 373 • 2nd Edition: Chap. 7, pp. 339-343, 349-354, 400 - 401, 407 - 420

5

Revision

6

Calculus 3

Kinematics

Optimization

Integration as the inverse of differentiation

Integration by Substitution

The Definite Integral The area under a graph

Power Point Presentations, worksheets

· For a particle in straight line motion, the displacement, s, velocity, v, and acceleration, a, are related as follows:

·

·

· Newtons dot notation

·

· or

· Optimization refers to determining maxima and/or minima of quantities in particular (engineering) problems

· Integration is the reverse process of differentiation

· Using known results of differentiation, some associated results for integration can be derived

· Now introduce integration by substitution.

· Definite integral and area under the graph : ’Definite Integration’ is when we integrate between two ’limits’, say a and b.

· Find the area bounded by two curves, y = f ( x ) and y = g ( x ) say, which intersect at two points

· To find the area bounded by two curves, y = f ( x ) and y = g ( x ) say, which intersect at two points,solve f ( x) = g ( x ) to find the x coordinates of the points of intersection, say x = a and x = b .

Tutorial Exercises: • Exercises 8(c): 4,5,7,8,12; Ex 9(c): 5,6,12

Engineering mathematics through applications by K. Singh: • 1st Edition: Chap. 8, pp. 388-409; Chap 9, pp 452-455 • 2nd Edition: Chap. 8, pp. 432-454; Chap 9, pp 499-502

7

Calculus 4 and Excel

Integration in Mechanics Integration by Parts

Partial fractions

The ‘cover-up’ rule

Improper fractions and partial fractions

Integration utilising partial fractions

The Trapezium Rule

Simpsons Rule

Applying software tools like Excel to solve engineering problems

Power Point Presentations, worksheets

Examples using Excel will be demonstrated here

· Sometimes you may have a function for which there isn’t a standard rule for finding the integral

· Consequently the exact value for a specified area under the graph cannot be found

· Instead you can find an approximate value for the area using Numerical Integration

· We are going to look at 2 methods:  The Trapezium Rule,

· Simpsons Rule

Tutorial exercises will be given to students based on Matlab and Excel.

Weekly Tests, Online Quizzes

Coursework needs to be done by all students

Tutorial Exercises: • Ex 8(f): 1-2; Ex 8(g): 1-4. • Ex 9(a): 2-4 ; Ex 9(b): 1, 3, 5

Engineering mathematics through applications by K. Singh: • 1st Edition: Chap. 9, pp. 426-440 & 568-574 • 2nd Edition: Chap. 9, pp. 473-487

Tutorial Exercises: • Exercises 12(a): 1-5, 7-9; Ex 12(b): 1-3; Ex 12(c): 1-2, 4-5; Ex 12(d): 1-6.

Book: Engineering mathematics through applications by K. Singh: • 1st Edition: Chap. 12, pp. 576-598 • 2nd Edition: Chap. 12, pp. 637-668

Revision for Coursework