Module FOUD0301: Calculus and Further Maths with Mechanics

Department: Foundation Year (Durham)

FOUD0301: Calculus and Further Maths with Mechanics

Prerequisites

• Any Core Foundation Maths module or equivalent modules

Corequisites

• None.

Excluded Combination of Modules

• FOUD0651, FOUD0551, FOUD0638

Aims

• To extend knowledge of Cartesian coordinates in two and three dimensions to include equations of circles and lines.
• To develop a knowledge of vectors and their applications in two and three dimensions to include equations of lines.
• To introduce and develop knowledge of the six trigonometrical functions and inverses.
• To introduce and develop understanding of trigonometric formulae and their uses.
• To introduce the concept of polar coordinates.
• To intoduce and develop knowledge of matrices and their applications
• To introduce and develop knowledge of complex numbers.
• To extend understanding of a range of standard techniques for differentiation and integration
• To introduce and develop a knowledge of first and second order differential equations and their applications.
• To improve confidence in algebraic and trigonometric manipulation.
• To develop students' abilities to apply mathematics to problems based on physical situations.

Content

• Cartesian equations: lines, normals, circles.
• Vectors : column and unit vectors, addition, subtraction and multiplication by scalar. Scalar(dot) and vector(cross) product and applications. Equations of lines and planes. Resolution.
• Trigonometry: six ratios, inverse functions, use of identities, solution of equations, sine and cosine rule Radian measure. Polar coordinates.
• Complex numbers: +, -, x, /, conjugate, polar form, Argand diagrams, De Moivre's theorem.
• Series expansions
• Motion: velocity , acceleration, equations with constant acceleration, projectiles, variable acceleration/force, F=ma. • Conservation of momentum, impulse.
• Hooke's law, SHM.
• First Order Differential equations including separating variables.
• Second Order Differential equations.
• Consolidation of Integration by parts and Integration by substitution
• Impicit and Parmetric differentiation
• Matrices (nxm): addition, subtraction, multiplication, determinant, transpose, inverse. Applications to simultaneous equations

Learning Outcomes

• By the end of the module the student will be able to
• Define the 6 trigonometrical functions (SSK1)
• State the sine and cosine rule (SSK2)
• Give standard Cartesian equations for circles and lines (SSK3)
• State the standard forms of general solutions of second order differntial equations (SSK4)
• State the rules of addition, subtraction and multiplication of matrices and for finding inverses

• By the end of this module the student will have acquired the skills to be able to:
• select and use trigonometric formulae and techniques as required in problems appropriate to the syllabus. (SSS1)
• confidently manipulate a range of Cartesian and vector equations in two and three dimensions. (SSS2)
• understand and use complex numbers in a range of situations as appropriate to the syllabus. (SSS3)
• Select and use a toolkit of differentiation and integration techniques in a range of situations as appropriate to the syllabus (SSS4)
• understand and use first and second order differential equations in a range of situations as appropriate to the syllabus. (SSS5)
• confidently manipulate a range of algebraic and trigonometric expressions as required in problems appropriate to the syllabus. (SSS6)
• Apply mathematics to a variety of problems based on physical situations (SSS7)

• By the end of the module students will be able to
• apply number in the tackling of numerical problems
• be able to demonstrate problem solving skills.
• Test 1 will assess: SSK 1, SSK2, SSS1, SSS2, SSS6, SSS7, KS1, KS2.
• Test 2 will assess SSS6, SSS7, KS1, KS2.
• Portfolio will assess SSK 1-5, SSS 1-7, KS 1-2
• End of module test will assess SSK 3-5, SSS 2-6, KS 1-2

Modes of Teaching, Learning and Assessment and how these contribute to the learning outcomes of the module

• Theory, initial concepts and techniques will be introduced during lectures.
• Much of the learning, understanding and consolidation will take place through the use of structured worksheets during seminars, tutorials and students' own time.
• Ability to recall, select and use knowledge and manipulative skills will be tested by: a coursework portfolio containing students solutions to questions or tasks set by the tutor on a weekly basis, mid-module invigilated tests and an end of module test.
• In addition to obtaining an overall weighted mark of 50% or above, students must obtain a mark of 50% or above for the following element/s -test 2 & end of module test

Teaching Methods and Learning Hours

Summative Assessment

Formative Assessment:

Students will be given self testing units on a weekly basis.

■ Attendance at all activities marked with this symbol will be monitored. Students who fail to attend these activities, or to complete the summative or formative assessment specified above, will be subject to the procedures defined in the University's General Regulation V, and may be required to leave the University