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Unit code: | MATH19681 |
Credit Rating: | 10 |
Unit level: | Level 1 |
Teaching period(s): | Semester 1 |
Offered by | School of Mathematics |
Available as a free choice unit?: | N |
Requisites
NoneAims
The programme unit aims to provide a first semester course in calculus and algebra to students in school of Electrical and Electronic Engineering.
Learning outcomes
On completion of this unit (1E1) successful students will be able to:
1. (Vectors) Find the position vector of a point given in 3D Cartesian coordinates . Multiply a vector by a scalar. Find the magnitude and direction of a vector. Add two or more vectors using the triangle law or using components. Calculate the scalar and vector products of two vectors. Find the angle between two vectors. Find the scalar and vector triple products of three vectors. Find the vector equation of a straight line through two points. Find the shortest distance from a point to a line and between two lines.
2. (Coordinates) Calculate the 2D polar coordinates of a point given the Cartesian coordinates and vice versa. Convert the equation of a curve in Cartesian coordinates to the equation in 2D polar coordinates and vice versa. Sketch a curve given its 2D polar equation. Convert the coordinates of a point from 3D Cartesian coordinates to 3D cylindrical or 3D spherical coordinates and vice versa. . Convert the equation of a curve in Cartesian coordinates to the equation in 3D polar coordinates (cylindrical or spherical) and vice versa. Calculate the point of intersection between a line and a surface expressed in any 3D coordinate system. Find the curve of intersection between two surfaces.
3. (Complex Numbers) Add, subtract, multiply and divide complex numbers in standard form. Locate a point in the Argand diagram. Find the modulus and argument of a complex number. Write a complex number using the exponential modulus/argument form. Write the trigonometric functions using exponentials of imaginary numbers. Use de Moivre's theorem to find powers of complex numbers.
4. (Hyperbolic Functions) Sketch the curves of all the hyperbolic functions. Use Osborne's rule to convert standard relation between trigonometric functions into the corresponding relations for hyperbolic functions. Differentiate expressions involving hyperbolic functions. Plot the inverse hyperbolic functions. Express the inverse hyperbolic functions in terms of natural logarithms. Differentiate the inverse hyperbolic functions.
5. (Differentiation) Differentiate products and quotients of functions. Differentiate functions of functions. Differentiate inverse trigonometric and hyperbolic functions. Calculate stationary points of functions and identify the type. Use implicit, logarithmic and parametric differentiation. Use differentiation to estimate small errors. Use L'Hopital's rule to find the limit of a quotient. Use the Newton Raphson method to solve equations numerically.
6. (Integration) Calculate an integral using substitution. Simplify an integral using integration by parts. Integrate rational functions using partial fractions. Calculate certain integrals with square roots of quadratic functions in the denominator.
Assessment methods
- Other - 25%
- Written exam - 75%
Assessment Further Information
Diagnostic Follow-up Coursework (if needed) (week 4): weighting within unit 10%.
Coursework 2: weighting within unit 5%. Computerised exercise.
Coursework 3: weighting within unit 5%. Computerised exercise .
Coursework 4: weighting within unit 5%. Computerised exercise.
2 hour (semester 1) examination Weighting within unit 75%.
Syllabus
5 lectures : Vectors : Vectors in component form ; vector addition, subtraction and multiplication by a scalar ; parallelogram and triangle of vectors ; vector equation of a straight line ; Scalar products ; vector products.
4 lectures : Coordinate Systems : Alternate coordinate systems in 2 and 3 dimensions i.e. cartesian, plane polar, cylindrical, spherical. Transformations between systems highlighting the role of the correct quadrant / octant and concentrating on points and position vectors.
4 lectures : Complex Numbers and Hyperbolic Functions : Definition of complex numbers : algebraic operations ; modulus, argument and Argand diagram ; trigonometric and exponential forms. De moivre's Theorem Definition of hyperbolic functions. Elementary properties. Inverse functions ; Osborne's Rule.
6 lectures : Differentiation : Working definition (rate of change, physical interpretation). Differentiation rules (parametric, implicit, logarithmic etc) Derivatives of logarithmic and hyperbolic functions. Applications to maxima and minima. l'Hopital's rule (including working defn of limit). Newton Raphson Method (application of differentiation).
3 lectures : Integration : Working definition of the integral. Fundamental theorem of calculus Physical interpretation. Definite integrals and areas under curves. Revision of integration techniques (polynomials etc). Integration by parts, by substitution and by partial fractions. (partial fractions themselves part of followup). Applications of integration. (this topic also to form 4 lectures from 1E2).
Recommended reading
KA Stroud, Engineering Mathematics, Palgrave
Croft et al., Introduction to Engineering Mathematics, Pearson
Study hours
- Assessment written exam - 2 hours
- Lectures - 24 hours
- Tutorials - 11 hours
- Independent study hours - 63 hours
Teaching staff
John Parkinson - Unit coordinator