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Mathematics 1M1


Unit code: MATH19661
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

None

Aims

The course unit aims to provide a basic course in calculus and algebra to students with A-level mathematics or equivalent in school of MACE.

Learning outcomes

On completion of this unit successful students will be able to:

1 - Calculate ordinary and partial derivatives using the chain, product, and quotient rules, as well as provided tables, and apply these derivatives to estimate the size of errors.

2 - Calculate indefinite integrals using substitution, integration by parts, and partial fractions, as well as provided tables, and apply these indefinite integrals to evaluate definite integrals using the fundamental theorem of calculus.

3 - Use Macaulay brackets to write formulas for functions originally defined piecewise.

4 - Apply the trapezium rule and Simpson's rule to approximate definite integrals.

5 - Apply the bisection method and Newton-Raphson method to approximate solutions for given equations.

6 - Add and subtract vectors, multiply vectors by a scalar, and understand the geometric interpretation of these operations as well as how to apply them to write a parametric equation for a line.

7 - Calculate the dot and cross products of vectors, and apply these operations to find the equation for a plane, length of a vector, angle between two vectors, area of a parallelogram, and volume of a parallelipiped.

8 - Find the intersection point or points of a line with another line, or a line with a plane, or determine that there is no intersection point.

9 - Carry out basic arithmetic (i.e. addition, subtraction, multiplication and division) with complex numbers, and understand how these operations correspond with geometric operations on the Argand diagram.

10 - Convert complex numbers back and forth between the exponential and standard forms. Use phasors to represent sinusoidal functions, and find the amplitude and phase of sums of sinusoidal functions.

11 - Calculate line integrals in two dimensions for differential forms and with respect to arc length.

12 - Calculate double integrals in Cartesian coordinates over domains that are unions of rectangles, and in polar coordinates over domains that are polar rectangles (i.e. represented as rectangles in polar coordinates).

13 - Understand Green's theorem, and apply it when appropriate to evaluate double integrals or line integrals.

14 - Find the general solution of first order ordinary differential equations which are separable, linear, homogeneous, or exact, and use the general solution to find the specific solution for given initial value problems.

15 - Understand the basic concepts of the mathematical theory of probability including "sample space", "event", "independent" events, the "intersection" of two events, and the "union" of two events, and apply these concepts in specific examples.

16 - Determine the probability that an event occurs, given that another event has occurred.

Assessment methods

  • Other - 20%
  • Written exam - 80%

Assessment Further Information

Diagnostic Followup Coursework (week 4) Weighting within unit 10%

Coursework 2 (week 11) Weighting within unit 10%

2 hour examination (semester 1) Weighting within unit 80%

Syllabus

4 lectures: Intermediate Calculus.
Review of product, quotient and chain rules for differentiation. Partial differentiation in 2 variables. Review of methods of integration (by parts, substitution and partial fractions.

4 lectures: Double integrals and line integrals.
Double integrals over rectangles and disks. Line integrals and integrals with respect to arc-length. Relation through Green's theorem.

2 lectures: Basic Numerical Methods.
Trapezoidal and Simpson's rule, inc. estimation of error. Bisection and Newton-Raphson methods for solving nonlinear equations.

4 lectures: Vectors.
Review of vectors in component form; vector addition, parallelogram and triangle of vectors. Vector equation of straight line.  Scalar and vector products. Triple Products. Applications including vector products and areas.

3 lectures: Complex numbers.
Definition, algebraic operations, modulus and argument.   Argand
Diagram, De Moivre's theorem.

4 lectures: Ordinary Differential Equations (ODEs).
Examples of First and Second order ODEs. Role of arbitrary constants.
Solution of first-order separable, linear and exact ODEs.  First order linear ODEs and integrating factors.

3 lectures: Probability.
Deterministic vs random (probabilistic) models. Random experiments and sample spaces. Definition and properties of probability; finite sample spaces. Conditional probability, independent events.

Recommended reading

KA Stroud, Engineering Mathematics, Palgrave

Croft et al., Mathematics for Engineers, Pearson

Study hours

  • Lectures - 24 hours
  • Tutorials - 11 hours
  • Independent study hours - 65 hours

Teaching staff

Sean Holman - Unit coordinator

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