Learning Outcomes:
On successful completion of this module, students should be able to:
1. Use mathematical notation and conventions accurately and carry out calculations involving numbers, fractions, percentages, units, averages, powers, roots, and exponents.
2. Define variables and translate scientific or practical problems into algebraic expressions, equations, and elementary mathematical models, keeping track of units and assumptions and interpreting the resulting answers.
3. Work with sets and functions, determine appropriate domains, compose and invert functions, and move between verbal, numerical, algebraic, and graphical representations.
4. Solve linear, simultaneous, quadratic, exponential, logarithmic, and elementary trigonometric equations and use the corresponding functions in applications.
5. Calculate and interpret limits, assess continuity, and calculate derivatives both from first principles and by using standard differentiation rules.
6. Interpret derivatives as slopes and rates of change and use them for linear approximation, graph analysis, optimisation, and problems involving position, velocity, and acceleration.
7. Calculate definite and indefinite integrals, apply the Fundamental Theorem of Calculus, and interpret integrals as signed area and accumulated change.
8. Select appropriate mathematical methods and digital tools, check the plausibility of results, and communicate mathematical reasoning clearly.
Indicative Module Content:
- Mathematical communication: conventions, notation, implications, and equivalence.
- Foundations: number systems, arithmetic laws, fractions, percentages, units, averages, powers, roots, and exponents.
- Mathematical modelling: variables, algebraic expressions, assumptions, equations, and interpretation.
- Sets and functions: notation, domains, composition, inverse functions, and graphical representations.
- Equations and graphs: linear, simultaneous, and quadratic equations; linear, polynomial, and non-polynomial graphs.
- Trigonometry: right-triangle geometry, sine, cosine, tangent, radians, the unit circle, and applications.
- Exponential and logarithmic functions: growth, logistic growth, decay, compound interest, logarithmic scales, equations, and applications.
- Limits and continuity: finite limits, limits at infinity, indeterminate forms, and continuity.
- Differentiation: rates of change, the derivative, first principles, differentiation rules, linear approximation, and natural growth.
- Optimisation: increasing and decreasing functions, critical points, higher derivatives, concavity, and motion.
- Integration: definite and indefinite integrals, antiderivatives, the Fundamental Theorem of Calculus, area, accumulated change, and applications.