Functions

[[L’Hôpital’s Rule]]

1. Limits and Continuity: Understanding the behaviour of functions as they approach specific points or infinity; foundational for defining derivatives and integrals.

2. Derivatives and Differentiation: Measures the rate of change of a function; key for understanding motion, growth, and optimisation.

3. Chain Rule: A method for differentiating composite functions; crucial for handling complex expressions.

4. Implicit Differentiation: Differentiating equations that define functions implicitly; useful for related rates and other applications.

5. Optimisation: Using derivatives to find maxima and minima of functions; applied in various fields to optimise performance and outcomes.

6. Integrals and Integration: Calculating areas under curves and accumulated quantities; essential for understanding total change.

7. Fundamental Theorem of Calculus: Links differentiation and integration, showing they are inverse processes.
8. Series and Sequences: Understanding sums of sequences and their convergence; important for approximations and various applications.
9. Multivariable Calculus: Extending calculus to functions of several variables; critical for real-world applications involving multiple factors.

10. Partial Derivatives: Differentiating functions of several variables with respect to one variable at a time; foundational for multivariable analysis.

11. Vector Calculus: Deals with vector fields and operations on them; used in physics and engineering.

12. Differential Equations: Equations involving derivatives; essential for modelling dynamic systems in science and engineering.