Topic 9 – Option: Calculus

Maclaurin’s Series

1. Maclaurin's series expansion
2. Series expansion for e^x
3. Series expansion for sin(x) and cos(x)
4. Series expansion for ln(1+x)
5. Exam Questions - Maclaurin’s series

Taylor’s Series

1. Exam Questions - Taylor’s series

Infinite sequences

1. Infinite sequences of real numbers and their convergence or divergence

Infinite series

1. Convergence of infinite series

Tests for convergence

1. Limit comparison test
2. Ratio test
3. Integral test

Types of series

1. The p-series
2. Series that converge absolutely
3. Series that converge conditionally
4. Alternating series

Differentiation – Continuity

Rolle’s Theorem

Mean Value Theorem

L’Hopital’s Rule

Integration as a limit of a sum

Separating the Variables

1. Differential Equations - Finding a general and a particular solution
2. Working with constants in log types
3. Differential Equations - Exponential and trig type
4. Separating the variables and sketching a family of curves

Exact Equations (Integrating Factors)

1. Exact equations where one side is the exact derivative of a product
2. Solving equations of the form dy/dx + Py = Q using an integrating factor
3. Exam Questions - Exact equations (integrating factors)

Substitution Types

1. Using substitution to reduce a differential equation to a known form
2. Exam Questions - Substitution types