5. Calculus

Course Description
This course develops a rigorous understanding of differential and integral calculus within the IBDP Analysis & Approaches framework. Students will learn both the theoretical foundations and practical techniques needed to analyse functions, solve optimisation and kinematics problems, work with areas and volumes, and apply calculus to real‐world contexts. HL students will extend these skills to first principles, higher‐order techniques, series expansions, and differential equations—preparing them for further study in STEM disciplines.

Learning Outcomes

SL Students will be able to:

• Define the derivative \(f'(x)\) as the instantaneous rate of change and compute it for polynomial and simple composite functions.
• Determine intervals of increase/decrease and locate turning points and inflection points using first and second derivative tests.
• Find equations of tangents and normals to curves.
• Apply differentiation to solve optimisation and kinematics problems.
• Understand antiderivatives and compute indefinite integrals, including reverse chain‐rule (substitution).
• Evaluate definite integrals to find areas under curves and areas between two curves.
• Use the second derivative to analyse concavity and acceleration in motion contexts.

HL Students will additionally be able to:

• Derive the derivative from first principles and extend to higher‐order derivatives.
• Evaluate limits, including indeterminate forms, using L’Hôpital’s Rule.
• Perform implicit differentiation and solve related‐rates and advanced optimisation problems.
• Integrate complex functions using partial fractions and integration by parts (including repeated parts).
• Compute volumes of revolution about both the $x$‐ and $y$‐axes using slicing and shell methods.
• Solve first-order differential equations (separable, integrating factor, homogeneous via $y=vx$) and implement numerical approximations (Euler’s method).
• Expand functions in Maclaurin series and determine their radius of convergence.