Mathematics: Analysis and Approaches SL

Calculus

Limits, derivatives, tangents, optimization, kinematics, integration, area, and mathematical modelling with rates of change.

Understand gradients and limits Connect secant gradients, average rate of change, and the intuition behind limits. Differentiate with the power rule Differentiate polynomial and power functions using derivative notation and the power rule. Find tangent and normal equations Find tangent and normal gradients and equations using derivatives at a point. Analyse increasing and decreasing intervals Use derivatives to classify stationary points and describe increasing or decreasing behaviour. Solve optimization problems Model and solve maximum or minimum problems using derivatives and clear interpretation. Use calculus in kinematics Connect displacement, velocity, and acceleration using differentiation in motion contexts. Integrate with the power rule Find indefinite and definite integrals using the reverse power rule and correct constant notation. Find area under curves Use definite integrals to calculate signed area and interpret area under a curve. Interpret limits and derivative behaviour Estimate limits and connect the signs of derivatives with increasing, decreasing, stationary, and curvature behaviour. Differentiate composite functions Differentiate fractional powers, trigonometric, exponential, logarithmic, product, and quotient functions with exact notation. Analyse curves and optimize Use stationary-point tests, concavity, inflexion, domain boundaries, and purposeful GDC extrema to analyse and optimize functions. Find speed from a distance-time graph Read two points off a distance-time graph and find the constant speed as distance over time (the gradient). Model motion and boundary conditions Connect displacement, velocity, acceleration, total distance, sign changes, and constants fixed by boundary conditions. Integrate functions and regions Integrate reverse-chain and linear-composite forms and distinguish signed integrals from geometric areas between curves.