Syllabus breakdown

AI SL Calculus syllabus points

Official syllabus requirements assessed across Paper 1 and Paper 2 in Topic 5.

  1. 5.1Introduction to the derivative as an instantaneous rate of change and gradient of a curve at a point
  2. 5.2Derivative of power functions: $\frac{d}{dx}(ax^n) = anx^{n-1}$ ($n \in \mathbb{Q}$), sum and difference rule
  3. 5.3Equations of tangents and normals to curves at specified points
  4. 5.4Increasing and decreasing functions: $f'(x) > 0$ and $f'(x) < 0$
  5. 5.5Stationary points (local maxima and minima) where $f'(x) = 0$, and optimization in applied contexts
  6. 5.7 & 5.8Definite integrals and the Trapezoidal Rule: $\int_a^b y dx \approx \frac{h}{2}\left((y_0 + y_n) + 2(y_1 + \dots + y_{n-1})\right)$ for irregular terrain and lake boundaries

Interactive guides

Topic 5 Interactive Study Guides

Each guide features full notes, 10 progressive practice exercises with complete markschemes, formula sheets with common pitfalls, and an exam mini test.

Lesson 5.1 - 5.3

Derivatives & Tangents

Power rule differentiation, interpreting instantaneous rate of change, evaluating gradient of curves, and linear equations of tangent lines.

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Lesson 5.4 - 5.6

Optimization in Context

Finding stationary points ($f'(x) = 0$), identifying maxima and minima, minimizing material costs, and maximizing profits or enclosed areas.

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Lesson 5.7 & 5.8 (Core AI Focus)

Trapezoidal Rule & Integration

Numerical integration across tabular surveyed coordinates, calculating step size $h$, estimating cross-sectional areas of lakes and tunnels, and curvature error analysis.

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Exam insights

Key calculus concepts in Math AI

  • Distinguishing Average vs Instantaneous Rate: Average rate of change is the secant slope $\frac{f(b) - f(a)}{b - a}$; instantaneous rate of change is the derivative $f'(x)$.
  • GDC Numerical Derivative: On Paper 2, find gradient values using nDeriv(f(x), x, c) to check your analytical differentiation.
  • Trapezoidal Rule Over/Under Estimation:
    • If the curve is concave down ($\cap$, $f''(x) < 0$), straight trapezoid chords lie below the curve $\implies$ underestimation.
    • If the curve is concave up ($\cup$, $f''(x) > 0$), straight chords lie above the curve $\implies$ overestimation.

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