H2 Maths Calculus | Free Notes & Worked Examples

Q: What does H2 Maths Notes (JC 1-2): 5) Calculus cover?
A: Differentiation, Maclaurin series, integration techniques, definite integrals, and differential equations aligned with the 2026 H2 syllabus.

Before you begin
Revisit your IP and Additional Maths rules for basic differentiation and integration so the JC techniques extend naturally. Set the GC to Exam Mode and know where derivative, integral, and root-finding menus live.

For the full topic map, paper weightings, and official PDF link, see our H2 Maths Syllabus 2026-27 overview.

• Calculus studies change and accumulation: Ask whether the question wants a rate or a total.
• Differentiation finds gradients; integration finds accumulated amounts: Choose the tool before doing algebra.
• H2 questions often combine functions, graphs, and interpretation: Write the meaning of the derivative or integral in words.

Concrete example: If a velocity-time graph is given, differentiation gives acceleration at an instant, while integration gives displacement over a time interval.

Status: SEAB's current H2 Mathematics (9758) syllabus PDF is labelled for 2026. Topic 5 coverage is within Section A Pure Mathematics, which is assessed in Paper 1 (100 marks) and Paper 2 Section A (40 marks).

Technique-choice checkpoint

Before starting algebra, decide what the question is asking calculus to do. This prevents using a familiar rule on the wrong object.

Worked check: if a velocity is $v(t)=3t^2-2t$, then acceleration at $t=4$ uses differentiation: $a(t)=v'(t)=6t-2$

5.1 | Differentiation

Core skills

• Differentiate polynomials, rational functions, exponentials, logarithms, and trig combinations.
• Use product, quotient, and chain rules fluently; annotate which rule you applied.
• Interpret gradients and tangents in context (rates of change, normals to curves).

Workflow checklist

1. Simplify the function before differentiating if it avoids quotient rule clutter.
2. State the derivative step-by-step: $\frac{\mathrm{d}}{\mathrm{d}x} [u(x)]$ then substitute.
3. For stationary points solve $f'(x) = 0$

Example 1 -- Tangent gradient

Given $y = x e^{-x}$, find the gradient at $x = 2$.

1. Differentiate: $\frac{\mathrm{d}y}{\mathrm{d}x} = e{-x} + x(-e^{-x}) = e^{-x}(1 - x)$

JC habits

• Record domain restrictions before using logarithmic differentiation.
• Quote implicit differentiation steps with differentiate both sides so markers follow the logic.
• When applying related rates, state the relationship then differentiate with respect to time.

5.2 | Maclaurin Series

Core ideas

• Maclaurin expands $f(x)$ about $x = 0$; the coefficients come from derivatives at zero.
• Memorise baseline series up to the cubic term:
  • $e^x = 1 + x + \frac{x^2}{2!} + \frac{x^3}{3!} + \cdots$

Example 2 -- Series by substitution

Find the Maclaurin series for the exponential with exponent -2x, keeping terms through the cubic order.

1. Start from $e^x$ baseline.
2. Substitute $-2x$ for $x$ and keep terms up to $x^3$; the truncated polynomial is:

$$1 - 2x + 2x^2 - \frac{4}{3} x^3$$

Tips

• Always state the interval of validity if the question requests it.
• Factor common powers of $x$ before truncating to avoid mistakes in higher terms.
• When multiplying series, keep terms up to the requested order only.

5.3 | Integration Techniques

Tools to master

• Algebraic simplification before integration (completing the square, partial fractions).
• Substitution: choose $u(x)$ so $\frac{\mathrm{d}u}{\mathrm{d}x}$ appears in the integrand.
• Integration by parts: $\int u \ dv = uv - \int v \ \mathrm{d}u$

Integration technique checkpoint

Before integrating, identify the obstacle that stops the expression from matching a basic standard form.

Worked check: for $\int x e^{x^2},\mathrm{d}x$, the obstacle is a composite exponential with the derivative of $x^2$ nearby. Let $u=x^2$

$$\int x e^{x^2},\mathrm{d}x = \frac{1}{2}\int e^u,\mathrm{d}u = \frac{1}{2}e^{x^2}+C.$$

Misconception check: do not choose the technique by the chapter heading alone. Choose it from the obstacle in the integrand, then check that the next line is simpler than the previous one.

Example 3 -- By parts

Evaluate $\int x e^x \mathrm{d}x$.

1. Let $u = x$, $dv = e^x \mathrm{d}x$.
2. Then $\mathrm{d}u = \mathrm{d}x$, $v = e^x$

Example 4 -- Partial fractions

Integrate $\int \frac{3x + 5}{x^2 - x - 2} \mathrm{d}x$.

1. Factor denominator: $x^2 - x - 2 = (x - 2)(x + 1)$.
2. Decompose: $\frac{3x + 5}{(x - 2)(x + 1)} = \frac{A}{x - 2} + \frac{B}{x + 1}$

5.4 | Definite Integrals and Applications

Key outcomes

• Evaluate definite integrals via Fundamental Theorem of Calculus: $\int_a^b f'(x) \mathrm{d}x = f(b) - f(a)$

Example 5 -- Curve area

Area between $y = x^2$ and $y = 2x$ from $x = 0$ to $x = 2$

$$\int_0^2 (2x - x^2) \mathrm{d}x = \bigl[ x^2 - \frac{x^3}{3} \bigr]_0^2 = 4 - \frac{8}{3} = \frac{4}{3}.$$

Example 6 -- Volume of revolution

Rotate $y = e^{-x}$ between $x = 0$ and $x = 1$ about the x-axis:

$$V = \pi \int_0^1 \left( e^{-x} \right)^2 \mathrm{d}x = \pi \int_0^1 e^{-2x} \mathrm{d}x = \pi \bigl[ -\frac{1}{2} e^{-2x} \bigr]_0^1 = \frac{\pi}{2} (1 - e^{-2}).$$

Habits

• Include absolute values around logarithms when evaluating between limits.
• Sketch the region before integrating to confirm which function sits on top.
• For symmetry, argue whether the integrand is even/odd to simplify limits.

5.5 | Differential Equations

MOE focus

• Solve first-order separable equations by rearranging into $g(y) dy = f(x) \mathrm{d}x$.
• Interpret solutions with constants of integration fitted to initial conditions.
• Model growth/decay problems and interpret steady states.

Example 7 -- Separable equation

Solve $\frac{dy}{\mathrm{d}x} = 3y$ with $y(0) = 5$.

1. Separate: $\frac{1}{y} dy = 3 \mathrm{d}x$.
2. Integrate: $\ln|y| = 3x + C$

Example 8 -- Logistic flavour

If $\frac{dy}{dt} = y(4 - y)$, separate to obtain $\frac{1}{y(4 - y)} dy = dt$

Presentation tips

• Always state the separation step explicitly before integrating.
• Keep constants tidy; e.g. write $y = Ae^{kx}$ and substitute data once.
• Interpret the final solution (growth, equilibrium value, long-term behaviour) in words.

Practice Quiz

Gauge how well you can connect differentiation, integration, and DE modelling techniques across the whole chapter.

5.6 | Quick Revision Checklist

• Differentiate products, quotients, and implicit relations confidently.
• Recall baseline Maclaurin series and transform them through substitution or multiplication.
• Execute substitution, parts, and partial fractions with calculator cross-checks.
• Evaluate definite integrals for areas or volumes with clear working and limits.
• Solve separable differential equations and describe the meaning of constants.

Want weekly guided practice on Calculus? Our H2 Maths tuition programme builds fluency in this topic through structured problem sets and exam-style drills.

Sources

• SEAB H2 Mathematics syllabus (9758), examinations from 2026 - Topic 5 Calculus (sub-topics 5.1-5.5 covering differentiation, Maclaurin series, integration techniques including partial fractions, definite integrals/area/volume, and differential equations). The official PDF link is recorded in the front matter citation for this note.

Next step: use the H2 Maths notes hub to deep-dive into Sub-topic 5.1 - Differentiation and the rest of the calculus sequence.