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.
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
Simplify the function before differentiating if it avoids quotient rule clutter.
State the derivative step-by-step: dxd[u(x)] then substitute.
For stationary points solve
f′(x)=0
and classify with the first or second derivative test.
Example 1 -- Tangent gradient
Given y=xe−x, find the gradient at x=2.
Differentiate: dxdy=e−x+x(−e−x)=e−x(1−x).
Substitute x=2 to get gradient e−2(1−2)=−e−2.
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:
ex=1+x+fracx22!+3!x3+⋯.
sinx=x−3!x3+⋯.
cosx=1−2!x2+⋯.
ln(1+x)=x−2x2+3x3+⋯
Example 2 -- Series by substitution
Find the Maclaurin series for the exponential with exponent -2x, keeping terms through the cubic order.
Start from ex baseline.
Substitute −2x for x and keep terms up to x3; the truncated polynomial is:
1−2x+2x2−34x3
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 dxdu appears in the integrand.
Integration by parts: ∫udv=uv−∫vdu; select u that simplifies on differentiation.
Standard forms: recognise ∫x1dx=ln∣x∣+C, trig substitutions, and inverse trig derivatives.
Example 3 -- By parts
Evaluate ∫xexdx.
Let u=x, dv=exdx.
Then du=dx, v=ex.
Apply by-parts: ∫xexdx=xex−∫exdx=xex−ex+C
Example 4 -- Partial fractions
Integrate ∫x2−x−23x+5dx.
Factor denominator: x2−x−2=(x−2)(x+1).
Decompose: (x−2)(x+1)3x+5=x−2A+x+1B.
Solve A=2, B=1.
Integrate to get 2ln∣x−2∣+ln∣x+1∣+C.
5.4 | Definite Integrals and Applications
Key outcomes
Evaluate definite integrals via Fundamental Theorem of Calculus: ∫abf′(x)dx=f(b)−f(a).
Interpret area under curves (signed area) and between curves by subtracting top minus bottom.
Compute volumes of revolution with π∫aby2dx (about the x-axis) or π∫abx2dy
Example 5 -- Curve area
Area between y=x2 and y=2x from x=0 to x=2:
∫02(2x−x2)dx=[x2−3x3]02=4−38=34.
Example 6 -- Volume of revolution
Rotate y=e−x between x=0 and x=1 about the x-axis:
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)dx.
Interpret solutions with constants of integration fitted to initial conditions.
Model growth/decay problems and interpret steady states.
Example 7 -- Separable equation
Solve dxdy=3y with y(0)=5.
Separate: y1dy=3dx.
Integrate: ln∣y∣=3x+C.
Exponentiate: y=Ke3x.
Apply initial condition: 5=Ke0⇒K=5.
Solution: y=5e3x.
Example 8 -- Logistic flavour
If dtdy=y(4−y), separate to obtain y(4−y)1dy=dt. Use partial fractions to integrate and solve for y(t), then apply the given initial condition in exam questions.
Presentation tips
Always state the separation step explicitly before integrating.
Keep constants tidy; e.g. write y=Aekx 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.
