H2 Maths Notes (JC 1-2): 5) Calculus

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

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 \( f'(x) = 0 \) and classify with the first or second derivative test.

Example 1 -- Tangent gradient

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

1. Differentiate: \( \frac{dy}{dx} = e{-x} + x(-e^{-x}) = e^{-x}(1 - x) \).
2. 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:
  • \( e^x = 1 + x + frac{x^2}{2!} + \frac{x^3}{3!} + \cdots \).
  • \( \sin x = x - \frac{x^3}{3!} + \cdots \).
  • \( \cos x = 1 - \frac{x^2}{2!} + \cdots \).
  • \( \ln(1 + x) = x - \frac{x^2}{2} + \frac{x^3}{3} + \cdots \) valid for \( \lvert x \rvert < 1 \).

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 \( \frac{du}{dx} \) appears in the integrand.
• Integration by parts: \( \int u \ dv = uv - \int v \ du \); select \( u \) that simplifies on differentiation.
• Standard forms: recognise \( \int \frac{1}{x} dx = \ln\lvert x \rvert + C \), trig substitutions, and inverse trig derivatives.

Example 3 -- By parts

Evaluate \( \int x e^x dx \).

1. Let \( u = x \), \( dv = e^x dx \).
2. Then \( du = dx \), \( v = e^x \).
3. Apply by-parts: \( \int x e^x dx = x e^x - \int e^x dx = x e^x - e^x + C \).

Example 4 -- Partial fractions

Integrate \( \int \frac{3x + 5}{x^2 - x - 2} dx \).

1. Factor denominator: \( x^2 - x - 2 = (x - 2)(x + 1) \).
2. Decompose: \( \frac{3x + 5}{(x - 2)(x + 1)} = \frac{A}{x - 2} + \frac{B}{x + 1} \).
3. Solve \( A = 2 \), \( B = 1 \).
4. Integrate to get \( 2 \ln\lvert x - 2 \rvert + \ln\lvert x + 1 \rvert + C \).

5.4 | Definite Integrals and Applications

Key outcomes

• Evaluate definite integrals via Fundamental Theorem of Calculus: \( \int_a^b f'(x) dx = f(b) - f(a) \).
• Interpret area under curves (signed area) and between curves by subtracting top minus bottom.
• Compute volumes of revolution with \( \pi \int_a^b y^2 dx \) (about the x-axis) or \( \pi \int_a^b x^2 dy \) (about the y-axis).

Example 5 -- Curve area

Area between \( y = x^2 \) and \( y = 2x \) from \( x = 0 \) to \( x = 2 \):

\[ \int_0^2 (2x - x^2) dx = \left[ x^2 - \frac{x^3}{3} \right]_0^2 = 4 - \frac{8}{3} = \frac{4}{3}. \]

Example 6 -- Volume of revolution

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

\[ V = \pi \int_0^1 \left( e^{-x} \right)^2 dx = \pi \int_0^1 e^{-2x} dx = \pi \left[ -\frac{1}{2} e^{-2x} \right]_0^1 = \frac{\pi}{2} (1 - e^{-2}). \]

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 \( g(y) dy = f(x) dx \).
• Interpret solutions with constants of integration fitted to initial conditions.
• Model growth/decay problems and interpret steady states.

Example 7 -- Separable equation

Solve \( \frac{dy}{dx} = 3y \) with \( y(0) = 5 \).

1. Separate: \( \frac{1}{y} dy = 3 dx \).
2. Integrate: \( \ln|y| = 3x + C \).
3. Exponentiate: \( y = K e^{3x} \).
4. Apply initial condition: \( 5 = K e^0 \Rightarrow K = 5 \).
5. Solution: \( y = 5 e^{3x} \).

Example 8 -- Logistic flavour

If \( \frac{dy}{dt} = y(4 - y) \), separate to obtain \( \frac{1}{y(4 - y)} dy = dt \). 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 \( y = Ae^{kx} \) and substitute data once.
• Interpret the final solution (growth, equilibrium value, long-term behaviour) in words.

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.

Next step: expand sub-topics 5.1 to 5.5 with full worked examples, including optimisation problems, series proofs, and modelling drills.