Revise Differentiation: tangents, normals and optimization for IB Mathematics: Applications and Interpretation (5.2)

A tangent is a straight line that 'just touches' a curve at a point, sharing the curve's gradient there. The derivative $f'(x)$ gives that gradient at any value of $x$. A normal is the line perpendicular to the tangent at the same point, so its gradient is the negative reciprocal of the tangent's. To write either line you need just two things: a point on it, and its gradient.

At a point $(a, f(a))$ on the curve $y=f(x)$:

Gradient of the tangent: $m_T = f'(a)$

Gradient of the normal: $m_N = -\dfrac{1}{m_T} = -\dfrac{1}{f'(a)}$

Equation of a line through $(x_1,y_1)$: $y - y_1 = m(x - x_1)$

The sign of the derivative tells you which way a curve is heading. Where $f'(x)>0$ the function is increasing (sloping up); where $f'(x)<0$ it is decreasing (sloping down). The dividing lines between these regions are the stationary points, where the gradient is exactly zero and the tangent is horizontal. You find them by solving $f'(x)=0$, and you classify each one as a local maximum (a peak, $f'$ going $+\to-$), a local minimum (a trough, $f'$ going $-\to+$), or a point of inflexion (no sign change). In AI SL the second derivative test is not required — a first-derivative sign check or the shape of the GDC graph is enough.

Optimisation is where differentiation earns its keep: finding the largest area, the greatest volume, the least cost or material. The strategy never changes. Write the quantity you care about — the objective — as a function. If it depends on more than one variable, use the constraint (a fixed perimeter, a fixed volume, a fixed length of fencing) to eliminate all but one variable. Then differentiate, solve $f'(x)=0$ to find the stationary point, justify that it is the maximum or minimum you want, and finally answer the exact quantity the question asked for, with units. The hardest step for most candidates is reducing to one variable — get that right and the calculus is routine.

In an exam the marks are analytic: each is tied to a specific line of working. A method mark (M) rewards a correct approach even if the arithmetic later slips; an accuracy mark (A) rewards a correct result and is usually dependent on the method mark being earned. Follow-through (FT) means a correct final step performed on your own earlier (wrong) value still scores, ISW ('ignore subsequent working') means a correct answer is not un-marked by a later fumble, and equivalent correct forms are accepted. All of that protection exists only if your method is on the page. Study how each mark below is earned by a specific line.

Your GDC is indispensable for checking analytic work and for tackling harder calculator-paper problems. Graph a function to read off maxima and minima directly, or use the built-in maximum/minimum finder over a sensible interval. The numerical derivative feature (often nDeriv or d/dx) gives the gradient at a point without algebra, which is perfect for verifying a tangent gradient. And when you optimise, always check the domain: lengths, areas and volumes must be positive, so restrict your search to sensible values of the variable.

Everything here is one skill — differentiate, set the derivative to zero, interpret the result — applied to two contexts: the geometry of a curve (tangents, normals, stationary points) and the optimisation of a real quantity. That same skill returns immediately in integration, where you reverse differentiation, and in the modelling questions that dominate Paper 2. Master the habit — model the quantity, reduce to one variable, solve $f'=0$, then justify and interpret — and the rest of calculus becomes variations on a method you already own.