IB Maths: Tangents, Normals & Stationary Points FAQ

It's tested across Papers 1 and 2 in both AA and AI, usually worth 5-9 marks per question. You'll be asked to find a gradient function, evaluate it at a point, then build a tangent or normal equation, or classify stationary points using the second derivative test.

A tangent touches the curve at one point and has the same gradient as the curve there — gradient equals $f^{\prime }(x)$. A normal is perpendicular to the tangent at that same point, so its gradient is $-\frac{1}{f^{\prime }(x)}$. Both use the point-gradient line equation.

Differentiate to get $f^{\prime }(x)$, substitute the $x$-value to find the gradient $m$, find the $y$-coordinate from $f(x)$, then substitute into $y-y_1=m(x-x_1)$. Simplify to the required form — usually $y=mx+c$.

Set $f^{\prime }(x)=0$ and solve for $x$ to locate stationary points, then find the matching $y$-values. Classify each using $f^{\prime \prime }(x)$: positive means a local minimum, negative means a local maximum, zero means you need a first-derivative sign check or point of inflexion test.

The most common one I mark down every year: using $f^{\prime }(x)$ itself as the normal gradient instead of $-1/f^{\prime }(x)$. Second is forgetting to find the $y$-coordinate before writing the line equation, leaving an answer with $x_1$ still undefined.

It depends on the paper. Paper 1 in AA is non-calculator, so you differentiate and solve by hand. Paper 2 (both AA and AI) allows a GDC, and AI students can often verify a gradient or find an intersection point numerically instead of algebraically.

Optimisation questions ask you to maximise or minimise a real quantity — area, volume, cost — by writing an expression in one variable, differentiating, setting it to zero, and checking it's a maximum or minimum with the second derivative. It's the same stationary-point method applied in context.

The core method — differentiate, set to zero, classify — is identical at SL and HL. HL just adds harder functions to differentiate first: composite, implicit and parametric curves, so the calculus itself, not the tangent/normal logic, is what gets tougher.

Yes — AI leans more on GDC-supported numerical differentiation and real-world modelling contexts, while AA expects fully worked algebraic differentiation by hand, especially in Paper 1. Both syllabuses use the identical tangent/normal formulas and classification rules underneath.

Full marks come from method discipline, not raw ability — showing every step (derivative, substitution, gradient, equation, classification) even when the arithmetic is easy. Most lost marks in my own classes come from skipping the y-coordinate step or leaving stationary points unclassified.

Differential and integral calculus together typically make up around 25-30% of AA and roughly 20% of AI content, based on the syllabus weightings in the current IB Mathematics guides. Tangents, normals and stationary points are the most frequently recurring sub-skill within that block.

A mix works best: concise Revision Notes to relearn the method, Topical Worksheets to drill the mechanics until they're automatic, and a full Mock Paper to test the topic under real time pressure alongside everything else. All three are available by topic on revisionprep.com.