| layout | page |
|---|---|
| title | TMUA — calculus topics & question styles |
| permalink | /tmua-calculus/ |
The Test of Mathematics for University Admission (TMUA) is a multiple-choice admissions test (two papers). It rewards fast, rigorous reasoning, not long written proofs. Official past papers and practice materials come from Cambridge University Press & Assessment / test administrators — use those for authentic questions. This page is a topic map aligned with calculus-heavy preparation, not a copy of any exam.
- Questions are usually short: pick the correct option among fixed choices, often after one algebraic twist, a sign case, or a “which statement is always true?” logical filter.
- Graphical sense matters: increasing/decreasing intervals, where derivatives vanish, concavity, asymptotes, and domain restrictions (logs, roots, reciprocals).
- Numerical / inequality reasoning: compare values without heavy computation (e.g. using monotonicity of (e^x), (\ln x)).
- Power rule, linear combinations, constants.
- Product rule and quotient rule without hesitation.
- Chain rule on nested expressions (incl. (e^{kx}), (\ln(kx)), (\sin(kx+\phi)), ((ax+b)^n)).
- Implicit differentiation mood (e.g. circle (x^2+y^2=R^2), related “(dy/dx) at a point” structure).
- Standard derivatives: polynomials, sin/cos/tan, exp, log, (\sqrt{x}) (domain (x>0)).
- Tangent / normal lines from a point and slope.
- Stationary points: classify with first derivative sign change or second derivative where smooth.
- Max/min on an interval: check endpoints as well as critical points.
- Increasing / decreasing on an interval from sign of (f').
- Concavity and inflection from sign of (f'') (where defined).
- Rates of change phrased as word problems (units + chain structure).
- Antiderivatives of standard forms (polynomial, (1/x), (e^{kx}), sin/cos with constants).
- Definite integrals as signed area; splitting intervals when the function changes sign.
- FTC-flavoured reasoning: if (F'=f), interpret (\int_a^b f) as (F(b)-F(a)).
- Substitution / recognition (no need for exam-length integration by parts for every TMUA item, but pattern-spotting helps).
- Limits motivated by graph behaviour and algebraic simplification (remove removable discontinuities).
- Vertical asymptotes where denominator (\to 0) (watch sign from left/right).
- Growth comparisons (e.g. polynomial vs exponential for large (x)) at a qualitative level.
- Piecewise definitions / |x| — corners, different left/right slopes.
- Domain of (\ln), (\sqrt{\cdot}) — “where is the expression real?”
- Rational functions — asymptotes, holes only if curriculum expects simplifying factors.
- Geometric series intuition ((|r|<1)) and partial sums.
- Taylor / Maclaurin mood: “local polynomial approximation” and which derivative counts match (often as reasoning, not long series manipulation).
- “Exactly one statement is true” — eliminate ambiguity options using counterexamples or monotonicity.
- Transform a messy expression until a derivative or integral is recognizable.
- Compare (f(a)), (f(b)) using (f') sign on ([a,b]).
- Parameter problems: find (k) so that a min/max or tangent condition holds (solve small equations cleanly).
- Official TMUA specimen & past papers (timed, full papers).
- Error log: classify mistakes (algebra, domain, sign, chain rule order, FTC bounds).
- Pair with this game on derivatives-as-slope and area/Riemann intuition — visual hooks only; TMUA timing still needs pen-and-paper drill.
- [MAT — calculus & reasoning]({% link mat-calculus.md %}) — Oxford-style MAT lens (separate from TMUA).
- [AP Calculus BC — prep]({% link ap-calculus-bc.md %}) — US BC syllabus & polar/formulas.
- [AP Physics C — prep]({% link ap-physics-c.md %}) — calculus-first physics map.
- [Math concepts & snippets]({% link math-concepts.md %}) — game-wide article index.
- [Gameplay]({% link gameplay.md %}) — how levels map to graphs and derivatives in-game.
Unofficial study aid for First Principles players; not affiliated with TMUA administrators.