Practice test — Calculus 2.
20 questions drawn at random from the whole Calculus 2 programme, mixed as in a real exam sitting: scoring +1/−0.25/0. At the end you receive by email the correct answer to every question and your weak points by topic — with an account you see them straight away on this page (the guided solution unlocks with your first lesson).
Each test mixes every topic of one first-year course at an Italian university, like a real exam sitting.
Test in ItalianThe questions of this simulator are in Italian, as in the real exam.
Topics covered by the test
- Functions of several variables (limits, partial derivatives, gradient)
- Free and constrained maxima and minima (Lagrange)
- Double and triple integrals, change of variables
- Curves, line integrals and conservative fields
- Surfaces, flux, Gauss and Stokes
- Power series and Fourier series
- Second-order linear differential equations
Two worked examples from the test
Un campo vettoriale si dice conservativo su un dominio se...
Solution
Passo 1 — Definizione. è conservativo su se esiste una funzione scalare (detta potenziale) tale che in ogni punto di .
Passo 2. Questa proprietà è ciò che rende il lavoro di indipendente dal percorso.
Risposta corretta: «esiste una funzione potenziale tale che su ».
Il campo è conservativo? Verifica la condizione
Solution
Passo 1. . Calcolo e
Passo 2. Le due derivate incrociate coincidono: su un dominio semplicemente connesso (qui intero) questo garantisce la conservatività.
Verifica diretta. Il potenziale è : infatti ✓
Risposta corretta: «sì, è conservativo: entrambe le derivate incrociate valgono ».
In the online practice test you will find 18 more questions like these; the correct answer to every question arrives by email.
What the Calculus 2 exam is like
The written paper is almost always a double or triple integral with a change of variables, a constrained maxima and minima problem and an exercise on vector fields or surfaces; the oral revolves around the Gauss and Stokes theorems and the difference between a conservative and an irrotational field. It is a geometry exam more than a calculation one: anyone who draws the domain before integrating already has half the mark.
Where the exam is lost
Getting the domain of integration wrong, or forgetting the Jacobian in a change of variables: the factor r in polar coordinates, ρ² sin φ in spherical ones. The other classic is concluding that a field is conservative because it is irrotational, without saying anything about the domain: if the domain is not simply connected the implication fails.
How to read your score, in short.
What are the questions of the Calculus 2 test like?
20 multiple-choice questions (4 options), drawn at random from a larger bank and spread over the WHOLE course programme together, not one topic at a time: functions of several variables (limits, partial derivatives, gradient), free and constrained maxima and minima (Lagrange), double and triple integrals, change of variables, curves, line integrals and conservative fields, surfaces, flux, Gauss and Stokes, power series and Fourier series, second-order linear differential equations.
Is there a penalty for wrong answers?
Yes: +1 point for a correct answer, −0.25 for a wrong one, 0 for a question left blank — as in the TOLC (the Italian university admission tests). You should answer only when you are reasonably sure: it is the same logic as a real multiple-choice exam.
What does the final score mean?
15 out of 20 or more: you are ready, the programme is solid. 10-14: needs consolidating — review the topics of the questions you got wrong and try again. Under 10: to be reviewed calmly before you sit the test again.
Course still weak?
Let us look at your results together — the diagnostic is free — and I will tell you honestly what it takes to close the gap.