Practice test — Calculus 1.
20 questions drawn at random from the whole Calculus 1 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.
Want to review the theory first? Read Limits explained properly.
Want to see the reasoning before you try? Look at the secant turning into a tangent, in an interactive figure.
Test in ItalianThe questions of this simulator are in Italian, as in the real exam.
Topics covered by the test
- Limits and indeterminate forms
- Derivatives and curve sketching
- Integrals (indefinite, definite, improper)
- Sequences and numerical series
- Continuity and theorems (Weierstrass, zeros, Lagrange)
- Taylor and Maclaurin expansions
- First-order differential equations
Two worked examples from the test
Cosa garantisce il teorema di Weierstrass per una funzione continua su un intervallo chiuso e limitato ?
Solution
Passo 1 — Enunciato. Il teorema di Weierstrass afferma che ogni funzione continua su un intervallo chiuso e limitato ammette (almeno) un punto di massimo assoluto e uno di minimo assoluto.
Passo 2. Le ipotesi sono essenziali: su un intervallo aperto o illimitato una funzione continua può non avere massimo o minimo (es. su ).
Risposta corretta: «l'esistenza di massimo e minimo assoluti di su ».
è continua su , con e . Cosa si può concludere?
Solution
Passo 1 — Teorema degli zeri (Bolzano). Se è continua su e (segni opposti agli estremi), allora esiste almeno un con .
Passo 2. Qui e : il prodotto è negativo, quindi le ipotesi sono soddisfatte e la conclusione vale.
Risposta corretta: «esiste almeno uno zero di in , per il teorema degli zeri». Il teorema richiede solo la continuità, non la derivabilità.
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 1 exam is like
Almost everywhere the exam is a written paper of two or three hours with four or five exercises — a limit or a curve sketch, an integral, a series and often a differential equation — followed by an oral on definitions and statements of theorems. The written part is passed with technique, the oral with hypotheses: knowing how to apply l’Hôpital’s rule is not enough if you cannot say when you are allowed to.
Where the exam is lost
Where Calculus 1 is almost always lost is not the hard integral: it is the Year 3 high-school algebra inside the calculus exercise. Next come Taylor expansions truncated at the wrong order, which lead the limit to a plausible but false result, and convergence tests applied to series that do not meet their hypotheses.
How to read your score, in short.
What are the questions of the Calculus 1 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: limits and indeterminate forms, derivatives and curve sketching, integrals (indefinite, definite, improper), sequences and numerical series, continuity and theorems (Weierstrass, zeros, Lagrange), Taylor and Maclaurin expansions, first-order 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.