Free · Year 5 of high school · 50 minutes · 15 questions · no penalty

Practice maths class test — Year 5: curve sketching.

15 questions drawn at random from a bank of 25, on the block of the programme your teacher is finishing: every attempt is a different test. No penalty for wrong answers, so answer even if you are not sure. At the end you receive by email the correct answer to every question and your weak topics — with an account you see them straight away here on the page. The step-by-step solution unlocks with your first lesson.

A class test is the written test an Italian high-school class sits after each block of the programme; this practice version is multiple choice.

Test in ItalianThe questions of this simulator are in Italian, as in the real exam.

Test prepared by Gaetano Livornese · Aerospace engineer, Politecnico di Milano · maths and physics tutor · updated 13 October 2026

Programme covered by the test

Domain, symmetries and intercepts with the axes
Sign of the function within the domain
Vertical, horizontal and oblique asymptotes
Monotonicity and stationary points from the first derivative
Relative and absolute maxima and minima on a closed interval
Concavity and inflection points from the second derivative
Reading the graph of f and the graph of f′
Optimisation problems with a constraint

Two worked examples from the test

Question 1 · Studio di funzione completo e ottimizzazione · Studio di funzione completo

Determina il dominio della funzione

$f (x) = \frac{ln ( 4 - x )}{x + 3}$

A $[- 3, 4)$
B $(- \infty, 4)$
C $(- 3, 4)$ correct answer
D $(- 3, 4]$

Solution

Logaritmo. L'argomento deve essere positivo: $4 - x > 0$ , cioè $x < 4$ .

Radice al denominatore. Non basta $x + 3 \geq 0$ : essendo sotto la frazione va chiesta strettamente positiva, $x + 3 > 0$ , cioè $x > - 3$ .

Sistema. Le due condizioni insieme danno $- 3 < x < 4$ .

Dominio: $(- 3, 4)$ . Includere $x = - 3$ annullerebbe il denominatore, includere $x = 4$ annullerebbe l'argomento del logaritmo.

Question 2 · Studio di funzione completo e ottimizzazione · Studio di funzione completo

Per quali valori di $x$ risulta $f (x) > 0$ , se

$f (x) = \frac{ln x}{x - 3} ?$

A $x < 1$ oppure $x > 3$
B $1 < x < 3$
Csoltanto $x > 3$
D $0 < x < 1$ oppure $x > 3$ correct answer

Solution

Dominio. Il logaritmo esiste per $x > 0$ e il denominatore chiede $x \neq = 3$ : $D = (0, 3) \cup (3, + \infty)$ . Ogni conclusione vale solo dentro il dominio.

Segno del numeratore. $ln x > 0$ per $x > 1$ , $ln x < 0$ per $0 < x < 1$ .

Segno del denominatore. $x - 3 > 0$ per $x > 3$ .

Tabella dei segni. - $0 < x < 1$ : negativo su negativo $\Rightarrow f > 0$ - $1 < x < 3$ : positivo su negativo $\Rightarrow f < 0$ - $x > 3$ : positivo su positivo $\Rightarrow f > 0$

Risposta: $0 < x < 1$ oppure $x > 3$ . Scrivere $x < 1$ è l'errore tipico: i valori negativi non appartengono al dominio e non vanno mai inclusi nel risultato dello studio del segno.

In the online practice test you will find 13 more questions like these; the correct answer to every question arrives by email.

Prefer paper? Download the test as a PDF (15 questions, 5 pages · 92 KB, ready to print). You receive the solutions by email after taking it online.

Download the PDF

What a maths test on curve sketching looks like

A full sketch to take all the way to the graph — domain, symmetries, sign, limits and asymptotes, first derivative with monotonicity and extremes, second derivative with concavity and inflection points — plus, usually, an optimisation problem with a constraint.

The mistake that costs the grade

Looking for maxima and minima outside the domain, or declaring a stationary point a maximum without studying the sign of f′ around it. In optimisation problems, forgetting to check that the solution found is geometrically acceptable — a negative length must be discarded.

How to use this practice test

Take it once cold, without revising and with the timer on: it is there to show you where you stand, not to look good. Then look at your weak topics, revise only those — usually two or three, not the whole chapter — and retake it after a few days. The 15 questions are drawn from a bank of 25, so the second attempt is not the same test and the scores are genuinely comparable.

Is the score not enough?

See which topics lost you the most marks, then read the step-by-step guide to curve sketching — or we look at your results together: the diagnostic is free.

Book the free diagnostic call The other Year 5 tests