Name: using derivatives to read a graph
Uploaded: 2026-01-12T12:24:56.000Z
Duration: 1 min 22 s
Channel: JoVE

Consideriamo una funzione; le sue prime e seconde derivate vengono utilizzate per disegnare la sua curva.La prima derivata è zero quando x è uguale a quattro, e indefinita quando x è uguale a zero o x è a sei. Questi punti critici definiscono gli intervalli per l'analisi.Il segno della prima derivata determina se la curva è crescente o decrescente.Un cambio di segno da negativo a positivo in x uguale a zero dà un minimo locale.Un cambiamento da positivo a negativo in x uguale a quattro dà un massimo locale. Se il segno non cambia a x uguale a sei, non esiste alcun estremo in quel punto.La seconda derivata è negativa sugli intervalli da meno infinito a 0 e da 0 a 6, il che mostra che il grafico è concavo verso il basso su questi intervalli.Ma la seconda derivata è positiva sull'intervallo x maggiore di sei, il che dà il grafico concavo verso l'alto su questo intervallo.Ad esempio, consideriamo una funzione di profitto. La prima derivata monitora l'aumento o la diminuzione del profitto e ne identifica i valori estremi.La seconda derivata mostra come cambia il tasso di profitto, il che aiuta a delineare la funzione.

studio qualitativo delle curve e delle derivate

Curve Sketching and Derivatives

Comprendere il comportamento di una funzione attraverso la derivata prima e la derivata seconda &#xE8; essenziale per analizzarne il grafico. Le derivate forniscono indicazioni su dove una funzione cresce o decresce, dove assume massimi o minimi locali e come varia la curvatura nei diversi intervalli.La derivata prima di una funzione rivela la pendenza della retta tangente in un punto qualunque. I punti in cui la derivata prima &#xE8; nulla oppure non &#xE8; definita sono detti punti critici, poich&#xE9; spesso indicano possibili estremi o cambiamenti di comportamento. Un minimo locale si verifica quando la derivata prima passa da negativa a positiva, mentre un massimo locale si ha quando passa da positiva a negativa. Se, in corrispondenza di un punto critico, la derivata prima non cambia segno, non &#xE8; presente alcun estremo.La derivata seconda fornisce informazioni sulla concavit&#xE0; della funzione. Quando la derivata seconda &#xE8; negativa, il grafico &#xE8; concavo verso il basso, il che indica che la derivata prima sta diminuendo. Ne risulta una curva che si flette verso il basso. Viceversa, una derivata seconda positiva implica che la funzione &#xE8; concava verso l&#x2019;alto: la pendenza della derivata prima aumenta e la curva si flette verso l&#x2019;alto. I cambiamenti di concavit&#xE0; suggeriscono la presenza di punti di flesso, nei quali la curva passa da un tipo di concavit&#xE0; a un altro.Esaminando i segni e l&#x2019;andamento delle derivate prima e seconda su intervalli diversi, &#xE8; possibile costruire uno schizzo accurato della funzione. Questo approccio integrato consente di individuare le regioni di crescita o decrescita, di localizzare massimi e minimi locali e di determinare la forma complessiva del grafico della funzione.ApplicazioniAd esempio, si consideri un modello di profitto rappresentato da una funzione quadratica: \begin{equation} P(x) = -2x^2 + 12x - 10 \end{equation}Il grafico rappresenta una curva del profitto, in cui x indica la spesa pubblicitaria (in migliaia di USD) e P(x) rappresenta il profitto (anch&#x2019;esso in migliaia di USD). La curva assume la forma di una U rovesciata, simile a una collina. In termini matematici, tale forma &#xE8; descritta come concava verso il basso.A bassi livelli di spesa pubblicitaria, l&#x2019;aumento dell&#x2019;investimento incrementa le vendite e il profitto in modo efficace. Tuttavia, man mano che la spesa continua a crescere, il profitto aggiuntivo generato da ogni ulteriore unit&#xE0; di spesa pubblicitaria inizia a ridursi. Ci&#xF2; riflette rendimenti decrescenti. Se la spesa diventa eccessiva, il profitto alla fine diminuisce, perch&#xE9; il costo della pubblicit&#xE0; aggiuntiva supera i ricavi che essa genera.La pendenza della curva &#xE8; descritta dalla derivata prima, e il profitto massimo si realizza nel punto in cui tale pendenza si annulla. A questo livello di spesa, la curva raggiunge il suo valore pi&#xF9; elevato.La derivata seconda della funzione &#xE8; un numero costante negativo. Una derivata seconda negativa indica che l&#x2019;intera curva &#xE8; concava verso il basso, confermando che tale concavit&#xE0; si mantiene in ogni punto.

Studio qualitativo delle curve e delle derivate

Comprendere il comportamento di una funzione attraverso la derivata prima e la derivata seconda &#xE8; essenziale per analizzarne il grafico. Le derivate ...

Education

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JoVE Core

Calculus

Analisi matematica

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Unità

Applications of Differentiation

Applicazioni della differenziazione

KEY TERMS AND CONCEPTS

TERMINI E CONCETTI CHIAVE

using derivatives to read a graph

Derivatives help students read the graph of a function by showing where the graph rises, falls, and bends. The first derivative tells the slope of the tangent line at each point. The second derivative shows how the curve changes shape over an interval.

Points where the first derivative is zero or undefined are critical points. These points are important because they may mark a local maximum, a local minimum, or no extremum at all. A local minimum appears when the first derivative changes from negative to positive. A local maximum appears when it changes from positive to negative.

If the first derivative does not change sign at a critical point, then the function does not have an extremum there. The second derivative adds information about concavity, which describes whether the graph bends upward or downward. A negative second derivative means the graph is concave downward. A positive second derivative means the graph is concave upward.

When the second derivative is negative, the slope of the first derivative is decreasing, so the curve bends downward. When the second derivative is positive, the slope of the first derivative is increasing, so the curve bends upward. A change in concavity can signal an inflection point, where the graph switches from one type of bending to the other.

By checking the signs of the first and second derivatives on different intervals, students can sketch a function more accurately. This process identifies where the function increases or decreases, where it has local maxima and minima, and how its overall shape develops. The same ideas apply to a quadratic profit model where x is advertising spend in thousands of USD and P(x) is profit in thousands of USD.

In that example, the graph looks like a hill, or an inverted U shape. That shape is concave downward, so the profit curve bends downward throughout. At low advertising levels, more spending raises sales and profit efficiently. As spending keeps rising, the extra profit from each additional unit of advertising gets smaller, which shows diminishing returns.

If advertising becomes too high, profit can start to fall because the cost of the extra advertising is greater than the revenue it brings in. The maximum profit occurs where the first derivative is zero, because the slope of the curve becomes flat there. The second derivative is a constant negative number, which confirms that the entire profit function is concave downward.

Using Derivatives to Read a Graph

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Using First Derivatives to Find Graph Changes

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Using Second Derivatives to Read Curvature

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Classifying Critical Points with the Second Derivative

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curve sketching and derivatives

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Using Derivatives to Resolve Limit Ambiguity

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Using Derivatives to Sketch Function Graphs

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key terms and concepts

Learn how the first and second derivatives reveal increase, decrease, local maxima, minima, concavity, and inflection points in function graphs.

Using Derivatives to Read a Graph | JoVE K12

The JoVE Core Calculus collection provides a comprehensive visual guide to the entire calculus sequence, seamlessly connecting mathematical theory to real-world applications in physics and economics. Through dynamic animated lessons, the course transitions smoothly from foundational single-variable concepts to advanced multivariable topics. Students will explore everything from standard differentiation and integration to spatial vectors, multi-dimensional derivatives, and complex vector fields. By breaking down abstract coordinate systems, infinite series, and spatial motion, this structured approach builds the robust analytical skills and advanced mastery essential for STEM success.

calculus

Using Derivatives to Read a Graph

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