Name: newton’s method for finding roots
Uploaded: 2026-01-12T12:24:54.000Z
Duration: 1 min 30 s
Channel: JoVE

Il Metodo di Newton è una tecnica iterativa per trovare radici approssimative di funzioni differenziabili a valori reali.Aiuta a risolvere equazioni non lineari troppo complesse per i metodi algebrici standard.Ad esempio, il Metodo di Newton può stimare il tasso di interesse da un'equazione non lineare che modella il rimborso del prestito auto. Queste equazioni sono scritte come y uguale a f di x e spesso sono rappresentate graficamente per sviluppare la formula.Il processo inizia con una stima iniziale, basata su una stima approssimativa della radice.Nel punto ipotetico, viene tracciata una retta tangente usando la pendenza della funzione. L'intercetta x di questa linea diventa una nuova stima, visivamente più vicina alla radice reale.Questa nuova stima deriva da approssimazione lineare. È uguale alla stima iniziale meno il valore della funzione diviso per la sua derivata in quella stima.Il processo viene ripetuto utilizzando la nuova stima. Ad ogni ripetizione, i valori spesso si avvicinano alla radice reale.Questo porta alla formula generale: la nuova stima è uguale alla stima precedente meno il valore della funzione diviso per la sua derivata.Ogni passaggio affina l'approssimazione, rendendo il Metodo di Newton uno strumento iterativo efficace per risolvere equazioni non lineari.

metodo di newton

Newton’s Method

Il metodo di Newton &#xE8; una potente tecnica iterativa per approssimare le radici di funzioni a valori reali e differenziabili, soprattutto quando le soluzioni analitiche risultano poco pratiche. Questo approccio &#xE8; ampiamente utilizzato nel calcolo scientifico, nell&#x2019;ingegneria e in finanza, dove le equazioni possono essere troppo complesse per essere affrontate con i metodi algebrici tradizionali. Il metodo si basa su un processo iterativo che raffina una stima iniziale utilizzando la derivata della funzione per avvicinarsi progressivamente alla soluzione effettiva. In termini matematici, segue la formula ricorsiva:<figure class="image image-style-align-center image_resized" style="display:table;margin-left:auto;margin-right:auto;width:50%;"><img style="aspect-ratio:300/48;" src="https://cloudfront.jove.com/files/ftp_upload/19849/19849_PT_Equation_1_editor_19849.svg" alt="19849_PT_Equation_1.svg" /></figure>dove:x_n = approssimazione corrente della radicef(x_n) = valore della funzione in x_nf&#x2032;(x_n) = derivata della funzione in x_nx_n+1 = approssimazione successiva, calcolata utilizzando la stima corrente.Ogni iterazione avvicina l&#x2019;approssimazione alla radice effettiva, purch&#xE9; la stima iniziale sia ragionevolmente vicina e la funzione presenti un comportamento regolare.Un&#x2019;applicazione pratica del metodo di Newton riguarda la modellazione finanziaria, ad esempio la stima dei tassi d&#x2019;interesse a partire da equazioni di rimborso non lineari. In tali contesti, le equazioni possono non prestarsi a soluzioni esplicite, ma il metodo di Newton pu&#xF2; convergere in modo efficiente verso una radice con un numero ridotto di passi di calcolo, a condizione che venga scelta una stima iniziale adeguata.Grazie alla sua efficienza e alle sue propriet&#xE0; di convergenza rapida, il metodo di Newton-Raphson rimane una delle tecniche pi&#xF9; potenti per la ricerca di radici e la risoluzione di equazioni nella matematica applicata e nelle scienze computazionali.Nonostante i suoi vantaggi, il metodo di Newton non garantisce la convergenza in tutti i casi. Se la derivata f&#x2032;(x_n) &#xE8; nulla o molto vicina a zero, la formula di aggiornamento pu&#xF2; determinare una divisione per un numero molto piccolo, causando instabilit&#xE0; numerica. Inoltre, una stima iniziale inadeguata pu&#xF2; far divergere il metodo o indurlo a entrare in un ciclo, invece di avvicinarsi a una radice. In aggiunta, per funzioni con punti di flesso, estremi locali o discontinuit&#xE0; della derivata, il metodo pu&#xF2; non riuscire ad avvicinarsi alla radice oppure pu&#xF2; convergere verso una soluzione non desiderata. Per questo motivo, un&#x2019;analisi accurata della funzione e una scelta appropriata della stima iniziale sono essenziali per garantire la corretta applicazione del metodo di Newton.

Metodo di Newton

Il metodo di Newton &#xE8; una potente tecnica iterativa per approssimare le radici di funzioni a valori reali e differenziabili, soprattutto quando le ...

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newton’s method for finding roots

Newton’s Method is an iterative way to find the roots of real-valued, differentiable functions. A root is the value of x that makes a function equal to zero. This method is useful when an exact algebraic answer is hard to find.

The process starts with an initial guess. Newton’s Method then uses the function value and its derivative, which is the slope of the function at that point, to build a better estimate. The next approximation is found from the recursive formula x n+1 = x n - f(x n) / f'(x n). Each step can move the estimate closer to the true root when the first guess is reasonable and the function behaves well.

This method is widely used in scientific computing, engineering, and finance. One practical example is financial modeling, where it can help estimate interest rates from nonlinear repayment equations. These problems often do not have a simple explicit solution, so Newton’s Method can reach a root in only a few steps.

Newton’s Method does not always converge. If the derivative f'(x n) is zero or very close to zero, the formula can become unstable because it divides by a small number. A poor starting guess can also make the method diverge or cycle instead of approaching a root.

The method can also fail near inflection points, local extrema, or places where the derivative is discontinuous. In some cases, it may converge to the wrong solution. That is why a careful look at the function and a well-chosen starting point are important for successful root finding.

Newton’s Method for Finding Roots

absolute and local extreme values

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Absolute and local extreme values show the highest and lowest points on a function graph. Absolute maximum and absolute minimum values describe the highest and lowest values across the entire domain. Local maximum and local minimum values describe the highest and lowest values within a smaller part of the graph. Periodic functions such as sine and cosine can have extreme values at many points. Their repeating pattern creates multiple local extrema. In these cases, the derivative becomes zero...

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Finding Absolute Extrema on a Closed Interval

Absolute extrema on a closed interval are found by checking critical numbers and the endpoints. A function’s maximum and minimum values help describe its overall behavior across its domain. These values are called extrema, and they can be local or absolute. Local extrema are peaks and valleys in a limited region. Absolute extrema are the highest or lowest points on the entire interval. Critical numbers are the domain values where the derivative is zero or does not exist. According to Fermat’s...

rolle’s theorem

Rolle’s Theorem and Zero Slope Points

Rolle’s Theorem explains when a differentiable function must have a flat tangent inside an interval. It applies to a real-valued function on a closed interval, which means the endpoints are included. The function must also be continuous on the whole interval and differentiable on the open interval, where the interior points lie. The theorem uses one more key condition. The function must have the same value at both endpoints. When that happens, Rolle’s Theorem guarantees at least one point...

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first derivatives and the shape of a graph

first-derivatives-and-the-shape-of-a-graph

Using First Derivatives to Find Graph Changes

The first derivative shows how a function changes across its domain. In calculus, the derivative is written as f'(x). It tells us whether the graph is rising, falling, or turning at a given point. A function is increasing on an interval when its derivative is positive. That means the slope of the tangent line is positive. A function is decreasing on an interval when its derivative is negative. In that case, the tangent line slopes downward. These sign changes help describe the overall shape...

first derivative test: problem solving

Finding Turning Points with the First Derivative

The first derivative test is used to find turning points on a smooth curve. It shows where a function changes from increasing to decreasing, or from decreasing to increasing. In this example, the curve can model an asset price that falls to a low point, rebounds, and then gradually declines again. The process starts by finding the first derivative of the function. Because the function is a product of a polynomial term and an exponential term, the product rule is needed. After differentiating,...

second derivatives and the shape of a graph

second-derivatives-and-the-shape-of-a-graph

Using Second Derivatives to Read Curvature

The second derivative shows how a graph bends. It gives key information about curvature and how that curvature changes over an interval. These ideas help with graph analysis in math and in real-world settings such as road elevation, population growth, and economic trends. A function f(x) is concave upward on an interval when its graph lies above all of its tangent lines. In this case, the second derivative is positive. That means the slope of the tangent line is increasing, and the graph bends...

second derivative test: problem solving

Classifying Critical Points with the Second Derivative

The second derivative test helps classify critical points of a function. Critical points are places where the first derivative is zero or undefined. They are possible locations for local maxima and local minima, but they do not always become one or the other. The second derivative shows concavity, or whether a graph bends up or down. If f''(x) &gt; 0, the function is concave up, and the critical point is a local minimum. If f''(x) &lt; 0, the function is concave down, and the critical point is...

curve sketching and derivatives

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...

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indeterminate products

Limits of Zero Times Infinity

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guidelines for sketching a curve

Using Derivatives to Sketch Function Graphs

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slant asymptotes

Rational Functions with Slant Asymptotes

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optimization problems

Pipe Length Around a Corner

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application of differentiation to business

application-of-differentiation-to-business

Maximizing Bakery Revenue with Calculus

Calculus helps businesses make pricing decisions that can improve revenue. In this example, a bakery uses a cupcake model to find the best price and daily sales volume. The goal is to see how changes in price affect demand and total revenue. The demand function shows the link between price and how many cupcakes are sold each day. In this model, p(x) gives the price per cupcake when x cupcakes are sold. Lower prices lead to higher demand, so the function captures how sales change with price. <br...

newton’s method

Newton’s Method is an iterative way to find the roots of real-valued, differentiable functions. A root is the value of x that makes a function equal to zero. This method is useful when an exact algebraic answer is hard to find. The process starts with an initial guess. Newton’s Method then uses the function value and its derivative, which is the slope of the function at that point, to build a better estimate. The next approximation is found from the recursive formula x n+1 = x n - f(x n) /...

the antiderivative of a function

Finding Original Functions with Antiderivatives

Antiderivatives help reconstruct an original function from its derivative. In calculus, an antiderivative of a function f(x) is a function F(x) whose derivative gives back f(x). This idea works like reversing differentiation to recover the starting form. Antiderivatives are not unique because differentiation removes constant terms. For that reason, the general form includes an arbitrary constant C. That constant represents unknown initial conditions when a function is rebuilt from its...

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Antiderivative Graph Shape and Slope

Antiderivative graphs show how a function accumulates over time. In calculus, an antiderivative tracks that buildup, much like the changing position of a rolling ball as its velocity changes. The graph of the antiderivative reflects the total effect of the original function's values. The sign of the original function helps determine the antiderivative's direction. When the function is positive, the antiderivative rises because the accumulation grows. When the function is negative, the...

application of antiderivatives: linear motion

application-of-antiderivatives-linear-motion

Finding Car Stopping Distance with Integration

Integration connects acceleration, velocity, and displacement in linear motion. It works as the reverse of a derivative, which tells how one quantity changes with another. In physics, this makes it possible to move from a rate of change back to the motion itself. A car provides a clear example. It is moving at a steady 20 meters per second when an obstacle appears 800 meters ahead. To stop in time, the car slows with a constant acceleration. That acceleration tells how the car’s velocity...

key terms and concepts

Learn how Newton’s Method uses derivatives and repeated guesses to find roots of equations, plus why a good starting point matters for convergence.

Newton’s Method for Finding Roots | 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

Newton’s Method for Finding Roots

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