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

O Método de Newton é uma técnica iterativa para encontrar raízes aproximadas de funções diferenciáveis e de valor real.Ele ajuda a resolver equações não lineares que são complexas demais para métodos algébricos padrão.Por exemplo, o Método de Newton pode estimar a taxa de juros a partir de uma equação não linear que modela o pagamento do empréstimo de carro. Essas equações são escritas como y igual a f de x e frequentemente são mostradas graficamente para desenvolver a fórmula.O processo começa com um palpite inicial, baseado em uma estimativa aproximada da raiz.No ponto adivinhado, uma linha tangente é traçada usando a inclinação da função. O x-intercepto dessa linha torna-se uma nova estimativa, que visualmente está mais próxima da raiz real.Essa nova estimativa vem da aproximação linear. É igual à estimativa inicial menos o valor da função dividido pela derivada nessa estimativa.O processo é repetido usando a nova estimativa. A cada repetição, os valores frequentemente se aproximam da raiz real.Isso leva à fórmula geral: a nova estimativa é igual à estimativa anterior menos o valor da função dividido pela derivada.Cada etapa refina a aproximação, tornando o Método de Newton uma ferramenta iterativa eficaz para resolver equações não lineares.

método de newton

Newton’s Method

O M&#xE9;todo de Newton &#xE9; uma t&#xE9;cnica iterativa poderosa para aproximar as ra&#xED;zes de fun&#xE7;&#xF5;es reais e diferenci&#xE1;veis, particularmente quando solu&#xE7;&#xF5;es anal&#xED;ticas s&#xE3;o impratic&#xE1;veis. Essa abordagem &#xE9; amplamente utilizada em computa&#xE7;&#xE3;o cient&#xED;fica, engenharia e finan&#xE7;as, onde as equa&#xE7;&#xF5;es podem ser complexas demais para serem resolvidas por m&#xE9;todos alg&#xE9;bricos tradicionais. O m&#xE9;todo baseia-se em um processo iterativo que refina uma estimativa inicial, usando a derivada da fun&#xE7;&#xE3;o, para se aproximar progressivamente da solu&#xE7;&#xE3;o verdadeira. Matematicamente, ele segue a f&#xF3;rmula recursiva:<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>onde:x_n = aproxima&#xE7;&#xE3;o atual da raizf(x_n) = valor da fun&#xE7;&#xE3;o em x_nf&#x2032;(x_n) = derivada da fun&#xE7;&#xE3;o em x_nx_n+1 = pr&#xF3;xima aproxima&#xE7;&#xE3;o, calculada a partir da estimativa atual.Cada itera&#xE7;&#xE3;o se aproxima da raiz verdadeira, desde que a estimativa inicial esteja razoavelmente pr&#xF3;xima e a fun&#xE7;&#xE3;o se comporte bem.Uma aplica&#xE7;&#xE3;o pr&#xE1;tica do m&#xE9;todo de Newton &#xE9; na modelagem financeira, como na estimativa de taxas de juros a partir de equa&#xE7;&#xF5;es de pagamento n&#xE3;o lineares. Nesses contextos, as equa&#xE7;&#xF5;es podem n&#xE3;o se prestar a solu&#xE7;&#xF5;es expl&#xED;citas, mas o m&#xE9;todo de Newton pode convergir de forma eficiente para uma raiz com um n&#xFA;mero m&#xED;nimo de passos computacionais, desde que seja escolhida uma estimativa inicial adequada.Devido &#xE0; sua efici&#xEA;ncia e &#xE0;s r&#xE1;pidas propriedades de converg&#xEA;ncia, o m&#xE9;todo de Newton-Raphson permanece uma das t&#xE9;cnicas mais poderosas para encontrar ra&#xED;zes e resolver equa&#xE7;&#xF5;es em matem&#xE1;tica aplicada e nas ci&#xEA;ncias computacionais.Apesar de suas vantagens, o m&#xE9;todo de Newton n&#xE3;o garante a converg&#xEA;ncia em todos os casos. Se a derivada f&#x2032;(x_n) for zero ou muito pr&#xF3;xima de zero, a f&#xF3;rmula de atualiza&#xE7;&#xE3;o pode levar &#xE0; divis&#xE3;o por um n&#xFA;mero muito pequeno, causando instabilidade num&#xE9;rica. Al&#xE9;m disso, estimativas iniciais inadequadas podem levar o m&#xE9;todo a divergir ou a entrar em um ciclo, em vez de se aproximar de uma raiz. Ademais, para fun&#xE7;&#xF5;es com pontos de inflex&#xE3;o, extremos locais ou descontinuidades na derivada, o m&#xE9;todo pode n&#xE3;o conseguir se aproximar da raiz ou pode convergir para uma solu&#xE7;&#xE3;o indesejada. Por isso, uma an&#xE1;lise cuidadosa da fun&#xE7;&#xE3;o e uma estimativa inicial bem escolhida s&#xE3;o cruciais para assegurar a aplica&#xE7;&#xE3;o bem-sucedida do m&#xE9;todo de Newton.

Método de Newton

O M&#xE9;todo de Newton &#xE9; uma t&#xE9;cnica iterativa poderosa para aproximar as ra&#xED;zes de fun&#xE7;&#xF5;es reais e diferenci&#xE1;veis, ...

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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

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Using Second Derivatives to Read Curvature

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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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