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

El método de Newton es una técnica iterativa para encontrar raíces aproximadas de funciones diferenciables y de valor real.Ayuda a resolver ecuaciones no lineales que son demasiado complejas para los métodos algebraicos estándar.Por ejemplo, el método de Newton puede estimar el tipo de interés a partir de una ecuación no lineal que modela el reembolso de préstamos de coche. Estas ecuaciones se escriben como y igual a f de x y a menudo se muestran gráficamente para desarrollar la fórmula.El proceso comienza con una suposición inicial, basada en una estimación aproximada de la raíz.En el punto adivinado, se traza una línea tangente usando la pendiente de la función. La intersección x de esta línea se convierte en una nueva estimación, que visualmente está más cerca de la raíz real.Esta nueva estimación proviene de una aproximación lineal. Es igual a la estimación inicial menos el valor de la función dividido por su derivada en esa estimación.El proceso se repite usando la nueva estimación. Con cada repetición, los valores a menudo se acercan más a la raíz real.Esto conduce a la fórmula general: la nueva estimación es igual a la estimación anterior menos el valor de la función dividido por su derivada.Cada paso refina la aproximación, haciendo del Método de Newton una herramienta iterativa eficaz para resolver ecuaciones no lineales.

método de newton

Newton’s Method

El M&#xE9;todo de Newton es una potente t&#xE9;cnica iterativa para aproximar las ra&#xED;ces de funciones reales y diferenciables, especialmente cuando las soluciones anal&#xED;ticas no son factibles. Este enfoque se utiliza ampliamente en computaci&#xF3;n cient&#xED;fica, ingenier&#xED;a y finanzas, donde las ecuaciones pueden ser demasiado complejas para los m&#xE9;todos algebraicos tradicionales. El m&#xE9;todo se basa en un proceso iterativo que refina una estimaci&#xF3;n inicial utilizando la derivada de la funci&#xF3;n para acercarse progresivamente a la soluci&#xF3;n verdadera. Matem&#xE1;ticamente, sigue la 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>donde:x_n = aproximaci&#xF3;n actual de la ra&#xED;zf(x_n) = valor de la funci&#xF3;n en x_nf&#x2032;(x_n) = derivada de la funci&#xF3;n en x_nx_n+1 = siguiente aproximaci&#xF3;n, calculada utilizando la estimaci&#xF3;n actual.Cada iteraci&#xF3;n se acerca a la ra&#xED;z real siempre que la estimaci&#xF3;n inicial sea razonablemente cercana y la funci&#xF3;n se comporte correctamente.Una aplicaci&#xF3;n pr&#xE1;ctica del M&#xE9;todo de Newton es el modelado financiero, como la estimaci&#xF3;n de tipos de inter&#xE9;s a partir de ecuaciones de pago no lineales. En estos contextos, las ecuaciones pueden no admitir soluciones expl&#xED;citas, pero el M&#xE9;todo de Newton puede converger eficientemente a una ra&#xED;z con pasos computacionales m&#xED;nimos, siempre que se elija una estimaci&#xF3;n inicial adecuada.Gracias a su eficiencia y r&#xE1;pida convergencia, el M&#xE9;todo de Newton sigue siendo una de las t&#xE9;cnicas m&#xE1;s potentes para el c&#xE1;lculo de ra&#xED;ces y la resoluci&#xF3;n de ecuaciones en matem&#xE1;ticas aplicadas y ciencias computacionales.A pesar de sus ventajas, el M&#xE9;todo de Newton no garantiza la convergencia en todos los casos. Si la derivada f&#x2032;(x_n) es cero o muy cercana a cero, la f&#xF3;rmula de actualizaci&#xF3;n puede llevar a una divisi&#xF3;n por un n&#xFA;mero peque&#xF1;o, lo que causa inestabilidad num&#xE9;rica. Adem&#xE1;s, una estimaci&#xF3;n inicial inadecuada puede provocar que el m&#xE9;todo diverja o entre en un ciclo en lugar de aproximarse a una ra&#xED;z. Por otra parte, para funciones con puntos de inflexi&#xF3;n, extremos locales o discontinuidades en la derivada, el m&#xE9;todo puede no aproximarse a la ra&#xED;z o converger a una soluci&#xF3;n indeseada. Por ello, un an&#xE1;lisis cuidadoso de la funci&#xF3;n y una estimaci&#xF3;n inicial bien elegida son fundamentales para garantizar la aplicaci&#xF3;n exitosa del M&#xE9;todo de Newton.

Método de Newton

El M&#xE9;todo de Newton es una potente t&#xE9;cnica iterativa para aproximar las ra&#xED;ces de funciones reales y diferenciables, especialmente cuando ...

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Calculus

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Unidad

Applications of Differentiation

Aplicaciones de la derivación

KEY TERMS AND CONCEPTS

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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...

critical numbers and the closed interval method

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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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Using Derivatives to Resolve Limit Ambiguity

Indeterminate forms appear when a limit gives an expression that cannot be read directly. Common examples are 0 divided by 0 and infinity divided by infinity. These results do not show the true behavior of the function near the point of interest, so more analysis is needed. L’Hôpital’s Rule is a useful method for handling these ambiguous limits. It applies when two differentiable functions approach 0 or infinity at the same time near the point being studied. Instead of finding the limit of the...

indeterminate products

Limits of Zero Times Infinity

Limits with a zero-times-infinity product are indeterminate. In this type of limit, one factor approaches zero while the other factor grows toward positive or negative infinity. The result is not obvious at first, because the product may approach zero, infinity, or a finite nonzero value. To evaluate these product limits, the expression is first rewritten as a quotient. This form makes it possible to use L'Hôpital's Rule, which compares the derivatives of the numerator and denominator. A...

guidelines for sketching a curve

Using Derivatives to Sketch Function Graphs

Sketching a function graph uses key features of the function to show its overall shape. A function plots as a curve on the coordinate plane, with the input on the horizontal axis and the output on the vertical axis. The first step is to find the domain, which is the set of input values where the function is defined. The domain shows how far the graph can extend left and right. Intercepts are also important reference points. The x-intercept is where the graph crosses the horizontal axis, and...

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

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Finding Car Stopping Distance with Integration

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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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