Question

Suppose that the graph of the function $f(x)$ has slope -2 at the point $(1,2) .$ If the Newton-Raphson algorithm is used to find a root of $f(x)=0$ with the initial guess $x_{0}=1,$ what is $x_{1} ?$

Name: Problem 20, Chapter 11: Calculus and Its Applications
Uploaded: 2021-09-17T22:24:39-08:00
Duration: 2 min 21 s
Channel: Stanley Enemuo
Description: Suppose that the graph of the function f(x) has slope -2 at the point (1,2) . If the Newton-Raphson algorithm is used to find a root of f(x)=0 with the initial guess x0=1, what is x1 ?

   Suppose that the graph of the function $f(x)$ has slope -2 at the point $(1,2) .$ If the Newton-Raphson algorithm is used to find a root of $f(x)=0$ with the initial guess $x_{0}=1,$ what is $x_{1} ?$

Calculus and Its Applications

Larry J. Goldstein,… 14th Edition

Chapter 11, Problem 20

Instant Answer

Solved by Expert Stanley Enemuo

09/17/2021

Step 1

This means that the derivative of the function $f'(x)$ at $x=1$ is -2. Show more…

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Suppose that the graph of the function $f(x)$ has slope -2 at the point $(1,2) .$ If the Newton-Raphson algorithm is used to find a root of $f(x)=0$ with the initial guess $x_{0}=1,$ what is $x_{1} ?$

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Key Concepts

Derivative

A mathematical tool that represents the instantaneous rate of change or the slope of a function at a given point. In the context of the Newton-Raphson method, the derivative f'(x) is used to determine the slope of the tangent line at the current approximation point, which is essential for calculating the next iteration towards the root.

Newton-Raphson Method

An iterative numerical technique used to find approximations of the roots of a real-valued function. It works by using the tangent line at an estimated point to generate a closer approximation to the actual root, following the formula x??? = x? - f(x?) / f'(x?). This method converges rapidly under appropriate conditions if the function is sufficiently smooth and the starting guess is close to the actual root.

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