Use Newton’s method with initial approximation x1 =-1to find x2 , the second approximation to the root of the equation x4-x-1=0. Explain how the methodworks by first graphing the function and its tangent lineat (1, -1) .

Answered: Use Newton’s method with initial…

Functions and Change: A Modeling Approach to College Algebra (MindTap Course List)

6th Edition

ISBN:9781337111348

Author:Bruce Crauder, Benny Evans, Alan Noell

Publisher:Bruce Crauder, Benny Evans, Alan Noell

Chapter2: Graphical And Tabular Analysis

Section2.1: Tables And Trends

Problem 1TU: If a coffee filter is dropped, its velocity after t seconds is given by v(t)=4(10.0003t) feet per...

See similar textbooks

Related questions

Question

Use Newton’s method with initial approximation x₁ =-1to find x₂ , the second approximation to the root of the equation x⁴-x-1=0. Explain how the methodworks by first graphing the function and its tangent lineat (1, -1) .

Expert Solution

Step 1

The given function is ,

$f (x) = x^{4} - x - 1$ with initial approximation $x_{1} = - 1$ .

To Find: Second approximation using Newton's method.

To Explain: Newton's method graphically by drawing tangent line at (1,-1).

Step 2

The given function is ,

$f (x) = x^{4} - x - 1$ .

$f' (x) = 4 x^{3} - 1$ .

Iteration rule of Newton's method:

$x_{n + 1} = x_{n} - \frac{f (x_{n})}{f' (x_{n})}$ .

Here first approximation is $x_{1} = - 1$ .

1st iteration:

$f (x_{1}) = f (- 1) = {(- 1)}^{4} - (- 1) - 1 = 1 f' (x_{1}) = f' (- 1) = 4 \cdot {(- 1)}^{3} - 1 = - 5 x_{2} = x_{1} - \frac{f (x_{1})}{f' (x_{1})} x_{2} = - 1 - \frac{1}{(- 5)} x_{2} = - 0.8$

Thus, the next approximation $x_{2} = - 0.8$ .

Step by step

Solved in 3 steps with 2 images

Knowledge Booster

$Implicit Differentiation$

Learn more about

Need a deep-dive on the concept behind this application? Look no further. Learn more about this topic, advanced-math and related others by exploring similar questions and additional content below.

Similar questions