# Question 12.7: A root of 3 sin x = x is near to x = 2.5. Use two iterations......

A root of   $3 \sin x=x$  is near to x = 2.5. Use two iterations of the Newton–Raphson technique to find a more accurate approximation.

Step-by-Step
The 'Blue Check Mark' means that this solution was answered by an expert.
Learn more on how do we answer questions.

The equation must first be written in the form f (x) = 0, that is

$f(x)=3 \sin x-x=0$

Then

$\begin{array}{ll}x_1=2.5 & \\f(x)=3 \sin x-x & f\left(x_1\right)=-0.705 \\f^{\prime}(x)=3 \cos x-1 & f^{\prime}\left(x_1\right)=-3.403\end{array}$

Then

$x_2=2.5-\frac{(-0.705)}{(-3.403)}=2.293$

The process is repeated with  $x_1$  = 2.293 as the initial approximation:

$x_1=2.293 \quad f\left(x_1\right)=-0.042 \quad f^{\prime}\left(x_1\right)=-2.983$

Then

$x_2=2.293-\frac{(-0.042)}{(-2.983)}=2.279$

Using two iterations of the Newton–Raphson technique, we obtain x = 2.28 as an improved estimate of the root.

Question: 12.8

Question: 12.1

## Verified Answer:

(a) If y = x², then by differentiation  \fr...
Question: 12.5

## Verified Answer:

Given y = x³ + 2x² then y′ = 3x² + 4x and y′′ = 6x...
Question: 12.6

## Verified Answer:

\begin{aligned}x_1 & =7.5 & & \...
Question: 12.10

## Verified Answer:

(a) \begin{gathered}\mathbf{a} \cdot \mathb...
Question: 12.9

## Verified Answer:

(a) If  \mathbf{a}=3 t^2 \mathbf{i}+\cos 2 ...
Question: 12.4

## Verified Answer:

y^{\prime}=4 x^3 \quad y^{\prime \prime}=12...
Question: 12.3

## Verified Answer:

Given y = x³, then y′ = 3x² and y′′ = 6x. Points o...
Question: 12.2

## Verified Answer:

(a) Given y = x² then y′ = 2x and y′′ = 2. We loca...