Question

In Exercises $75-78$use Newton’s method (see Example $8$)to approximate a root for the given function with the specified value of $x_0$. Terminate your sequence when $\left|x_{n-1}-x_n\right|<0.001$.

$f\left(x\right)=e^x+\sin x,x_0=0$.

Short answer

The required approximation to the root of the given equation$f\left(x\right)=e^x+\sin x,x_0=0$is :-

$x=-0.588532$.

Step-by-step solution

Step 1. Given Information

We have given the following function :-

$f\left(x\right)=e^x+\sin x,x_0=0$.

We have to estimate the root of the equation by using newton's method with specified initial approximation.

Step 2. Essential requirements to apply Newton's Method

We have given the following function :-

$f\left(x\right)=e^x+\sin x$.

Then its first derivative with respect to $x$is :-

$f\text{'}\left(x\right)=e^x+\cos x$

Also we have given the initial approximation is :-

$x_0=0$.

Also the formula for newton's method is as following :-

$x_{k+1}=x_k-\frac{f\left(x_k\right)}{f\text{'}\left(x_k\right)}$

Step 3. Apply Newton's formula

By applying Newton's formula first approximation to the root is :-

$$\begin{gathered} x_1=x_0-\frac{f\left(x_0\right)}{f\text{'}\left(x_0\right)} \\ \Rightarrow x_1=0-\frac{e^0+\sin 0}{e^0+\cos 0} \\ \Rightarrow x_1=0-\frac{1+0}{1+1} \\ {x}_1=\frac{-1}{2}=-0.5 \end{gathered}$$

Then second approximation is :-

$$\begin{gathered} x_2=-0.5-\frac{f\left(-0.5\right)}{f\text{'}(-0.5)} \\ \Rightarrow x_2=-0.5-\frac{e^{-0.5}+\sin (-0.5)}{e^{-0.5}+\cos (-0.5)} \\ \Rightarrow x_2=-0.5-\frac{0.607-0.4794}{0.607+0.8776} \\ \Rightarrow x_2=-0.58595 \end{gathered}$$

Then third approximation is :-

$$\begin{gathered} x_3=-0.58595-\frac{f\left(-0.58595\right)}{f\text{'}\left(-0.58595\right)} \\ \Rightarrow x_3=-0.58853 \end{gathered}$$

Then fourth approximation is :-

$$\begin{gathered} x_4=-0.58853-\frac{f\left(-0.58853\right)}{f\text{'}\left(-0.58853\right)} \\ \Rightarrow x_4=-0.588532 \end{gathered}$$

Here $$\begin{gathered} \left|x_4-x_3\right|=\left|-0.588532+0.58853\right| \\ \Rightarrow \left|x_4-x_3\right|=0.000002<0.001 \end{gathered}$$

So here we terminate the sequence.

Hence the required approximation to root of the given equation is :-

$x=-0.588532$