Answer to Question #139998 in Quantitative Methods for xerin

Question

find the roots using bisector. Stop until at 15th iteration or when approximate percent relative error is below 0.05%.

e^x+x=4

Answer 1

$$f(x)=e^x+x-4$$

$a=0, b=2$

$f(a)=f(0)=e^0 +0-4=-3$

$f(b)=f(2)=e^2 +2-4=e^2-2$

$f(a)f(b)=f(0)f(2)=-3(e^2-2)<0$

$$x_n=\dfrac{a_n+b_n}{2}$$

$a_{n+1}=x_n, b_{n+1}=b_n, f(a_n)f(x_n)\geq0$

$a_{n+1}=a_n, b_{n+1}=x_n, f(b_n)f(x_n)\geq0$

$|f(x_n)|\leq \varepsilon=>answer =x_n$

$$\begin{matrix} n & x_n & f(x_n)\\ 0 & 1 & -0.28171817 \\ 1 & 1.5 & 1.9868907 \\ 2 & 1.25 & 0.74034296 \\ 3 & 1.125 & 0.20521685 \\ 4 & 1.0625 & -0.04390406 \\ 5 & 1.09375 & 0.07919854 \\ 6 & 1.078125 & 0.01728845 \\ 7 & 1.0703125 & -0.01339680 \\ 8 & 1.07421875 & 0.00192349 \\ 9 & 1.072265625 & -0.00574223 \\ 10 & 1.0732421875 & -0.00191077 \\ 11 & 1.07373046875 & 0.00000601 \\ \end{matrix}$$

$$\big|\dfrac{1.07373046875-1.0732421875}{1.0732421875}\big|\cdot100\%\approx$$

$$\approx0.0455\%<0.05\%$$

Root is $1.07373046875$