Answer to Question #139426 in Quantitative Methods for Usman

Question

Find the root of the equation tanx + tanhx = 0 which lies in the interval (1.6, 3.0) correct to four significant digits using the method of false position.

Answer 1

Given $f(x)=\tan x+\tanh x$

$f(a)=f(1.6)=\tan(1.6)+\tanh(1.6)=-33.3109$

$f(b)=f(3)=\tan(3)+\tanh(3)=0.8525$

Using iterative Regula-Falsi Formula, the first approximation,

$$x_1=a-\dfrac{b-a}{f(b)-f(a)}f(a)=$$ $$=1.6-\dfrac{3-1.6}{0.8525+33.3109}(-33.3109)=2.9324$$

$f(2.9324)=0.7821$

$a=1.6, b=2.9324$

$$x_2=1.6-\dfrac{2.9324-1.6}{0.7821+33.3109}(-33.3109)=2.9018$$

$f(2.9018)=0.7495$

$a=1.6, b=2.9018$

$$x_3=1.6-\dfrac{2.9018-1.6}{0.7495+33.3109}(-33.3109)=2.8732$$

$f(2.8732)=0.7186$

$a=1.6, b=2.8732$

$$x_4=1.6-\dfrac{2.8732-1.6}{0.7186+33.3109}(-33.3109)=2.8463$$

$f(2.8463)=0.6891$

$a=1.6, b=2.8463$

$$x_5=1.6-\dfrac{2.8463-1.6}{0.6891+33.3109}(-33.3109)=2.8211$$

$f(2.8211)=0.6609$

$...$

$x_{149}=2.36550$

$x_{150}=2.36548$

$x_{151}=2.36546$

$...$

$x_{499}=2.36502$

Root of $f(x)$ is $2.365$