Question

Determine the root of the equation

Answer 1

Answer:

0.97087

Step-by-step explanation:

Given the function:

$f(x) = e^{2x} - x - 6$

Its first derivative is:

$f'(x) = 2e^{2x} - 1$

Using Newton's method:

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

and starting with the point:

$x_1 = 0.95$

we get:

$f(x_1) = e^{2(0.95)} - 0.95 - 6 = -0.26410$

$f'(x_1) = 2e^{2(0.95)} - 1 = 12.37178$

$x_2 = x_1 - \frac{f(x_1)}{f'(x_1} = 0.95 - \frac{-0.26410}{12.37178} = 0.97134$

$f(x_2) = e^{2(0.97134)}- 0.97134 - 6 = 0.00608$

$f'(x_2) = 2e^{2(0.97134)} - 1 = 12.95485$

$x_3 = 0.97134 - \frac{0.00608}{12.95485} = 0.97087$

$f(x_3) = e^{2(0.97087)} - 0.97087 - 6 = -2.6240413 \cdot 10^{-7}$