Answer:
0.97087
Step-by-step explanation:
Given the function:
[tex]f(x) = e^{2x} - x - 6[/tex]
Its first derivative is:
[tex]f'(x) = 2e^{2x} - 1[/tex]
Using Newton's method:
[tex]x_{n+1} = x_n - \frac{f(x_n)}{f'(x_n)}[/tex]
and starting with the point:
[tex]x_1 = 0.95[/tex]
we get:
[tex]f(x_1) = e^{2(0.95)} - 0.95 - 6 = -0.26410[/tex]
[tex]f'(x_1) = 2e^{2(0.95)} - 1 = 12.37178[/tex]
[tex]x_2 = x_1 - \frac{f(x_1)}{f'(x_1} = 0.95 - \frac{-0.26410}{12.37178} = 0.97134[/tex]
[tex]f(x_2) = e^{2(0.97134)}- 0.97134 - 6 = 0.00608[/tex]
[tex]f'(x_2) = 2e^{2(0.97134)} - 1 = 12.95485[/tex]
[tex]x_3 = 0.97134 - \frac{0.00608}{12.95485} = 0.97087[/tex]
[tex]f(x_3) = e^{2(0.97087)} - 0.97087 - 6 = -2.6240413 \cdot 10^{-7}[/tex]