f ( x ) = e x + x − 4 f(x)=e^x+x-4 f ( x ) = e x + x − 4 a = 0 , b = 2 a=0, b=2 a = 0 , b = 2
f ( a ) = f ( 0 ) = e 0 + 0 − 4 = − 3 f(a)=f(0)=e^0 +0-4=-3 f ( a ) = f ( 0 ) = e 0 + 0 − 4 = − 3
f ( b ) = f ( 2 ) = e 2 + 2 − 4 = e 2 − 2 f(b)=f(2)=e^2 +2-4=e^2-2 f ( b ) = f ( 2 ) = e 2 + 2 − 4 = e 2 − 2
f ( a ) f ( b ) = f ( 0 ) f ( 2 ) = − 3 ( e 2 − 2 ) < 0 f(a)f(b)=f(0)f(2)=-3(e^2-2)<0 f ( a ) f ( b ) = f ( 0 ) f ( 2 ) = − 3 ( e 2 − 2 ) < 0
x n = a n + b n 2 x_n=\dfrac{a_n+b_n}{2} x n = 2 a n + b n a n + 1 = x n , b n + 1 = b n , f ( a n ) f ( x n ) ≥ 0 a_{n+1}=x_n, b_{n+1}=b_n, f(a_n)f(x_n)\geq0 a n + 1 = x n , b n + 1 = b n , f ( a n ) f ( x n ) ≥ 0
a n + 1 = a n , b n + 1 = x n , f ( b n ) f ( x n ) ≥ 0 a_{n+1}=a_n, b_{n+1}=x_n, f(b_n)f(x_n)\geq0 a n + 1 = a n , b n + 1 = x n , f ( b n ) f ( x n ) ≥ 0
∣ f ( x n ) ∣ ≤ ε = > a n s w e r = x n |f(x_n)|\leq \varepsilon=>answer =x_n ∣ f ( x n ) ∣ ≤ ε => an s w er = x n
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 \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} n 0 1 2 3 4 5 6 7 8 9 10 11 x n 1 1.5 1.25 1.125 1.0625 1.09375 1.078125 1.0703125 1.07421875 1.072265625 1.0732421875 1.07373046875 f ( x n ) − 0.28171817 1.9868907 0.74034296 0.20521685 − 0.04390406 0.07919854 0.01728845 − 0.01339680 0.00192349 − 0.00574223 − 0.00191077 0.00000601
∣ 1.07373046875 − 1.0732421875 1.0732421875 ∣ ⋅ 100 % ≈ \big|\dfrac{1.07373046875-1.0732421875}{1.0732421875}\big|\cdot100\%\approx ∣ ∣ 1.0732421875 1.07373046875 − 1.0732421875 ∣ ∣ ⋅ 100% ≈
≈ 0.0455 % < 0.05 % \approx0.0455\%<0.05\% ≈ 0.0455% < 0.05% Root is 1.07373046875 1.07373046875 1.07373046875
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