1. **Problem:** Use the bisection method to find a root of the equation $x^3 - 4x - 9 = 0$ in the interval $(2, 3)$ up to 5 iterations, and calculate the corresponding error for each iteration to four decimal places. 2. **Formula and rules:** - The bisection method finds roots by repeatedly halving the interval where the function changes sign. - At each iteration, compute midpoint $c = \frac{a+b}{2}$. - Evaluate $f(c)$; if $f(a)f(c) < 0$, root lies in $[a,c]$, else in $[c,b]$. - Error estimate at iteration $n$ is $|b_n - a_n|/2$. 3. **Initial interval:** $a=2$, $b=3$. 4. **Iteration 1:** - $c_1 = \frac{2+3}{2} = 2.5$ - $f(2) = 2^3 - 4\times2 - 9 = 8 - 8 - 9 = -9$ - $f(2.5) = 2.5^3 - 4\times2.5 - 9 = 15.625 - 10 - 9 = -3.375$ - Since $f(2)f(2.5) > 0$, root lies in $[2.5, 3]$. - New interval: $a=2.5$, $b=3$. - Error estimate: $\frac{3-2.5}{2} = 0.25$. 5. **Iteration 2:** - $c_2 = \frac{2.5+3}{2} = 2.75$ - $f(2.75) = 2.75^3 - 4\times2.75 - 9 = 20.796875 - 11 - 9 = 0.7969$ - Since $f(2.5)f(2.75) < 0$, root lies in $[2.5, 2.75]$. - New interval: $a=2.5$, $b=2.75$. - Error estimate: $\frac{2.75-2.5}{2} = 0.125$. 6. **Iteration 3:** - $c_3 = \frac{2.5+2.75}{2} = 2.625$ - $f(2.625) = 2.625^3 - 4\times2.625 - 9 = 18.077 - 10.5 - 9 = -1.423$ - Since $f(2.5)f(2.625) > 0$, root lies in $[2.625, 2.75]$. - New interval: $a=2.625$, $b=2.75$. - Error estimate: $\frac{2.75-2.625}{2} = 0.0625$. 7. **Iteration 4:** - $c_4 = \frac{2.625+2.75}{2} = 2.6875$ - $f(2.6875) = 2.6875^3 - 4\times2.6875 - 9 = 19.393 - 10.75 - 9 = -0.3574$ - Since $f(2.625)f(2.6875) > 0$, root lies in $[2.6875, 2.75]$. - New interval: $a=2.6875$, $b=2.75$. - Error estimate: $\frac{2.75-2.6875}{2} = 0.0313$. 8. **Iteration 5:** - $c_5 = \frac{2.6875+2.75}{2} = 2.71875$ - $f(2.71875) = 2.71875^3 - 4\times2.71875 - 9 = 20.086 - 10.875 - 9 = 0.2148$ - Since $f(2.6875)f(2.71875) < 0$, root lies in $[2.6875, 2.71875]$. - New interval: $a=2.6875$, $b=2.71875$. - Error estimate: $\frac{2.71875-2.6875}{2} = 0.0156$. **Final approximate root after 5 iterations:** $c_5 = 2.7188$ (rounded to 4 decimals). **Errors at each iteration:** 0.2500, 0.1250, 0.0625, 0.0313, 0.0156.