Use the Bisection method to find a root of the equation x3 -4x-9 in the interval (2,3),
accurate to four decimal places.
To use the Bisection method, we need to find two values, a and b, in the given interval (2,3), such that f(a) and f(b) have opposite signs.
Let's evaluate f(2) and f(3):
f(2) = (2)^3 - 4(2) - 9 = -1
f(3) = (3)^3 - 4(3) - 9 = 9
Since f(2) is negative and f(3) is positive, we can conclude that there is a root between 2 and 3.
We can now apply the Bisection method to find the root accurately to four decimal places.
Let's define our initial values:
a = 2
b = 3
tolerance = 0.0001
Now let's calculate the midpoint of the interval:
c = (a + b) / 2 = (2 + 3) / 2 = 2.5
Evaluate f(c):
f(2.5) = (2.5)^3 - 4(2.5) - 9 = -1.375
Since f(2.5) is negative, the root is in the interval (a, c).
Calculate the absolute value of the difference between b and a:
abs(b - a) = abs(3 - 2) = 1
If abs(b - a) is less than or equal to the tolerance, we have found the root within the desired accuracy. If not, we repeat the steps using the new interval (a, c).
Now, the new values are:
a = 2
b = 2.5
Calculate the new midpoint:
c = (a + b) / 2 = (2 + 2.5) / 2 = 2.25
Evaluate f(c):
f(2.25) = (2.25)^3 - 4(2.25) - 9 = -0.578125
Since f(2.25) is negative, the root is in the interval (a, c).
Calculate the absolute value of the difference between b and a:
abs(b - a) = abs(2.5 - 2) = 0.5
Since the absolute value of the difference between b and a is less than the tolerance, we have found the root within the desired accuracy.
Therefore, the root of the equation x^3 - 4x - 9 within the interval (2,3), accurate to four decimal places, is approximately 2.25.