usingbisection method and fp,evaluat the solution set of x3-4x-9
To find the solution set of the equation x^3 - 4x - 9 = 0 using the bisection method, we need to evaluate the function f(x) = x^3 - 4x - 9.
Step 1: Define the function
f(x) = x^3 - 4x - 9
Step 2: Choose initial bracketing interval
Select two initial guesses, a and b, such that f(a) and f(b) have opposite signs. This ensures that there is a root of the equation between a and b.
Step 3: Calculate the midpoint of the interval
c = (a + b) / 2
Step 4: Evaluate f(c)
Calculate the value of f(c) by substituting c into the function f(x).
Step 5: Update the interval
If f(c) has the same sign as f(a), replace a with c. Otherwise, replace b with c.
Step 6: Repeat steps 3 to 5
Repeat steps 3 to 5 until the width of the interval becomes smaller than a chosen tolerance or a specific maximum number of iterations is reached.
Step 7: Find the solution set
Once the bisection method converges, the final interval contains the root(s) of the equation. The solution set is the value(s) of x that correspond to the endpoints of this interval.
Alternatively, you can use Python programming language to implement the bisection method and evaluate the solution set. Here's an example code snippet:
```python
def f(x):
return x**3 - 4*x - 9
def bisection_method(a, b, tol):
max_iterations = 100
iterations = 0
while abs(b - a) > tol and iterations < max_iterations:
c = (a + b) / 2
fc = f(c)
if f(a) * fc < 0:
b = c
else:
a = c
iterations += 1
return a, b
# Example usage:
a, b = bisection_method(1, 4, 1e-6)
solution_set = [a, b]
print("Solution set:", solution_set)
```
This code snippet finds the solution set of x^3 - 4x - 9 = 0 within a tolerance of 1e-6, starting with initial guesses of 1 and 4. The solution set is then printed as output.