Final answer:
Using the bisection method involves computing f(a) and f(b), finding the midpoint c, computing f(c), and iteratively narrowing the interval by checking the signs of the function values until the interval is sufficiently small or ten iterations are completed.
Step-by-step explanation:
To find a root of the function f(x) = 3x + sin(x) - e^x using the bisection method, we initially check the signs of f(a) and f(b), where a = 0 and b = 2.
The bisection method is iterative, requiring several steps to narrow down the interval where the root lies. In each iteration, the interval is halved by computing f(c), where c is the midpoint of the interval (a, b), and then selecting the subinterval where the sign of f(x) changes. This process is repeated until the interval is sufficiently small or the maximum number of iterations is reached.
Here's the procedure:
Compute f(a) and f(b).
Determine the midpoint c = (a + b)/2.
Compute f(c).
If f(a) × f(c) < 0, the root lies in (a, c); set b = c. If f(b) × f(c) < 0, the root lies in (c, b); set a = c.
Check if |b - a| is small enough or if 10 iterations have been completed. If not, repeat steps 2-4.
This method relies on the fact that if f(a) and f(b) have opposite signs, then by the Intermediate Value Theorem, f(x) must have at least one root between a and b. The bisection method is a form of numerical integration that provides an approximate solution to the location of zeros in functions.