Question

Use the Bisection Method to approximate, accurate to two decimal places, the value of the root of the given function in the given interval. $$ f(x)=\sin x-1 / 2 \text { on }[0.5,0.55] $$

Short answer

The root is approximately 0.52.

Step-by-step solution

Understand the Bisection Method

The Bisection Method is a numerical method to find roots of continuous functions. It repeatedly bisects an interval and selects a subinterval in which a root must lie for further processing. This process continues until the interval is very small.

Check the Function Values at the Endpoints

Calculate the function at the given interval's endpoints to ensure a root exists between them.\[ f(0.5) = \sin(0.5) - \frac{1}{2} \]\[ f(0.55) = \sin(0.55) - \frac{1}{2} \] Use a calculator to find these values. \[ f(0.5) \approx -0.020574 \] and \[ f(0.55) \approx 0.020737 \]. Since the signs of these values are different, a root exists in (0.5, 0.55).

Calculate the Midpoint

Find the midpoint of the interval \([0.5, 0.55]\): \[ m = \frac{0.5 + 0.55}{2} = 0.525 \].

Evaluate the Function at the Midpoint

Calculate \( f(0.525) = \sin(0.525) - \frac{1}{2} \). Using a calculator, \( f(0.525) \approx 0.000325 \), which is positive. Since \( f(0.5) \) was negative, the root must be in the interval \([0.5, 0.525]\).

Update the Interval

Since the function changes sign between \(0.5\) and \(0.525\), update the interval to \([0.5, 0.525]\).

Repeat the Bisection Steps

Continue the process: find a new midpoint, evaluate the function at this new midpoint, and update the interval again until the interval is narrower and the result is accurate to two decimal places. After a few repetitions, you'll find the interval sufficiently small and the root approximated correctly.

Key concepts

Numerical Methods

Numerical methods are powerful mathematical tools used for solving complex problems. They provide approximate solutions where analytical methods might be difficult to apply. In simpler terms, these methods are like algorithms that we use to get close to the answer when exact solutions are hard to find. They allow us to solve equations, find roots, and even compute integrals by using numerical calculations rather than writing down long equations.
The Bisection Method, which is one of these numerical methods, is particularly useful in solving equations where a function cannot easily be zeroed by simple algebra. Since it involves breaking down an interval into smaller pieces, numerical methods like the Bisection Method give us a step-by-step way to get to the root of the equation accurately.
By applying these methods, practitioners can calculate and approximate values that traditional algebraic methods might not handle efficiently. As such, numerical methods play a crucial role in modern computational mathematics, making them indispensable tools for engineers and scientists alike.

Root-Finding

Root-finding is the process of determining where a function equals zero. In mathematical terms, a root of a function is any value of the variable that makes the function's output zero. It is crucial for solving many real-world problems, such as finding the break-even point in economics or where a projectile hits the ground in physics.
The Bisection Method is one approach to finding roots of continuous functions. It is highly reliable when applied to functions that cross the x-axis. This method finds a subinterval where the function changes sign, ensuring a root is present. The reliability of the Bisection Method stems from the Intermediate Value Theorem, which states that if the function changes from positive to negative (or vice versa) over an interval, then a root lies within that interval.
Since root-finding is fundamental in mathematics and engineering, mastering methods like the Bisection Method allows for precise solutions in practical problems. The ability to accurately determine roots can greatly aid in modeling, simulations, and predicting outcomes.

Continuous Functions

A continuous function is one where small changes in the input lead to small changes in the output. Put simply, you can draw a continuous function on a graph without lifting your pencil. This property is essential in numerical methods because it guarantees that processes like the Bisection Method can be applied effectively.
Function continuity ensures that within any interval, if the function changes sign, there is at least one root. This characteristic underpins the Bisection Method, providing confidence that every little section of the interval might harbor a root if the function changes its sign between the two endpoints.
Understanding continuous functions is not only important for applying the Bisection Method but also for grasping other concepts in calculus and analysis. Continuous functions form the backbone of many areas in mathematics, making them crucial in understanding the behavior of functions in both theoretical and applied mathematics.