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. $$ f(x)=\cos x-\sin x \text { on }[0.7,0.8] $$

Short answer

The root is approximately 0.76.

Step-by-step solution

Understand the Bisection Method

The Bisection Method is a numerical method to find roots of a continuous function. It requires an initial interval where the function changes sign, ensuring that there is at least one root in that interval. The method involves repeatedly dividing the interval in half and selecting the subinterval that contains the root.

Verify Initial Conditions

Check that the function changes sign over the interval [0.7, 0.8]. Calculate \(f(0.7) = \cos(0.7) - \sin(0.7)\) and \(f(0.8) = \cos(0.8) - \sin(0.8)\). If \(f(0.7)\) and \(f(0.8)\) have opposite signs, there is a root in this interval.

Calculate Initial Function Values

Calculate the values: \(f(0.7) \approx 0.099\) and \(f(0.8) \approx -0.072\). Since these values have opposite signs, there is a root between 0.7 and 0.8.

Perform First Bisection

Find the midpoint of the interval [0.7, 0.8]: \(c = \frac{0.7 + 0.8}{2} = 0.75\). Calculate \(f(0.75) = \cos(0.75) - \sin(0.75)\). If \(f(0.75)\) changes sign relative to either \(f(0.7)\) or \(f(0.8)\), choose the subinterval where the sign changes as the new interval.

Analyze First Bisection Result

Calculate \(f(0.75) \approx 0.014\). Since \(f(0.75)\) and \(f(0.8)\) have opposite signs, update the interval to [0.75, 0.8].

Perform Second Bisection

Find the midpoint of [0.75, 0.8]: \(c = \frac{0.75 + 0.8}{2} = 0.775\). Calculate \(f(0.775) = \cos(0.775) - \sin(0.775)\). Determine the new interval based on the sign of \(f(0.775)\).

Analyze Second Bisection Result

Calculate \(f(0.775) \approx -0.028\). Since \(f(0.775)\) is negative and \(f(0.75)\) is positive, update the interval to [0.75, 0.775].

Perform Third Bisection

Find the midpoint of [0.75, 0.775]: \(c = \frac{0.75 + 0.775}{2} = 0.7625\). Calculate \(f(0.7625) = \cos(0.7625) - \sin(0.7625)\). Choose the subinterval based on the sign change.

Analyze Third Bisection Result

Calculate \(f(0.7625) \approx -0.007\). Since the sign changes between \(f(0.7625)\) and \(f(0.75)\), update the interval to [0.75, 0.7625].

Perform Fourth Bisection

Find the midpoint of [0.75, 0.7625]: \(c = \frac{0.75 + 0.7625}{2} = 0.75625\). Calculate \(f(0.75625) = \cos(0.75625) - \sin(0.75625)\).

Analyze Fourth Bisection Result

Calculate \(f(0.75625) \approx 0.0035\). Since the sign changes between \(f(0.75625)\) and \(f(0.7625)\), update the interval to [0.75625, 0.7625].

Evaluate Approximation Precision

The interval is now [0.75625, 0.7625]. Check if the midpoint, 0.759375, has the same sign change with the endpoints or provides sufficient precision, but stopping at a tolerance for a two-decimal accuracy.

Conclusion: Approximated Root

The root of the function is approximately at 0.76, accurate to two decimal places.

Key concepts

Numerical Methods

Numerical methods are techniques used to find approximate solutions to mathematical problems that are difficult or impossible to solve analytically. These methods are invaluable when dealing with equations or systems that have no closed-form solution. The Bisection Method is one such numerical method used for finding the roots of a function. It works by repeatedly dividing an interval in two and using the behavior of the function to narrow down where a root lies.

This method particularly excels at dealing with continuous functions where the Intermediate Value Theorem applies. It's widely used in engineering, physics, and computer science for tasks that require finding approximate solutions quickly. Numerical methods are crucial for modern-day problem-solving, as they allow us to handle complex calculations through iterative processes.

Root Finding

Root finding involves determining the values, or "roots," of a function where it crosses the x-axis (or the value at which the function equals zero). The Bisection Method is a straightforward approach to discovering these roots when an initial interval where the function changes sign is known. The Bisection Method is reliant on the assumption that the function is continuous and differs in sign at the endpoints of the interval.

Steps involved typically include:

• Identifying an interval where a sign change occurs, ensuring there's at least one root.
• Calculating the midpoint of the current interval.
• Evaluating the function at this midpoint.
• Determining which half of the interval contains the root based on sign changes.
• Repeating the splitting process until the interval is sufficiently small.

This method provides a simple, yet effective, way to ensure an approximate solution is found with a defined level of accuracy.

Continuous Function

A continuous function is a key prerequisite for the Bisection Method. A function is deemed continuous if, intuitively, you can draw it without lifting your pencil from the paper. Mathematically, small changes in the input produce small changes in the output, without any abrupt jumps or breaks.

The Intermediate Value Theorem applies to continuous functions and is fundamental to the functionality of the Bisection Method. The theorem states that if a function changes sign over an interval, and is continuous everywhere in that interval, then it must cross zero – at least once somewhere within that interval.

Understanding continuous functions is crucial when dealing with real-world applications where physical quantities often do not have abrupt changes. The continuity ensures that methods like Bisection can be applied with the certainty that a root exists if there's a sign change over the interval.