Question

Given functions \(f\) and \(g\) on an interval \(I\), how can the Bisection Method be used to find a value \(c\) where \(f(c)=g(c)\) ?

Short answer

Use the Bisection Method on the function \(h(x) = f(x) - g(x)\) to find where it equals zero.

Step-by-step solution

Convert to a Single Function

To use the Bisection Method, first define a new function based on the given functions: \[ h(x) = f(x) - g(x) \]This transforms the problem of finding \(c\) such that \(f(c) = g(c)\) into finding a root of \(h(x)\), i.e., where \(h(c) = 0\).

Identify Interval Where Sign Changes

Choose an interval \([a, b]\) where the function \(h(x)\) changes sign. This implies \(h(a) \cdot h(b) < 0\), indicating there is at least one root in the interval due to the Intermediate Value Theorem.

Apply the Bisection Method

Iterate the Bisection Method by calculating the midpoint \(m = \frac{a+b}{2}\). Evaluate \(h(m)\). Determine the new interval:- If \(h(a) \cdot h(m) < 0\), then the root is in \([a, m]\).- If \(h(m) \cdot h(b) < 0\), then the root is in \([m, b]\).- If \(h(m) = 0\), then \(m\) is the root.

Repeat Until Sufficiently Accurate

Continue the process of dividing the interval in half and selecting the subinterval where the sign changes until the interval \([a, b]\) is sufficiently small, i.e., \(b-a\) is less than some predetermined tolerance. Then \(c \approx m\) is the solution.

Key concepts

Root Finding

Finding the root of a function is like playing a detective game where you have to find the point where the function equals zero. This process is called root finding. For example, if you have a function like \( h(x) = f(x) - g(x) \), a root of this function is a value, \( c \), such that \( h(c) = 0 \). In simpler terms, you're looking for a point where the graph of the function touches the x-axis. This is crucial when trying to find where two functions, such as \( f(x) \) and \( g(x) \), are equal because if they are equal, then \( f(x) - g(x) = 0 \). In this context, finding the root of \( h(x) \) tells us where the original functions meet.

Intermediate Value Theorem

The Intermediate Value Theorem is a helpful mathematical principle that acts as a bridge for root finding. Suppose you have a continuous function \( h(x) \) on an interval \/([a, b]\/). If the function \( h(x) \) takes on different signs at \( a \) and \( b \), then there has to be at least one c between \( a \) and \( b \) where \( h(c) = 0 \).

So why is this theorem important when using the Bisection Method? Because it assures us that a root exists in an interval where the function changes sign. Without this theorem, we wouldn't have a reliable way to guarantee that we're closing in on the root. The theorem is like a safety net, ensuring that the Bisection Method won't lead us astray.

Function Transformation

Transforming functions is like rephrasing a sentence to make it easier to understand. In the context of the Bisection Method, we start with two functions, \( f(x) \) and \( g(x) \), and we're interested in finding where they're equal. By introducing a new function, \( h(x) = f(x) - g(x) \), we transform the problem into a root finding problem.

This transformation is extremely useful because it simplifies the task: instead of dealing with two separate functions, we're now working with a single function. This approach uses the principle that if \( h(x) \) becomes zero, it means that the original functions \( f(x) \) and \( g(x) \) intersect or are equal at that point.

Numerical Methods

Numerical methods are like problem-solving techniques that don't require exact calculations, making them handy when algebraic approaches become too complex. The Bisection Method is one of these numerical methods used for root finding.

Starts with picking an interval \([a, b]\) where the function changes sign. It then repeatedly divides the interval into smaller segments to zero in on the root. In each step, it calculates the midpoint \( m = \frac{a+b}{2} \) and evaluates \( h(m) \). Depending on the sign of \( h(m) \), the method updates the interval to either \([a, m]\) or \([m, b]\).

The Bisection Method continues this process, narrowing down the possibilities, until it finds an approximate value of the root within a desired accuracy. This iterative approach is particularly valuable when exact solutions are difficult or impossible to determine analytically.