Question

Roots of Equations - Method of False Position Draw and label the first three steps of the method of false position to find the root at x=1.5 given the interval [−3,3] in the following function.

Answer 1

The Method of False Position is a numerical technique used to find roots of equations and includes identifying an interval, making initial guesses, and iteratively computing and adjusting guesses based on function values. It involves calculations that may include square roots and cube roots, where usage of a calculator is often required.

Step-by-step explanation:

The Method of False Position, also known as Regula Falsi, is a numerical method used to find roots of equations, which can be crucial in solving equilibrium problems where square roots, cube roots, or higher roots are involved. The process involves several steps:

1. Identify an interval where the function changes sign (here it's given as [−3,3]).
2. Pick two points, a and b, such that f(a) and f(b) have opposite signs. These are your initial guesses.
3. Compute the false position (c) using the formula c = b - (f(b) * (b-a)) / (f(b) - f(a)) and evaluate f(c).
4. If f(c) has the same sign as f(a), let c become the new a; otherwise, let c become the new b.
5. Repeat the process with the new interval [a,b] until c is sufficiently close to the actual root.

Since we're provided an interval and a root to find at x=1.5, we would start by evaluating the function at the end points of the interval and then proceed with the subsequent steps. To perform these calculations, especially those involving roots, familiarity with your calculator's functions is important. If uncertain, one should consult their instructor.

Two-Dimensional (x-y) Graphing can be a valuable visual aid in understanding the position and movement toward the actual root during each step of the Method of False Position.