Finding roots for different functions is a vital application of mathematics. Unlike linear and quadratic equations, countless functions have roots that cannot be calculated using traditional methods. To solve this problem, Issac Newton, an English polymath who pioneered the exploration of the mathematical and physical worlds through the development of significant theories, devised a method in 1669. The method was significantly simplified by English mathematician Joseph Raphson, to the current form taught in secondary classrooms.
The algorithm for Newton’s method is concise and comprehensible. First we can find a point on the function graph arbitrarily, calling it (x₀, f(x₀)). Its tangent line, whose slope is the derivative of f(x) on x₀, intersects the x-axis on x₁. To find x₁, we can use the following equation:
f(x₀) - 0 = f'(x₀)(x₁ - x₀), derived from the equation y₁ - y₂ = m(x₁ - x₂)
Therefore,
Simplified to,
After finding x₁, we repeat the steps above to find x₂, x₃, x₄…xₙ. As n approaches infinity, xₙ gets infinitely close to the root, as illustrated in Figure 1.
The general equation is:
Figure 1. Illustration for Newton’s method (The Story of Mathematics, 2023)
Newton’s method is an application of the derivative, a basic concept in calculus that describes the instantaneous rate of change. The main idea behind Newton’s method is by solving the first derivative, we pretend the curved function is a linear function, making the root much easier to find. Though the first calculation will likely deviate from the actual root, iterating the process will get closer to the answer.
Understanding Newton’s method facilitates students’ learning process in calculus, as both bear the concept of “infinity”. For calculus, the delta x and delta y are infinitely small but would never reach zero. Their quotient is the rate of change at a certain point. Similarly, in Newton’s method, the predicted root is infinitely close to the actual root, but would never reach it.
Work Cited
The Story of Mathematics. (2023). Newton’s method - Process, Approximation, and Example. [online] Available at: https://www.storyofmathematics.com/newtons-method/ [Accessed 1 Jun. 2024].
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