Bisection Method and Absolute Error in Zeros of Continuous Functions

Notes

Roots of nonlinear equations in one variable Introduction Definition: A continuous function is a function whose graph is continuous without any breaks or jumps. ie., if we are able to draw the curve (graph) of a function without even lifting the pencil, then we say that the function is continuous. Continuous Functions Examples Discontinuous Functions Examples Zero of functions Definition: The zeros of a function are defined as the values of the variable of the function such that the function equals 0 Example(1): let 𝒇(𝒙) = 𝒙𝟐 − 𝟏, then 𝒙 = ±𝟏 are zeros of the function 𝒇. Since 𝒇(𝟏) = 𝟏𝟐 − 𝟏 = 𝟎 𝒇(−𝟏) = (−𝟏)𝟐 − 𝟏 = 𝟎 Example: let 𝒇(𝒙) = 𝒔𝒊𝒏𝒙, then 𝒙 = 𝟎, 𝝅, 𝟐𝝅, 𝟑𝝅, … are zeros of the function 𝒇. Since 𝒇(𝒏𝝅) = 𝒔𝒊𝒏(𝒏𝝅 ) = 𝟎, 𝒏 = 𝟎, 𝟏, 𝟐, 𝟑, …. Graphically The zeros of a function are the points on the x-axis where the graph cuts the x-axis. In other words, we can say that the zeros of a function are the x-intercepts of its graph. Example: a) 𝒙 = −𝟒, −𝟏, 𝟏, 𝟑 are zeros of the function b) 𝒙 = −𝟐 is the zero of the function c) The function has no zero Note: The zeros of the function f are the same as the (roots) solutions to the equat...