MATHEMATICS S5 UNIT 4: Solving Equations by Numerical Methods.

By Mukongori Categories: S5 MATHEMATICS

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About Course

1. Bisection Method (Interval Halving Method)

Principle: This method relies on the Intermediate Value Theorem. If a continuous function $f (x)$ has opposite signs at two points, $a$ and $b$ (i.e., $f (a) \cdot f (b) < 0$ ), then there must be at least one root between $a$ and $b.$

Recognize the Need for Numerical Methods: Understand why analytical (exact) solutions are not always feasible or possible for various types of equations (e.g., transcendental equations, high-degree polynomials).
Grasp the Core Concept of Iteration: Comprehend that numerical methods typically involve an iterative process to generate successive approximations that converge towards a solution.
Understand Error and Accuracy: Define and differentiate between various types of errors in numerical computation (e.g., truncation error, round-off error) and understand how to control and assess the accuracy of a numerical solution.
Define Stopping Criteria: Know how to set appropriate stopping criteria for iterative processes based on desired tolerance (e.g., absolute error, relative error, function value close to zero).
Proficiency in Specific Root-Finding Methodologies: Bisection Method, Newton-Raphson Method, Secant Method, Regula Falsi Method (False Position Method), Fixed-Point Iteration.

Course Content

Linear Interpolation and Extrapolation.

Linear Interpolation.

25:02
Linear Extrapolation.

21:10

Location of roots.

Iterative Methods.

Questions and Answers

End of Unit Assessment.

Final Unit Exam.

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A course by

Familiarity with solving linear and quadratic equations, as well as basic polynomial equations.
Derivatives: A thorough understanding of how to compute derivatives of various functions (polynomials, exponentials, logarithms, trigonometric functions) is absolutely crucial, especially for methods like the Newton-Raphson method.
Limits: Understanding the concept of a limit is fundamental to understanding convergence, which is a core theme in numerical methods.
Continuity: The concept of continuous functions is vital for methods like the Bisection Method and Regula Falsi, which rely on the Intermediate Value Theorem.

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