NUMERICAL SOLUTIONS OF EQUATIONS
Other note source NUMERICAL SOLUTIONS OF EQUATIONS Concise Theory & Formula Guide Cambridge AS & A Level Mathematics 1. INTRODUCTION Numerical methods find approximate solutions to equations that cannot be solved algebraically. Examples include x³ + x - 4 = 0 , eˣ = 2x + 1 , and sin(x) = x - 1 . Historical Fact: Quintic equations (degree 5 and higher) generally have no algebraic solutions, making numerical methods essential. 2. LOCATING ROOTS (Section 6.1) Root Definition α is a root of f(x) = 0 if f(α) = 0 Method 1: Graphical Approach Rearrange equation as g(x) = h(x) , sketch both graphs, and find intersection points. Each intersection represents a root. Method 2: Change of Sign Method Change of Sign Principle: If f(x) is continuous and f(a) · f(b) If f(a) 0 → Root exists between a and b Example: Change of Sign Problem: Show f(x) = x⁵ + x - 1 = 0 has a root between 0 ...
