Geometric Bites

The Maclaurin Series — A Clean Derivation Many smooth functions can be written as an infinite polynomial. When this expansion is centred at x = 0 , we obtain the Maclaurin series . This article derives the Maclaurin formula directly from repeated differentiation, showing precisely why the coefficients involve derivatives and factorials. 1) Begin with a General Power Series Suppose a function f(x) can be expressed as f(x) = a₀ + a₁x + a₂x² + a₃x³ + … + aᵣxʳ + … The constants aᵣ are real coefficients whose values we wish to determine. 2) Evaluate at x = 0 f(0) = a₀ so a₀ = f(0). 3) Differentiate Once f′(x) = a₁ + 2a₂x + 3a₃x² + … + r·aᵣxʳ⁻¹ + … Setting x = 0 eliminates all higher powers: f′(0) = a₁. Thus, a₁ = f′(0). 4) Differentiate Again f″(x) = 2·1·a₂ + 3·2·a₃x + … + r(r−1)aᵣxʳ⁻² + … Evaluate at x = 0 : f″(0) = 2! · a₂ Hence a₂ = f″(0) / 2!. 5) The General Pattern Differentiate repeatedly. After r differentiations, a...

Injective, Surjective, and Bijective Functions In mathematics, a function describes how elements of one set are assigned to elements of another. Three important properties capture how completely and uniquely a function connects its domain to its codomain: injective , surjective , and bijective . These properties tell us whether different inputs can share the same output and whether every possible output is used. Injective (One-to-One) A function is injective if different inputs always produce different outputs. No two distinct elements in the domain are allowed to map to the same result in the codomain. Formally, a function f : A → B is injective if: f(a₁) = f(a₂) ⇒ a₁ = a₂ Equivalently, if a₁ ≠ a₂ , then f(a₁) ≠ f(a₂) . Example: f(x) = e x from ℝ → ℝ is injective. Different inputs produce different outputs, but not every real number is reached, so it is not surjective. Surjective (Onto) A function is surjective if every element of the codomain is reach...

Function Composition: A Simple Way to Organise Your Mathematics Function composition is one of the most useful tools in mathematics. It allows us to combine several steps into a single process, keeping our work neat, organised, and easy to reuse. Rather than performing one operation after another by hand, composition lets us build those steps into a single function. Once you become comfortable with function composition, you never want to go back to doing everything manually. It reduces clutter, helps you work systematically, and allows you to achieve remarkable results with only a few lines of equations. What Is Function Composition? A function takes an input, performs an operation, and produces an output. Function composition takes this idea further: it links functions together so that the output of one becomes the input of the next. We write this as: (f ∘ g)(x) = f(g(x)) This means that g acts first, then f . The circle symbol ∘ simply means “do one funct...