Taylor SeriesJEE Advanced

Polynomials impersonating functions — interactive Mathematics simulation for IIT-JEE.

Concept

A smooth function equals an infinite polynomial built from its derivatives at one point: the Taylor series. Around 0 (Maclaurin): sin x = x − x³/3! + x⁵/5! − …. Each added term hugs the true curve further out — watch the polynomial 'grow arms'. The first terms ARE the standard limits and approximations physics uses everywhere.

Key formula

f(x) = \sum_{k=0}^{\infty}\frac{f^{(k)}(0)}{k!}x^k: \quad e^x = 1 + x + \tfrac{x^2}{2!} + \cdots, \quad \sin x = x - \tfrac{x^3}{3!} + \cdots

Derivation

Demand a power series $\sum a_kx^k$ match f's value and ALL derivatives at 0: differentiate k times, set x = 0 → $a_k = f^{(k)}(0)/k!$ .

sin's derivatives cycle (cos, −sin, −cos, sin) giving alternating odd terms. Truncation error ≈ the first dropped term (alternating series) — that's why 'sin θ ≈ θ' works below ~10°.

Scenarios to explore

Taylor & Maclaurin Series — Polynomials sneaking up on sin, cos, eˣ — term by term.

Real-world applications

Small-angle physics: pendulum SHM, optics paraxial approximations.
Limits without L'Hôpital: (1 − cosx)/x² → 1/2 by reading the x² coefficient.
Calculators & CPUs evaluate transcendentals from these polynomials.

JEE exam tips

Memorise the big five: eˣ, sinx, cosx, ln(1+x), (1+x)ⁿ — nearly every series limit is a plug-in.
For limits, expand to JUST enough order to cancel the denominator.
sinx < x < tanx for x ∈ (0, π/2) — first-term inequalities.

Common mistakes

Forgetting factorials in denominators.
Using ln(1+x)'s series outside |x| < 1 (it diverges — convergence has a radius).
Sign slips in the alternating patterns.

Exam traps to avoid

(1+x)ⁿ with non-integer n needs |x| < 1 — binomial series, not theorem.
Odd functions have only odd powers (sin), even functions only even (cos) — a free error check.