#C06 · Calculus & Analysis · Lv.4View in 3D Mind-Map

Taylor & Maclaurin Series Expansion

Brook Taylor & Colin Maclaurin · 1715 CE

f(x) = \sum_{n=0}^{\infty} \frac{f^{(n)}(a)}{n!}(x-a)^n

e^x = \sum_{n=0}^{\infty}\frac{x^n}{n!},\quad \sin x = x - \frac{x^3}{3!} + \frac{x^5}{5!} - \cdots

Reconstructs any smooth transcendental function (e^x, sin x, cos x) as an infinite polynomial using only its derivatives at a single point.

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Input x (rad)

Terms N

P_{3}(1.2) = \sum_{k=0}^{3} \frac{1.2^k}{k!} = 3.20800 \; (e^{1.2} = 3.32012)

3차 테일러 다항 근사

3.208000

실제 초월함수 e^(1.2)

3.320117

절단 오차 |R_N|

1.12e-1

Prerequisite Path (10 STEP)

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Click steps in order to follow the 3D neon path from basics to target.

STEP 1#A02Polynomial Expansion & Factoring

(a+b)^2 = a^2 + 2ab + b^2

STEP 2#A01Linear Equation & Balance Axiom

ax + b = 0 \implies x = -\frac{b}{a}

STEP 3#A03Laws of Exponents & Radicals

a^m \cdot a^n = a^{m+n}

STEP 4#A06Logarithm Laws & Change of Base

\log_a(xy) = \log_a x + \log_a y

STEP 5#A08Binomial Theorem & Pascal's Triangle

(a+b)^n = \sum_{k=0}^{n} \binom{n}{k} a^{n-k} b^k

STEP 6#C01Limits, Continuity & Euler's Number e

e = \lim_{n \to \infty} \left(1 + \frac{1}{n}\right)^n \approx 2.718281828

STEP 7#A07Arithmetic & Geometric Series Sum

S_n = \frac{a(1 - r^n)}{1 - r}

STEP 8#C02Derivative Definition & Instantaneous Rate

f'(x) = \frac{df}{dx} = \lim_{h \to 0} \frac{f(x+h) - f(x)}{h}

STEP 9#C04Fundamental Theorem of Calculus (FTC)

\int_a^b f(x)\

STEP 10#C06Taylor & Maclaurin Series Expansion

f(x) = \sum_{n=0}^{\infty} \frac{f^{(n)}(a)}{n!}(x-a)^n

🚀 Next Unlocks (Derived Formulas) (2)

#C07Euler's Formula & Euler's Identity #P07Central Limit Theorem & Confidence Intervals

Geometric & Intuitive Insight

덧셈·곱셈밖에 할 줄 모르는 컴퓨터가 어떻게 sin(x)나 e^x 를 계산할까요? 바로 테일러 급수를 통해 초월함수를 다항식(x - x³/6 + x⁵/120…)으로 바꾸어 사칙연산만으로 계산하는 것입니다!

Step-by-Step Derivation

Step 1.1단계: 다항식 P(x) = c₀ + c₁(x-a) + c₂(x-a)² + c₃(x-a)³ + … 이 x=a 에서 f(x)와 모든 n차 미분계수가 같도록 설정합니다.
Step 2.2단계: k번 미분한 뒤 x=a 를 대입하면 f^(k)(a) = k! · c_k 가 되므로 계수는 c_k = f^(k)(a) / k! 입니다.
Step 3.3단계: a=0 인 매클로린 급수에 e^x, sin x, cos x 의 미분값을 넣으면 각각의 멱급수가 도출됩니다.

AI & Coding Real-World Use

CPU/GPU fast Math.sin/exp approximation and XGBoost / Newton-Raphson 2nd-order optimization

#XGBoost 2nd-Order Loss#CPU/GPU Fast Math#Physics Linearization

const sinTaylor = (x: number) => x - (x**3)/6 + (x**5)/120 - (x**7)/5040;

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