Parametrization of CurvesDefinition 1. If $x$ and $y$ are given as functions $$x = f(t), \text{  } y=g(t)$$ over an interval $I$ of $t$-values, then the set of points $(x, y) = (f(t), g(t))$ defined by these equations is a parametric curve. The equations are parametric equations for the curve.The variable $t$ is a parameter for the curve, and its domain $I$ is the parameter interval. If $I$ is a..

Taylor Series함수 $f$를 양수의 수렴 반경을 가지고 $$f(x) = \sum_{n=0}^{\infty} a_n(x-a)^n$$ 이라고 하자. 이 수렴 구간에서 미분하면 $$f'(x) = a_1 + 2a_2(x-a) + 3a_3(x-a)^2 + \cdots + na_n(x-a)^{n-1} + \cdots , \\ f''(x) = 1\cdot 2 a_2 + 2 \cdot 3 a_3 (x-a) + 3\cdot 4 a_4 (x-a)^2 + \cdots , \\ f'''(x) = 1 \cdot 2 \cdot 3 a_3 + 2 \cdot 3 \cdot 4a_4(x-a) + 3\cdot 4 \cdot 5a_5 (x-a)^2 + \cdots , \\ \vdots$$ 임을 알 수 있다. 각 등식에 $x=a$..

Power SeriesDefinition 1. A power series about $x=a$ is a series of the form $$\sum_{n=0}^{\infty} c_n(x-a)^n$$ in which the center $a$ and the coefficients $c_0, c_1, ..., c_n$ are constants.Convergent Theorem for Power SeriesTheorem 1. If the power series $\sum_{n=0}^{\infty} a_n x^n$ converges at $x = c \neq 0$, then it converges absolutely for all $x$ with $|x| |d|$.Corollary. The convergen..

The $n$th-Term Test for a Divergent Series Theorem 1. If $\sum_{n=1}^{\infty} a_n$ converges, then $a_n \rightarrow 0$.Proof. Let $\sum_{n=1}^{\infty} a_n = \lim_{n \to \infty} s_n = L$, where $s_n$ is the partial sums of the series and $L$ is the sum of the series. Note that $$\lim_{n \to \infty} s_n = \lim_{n \to \infty} s_{n-1} = L \\ \Longrightarrow \lim_{n \to \infty} (s_n - s_{n-1}) = \lim..

Infinite SeriesDefinition 1. Given a sequence of numbers $\{ a_n \}$, an expression of the form $$a_1 + a_2 + \cdots + a_n + \cdots$$ is an infinite series. The number $a_n$ is the $n$th term of the series. The sequence $\{ s_n \}$ defined by $$s_n = \sum_{k=1}^n a_k$$ is the sequence of partial sums of the series, the number $s_n$ being the $n$th partial sum. If the sequence of partial sums con..

Sequence, 즉 수열은 숫자들의 나열이라고 정의할 수 있고 $$\{a_n\}_{n=1}^\infty$$로 표기되는 무한수열은 정의역이 자연수인 함수로 간주할 수 있다.Convergence and Divergence of SequencesDefinition 1. The sequence $\{ a_n \}$ converges to the number $L$ if for every positive number $\epsilon$ there corresponds an integer $N$ such that for all $n$, $$n > N \Longrightarrow |a_n - L| diverges. If $\{ a_n \}$ converges to $L$, we write $\lim_{n \righ..

Improper Integrals Definition 1. Integrals with infinite limits of integration are improper integrals of Type I.(1) If $f(x)$ is continuous on $[0, \infty)$, then $$\int_a^{\infty} f(x) dx = \lim_{b \rightarrow \infty} \int_a^b f(x) dx.$$ (2) If $f(x)$ is continuous on $(- \infty, b]$, then $$\int_{- \infty}^b f(x) dx = \lim_{a \rightarrow - \infty} \int_a^b f(x) dx.$$ (3) If $f(x)$ is continuou..

IrreducibleDefinition 1. A quadratic polynomial is irreducible if it cannot be written as the product of two linear factors with real coefficients. That is, the polynomial has no real roots. The Fundament든 Theorem of Algebra, 대수학의 기본정리에 의해 모든 실계수 다항식은 irreducible polynomial, 즉 linear or quadratic polynomial로 분해될 수 있다는 사실이 증명되어 있다. Method of Partial FractionsFor polynomials $f(x)$ and $g(x)$ with..

Integration by PartsTheorem 1. Let $f$ and $g$ be differentiable functions of $x$. Then $$\int f(x) g'(x) dx = f(x) g(x) - \int f'(x) g(x) dx.$$Proof. By the Product Rule, we have $$\frac{d}{dx} [f(x) \cdot g(x)] = f'(x)g(x) + f(x)g'(x) \\ \Longrightarrow \int \frac{d}{dx} [f(x) \cdot g(x)] = \int f'(x) g(x) dx + \int f(x) g'(x) dx \\ \Longrightarrow \int f(x) g'(x) = f(x) g(x) - \int f'(x) g(x)..

LinearizationDefinition 1. If $f$ is differentiable at $x=a$, then the approximating function $$L(x) = f(a) + f'(a)(x-a)$$ is the linearization of $f$ at $a$. The approximation $$f(x) \approx L(x)$$ of $f$ by $L$ is the standard linear approximation of $f$ at $a$. The point $x=a$ is the center of the approximation.미분가능한 함수 $f$에 대해 $(a, f(a))$에 접하는 접선의 방정식은 위와 같이 주어지며, 이를 선형 근사라고 부른다. Differentia..