고등수학까지만 공부하고 미적분학(Calculs) 를 독학으로 공부한 내용을 정리한 글입니다. 책은 Man Sik Min·Hyeong Chul Jeong·Hyejung Lee, 『CALCULUS』, 한티미디어 입니다.

0. Preface

시작부터 어질어질하다;;

용어 정리

• Calculs: 미적분학
• partial derivative: 편미분
• multiple integral: 중적분
• composite function: 합성함수
• inverse function: 역함수
• exponential function: 지수함수
• log function: 로그함수
• trigonometric function: 삼각함수
• limits of functions: 함수의 극한
• continuity: 연속성
• transcendental function: 초월함수
• mean value theorem: 중간값 정리
• indefinite: 부정
• revolution: 회전
• improper: 이상
• derivative: 미분(微分), 작을 미, 나눌 분
• normal line: 법선(法線), 법 법, e.g. 법률, 방법
• slope: 기울기
• local maximum: 극대(極大), 극진할·다할 극
• local minimum: 극소(極小)
• indefinite integral: 부정적분
• definite integral: 정적분
• curve: 곡선
• volume of revolution: 회전체의 부피
• arcs: 호(弧), 활 호
• plain: 평면
• improper integral: 이상적분
• multivariate function: 다변수 함수
• differential equation: 미분 방정식
• series: 급수(級數), 등급 급, 셈 수
• parametric calculus: 매개변수 미적분
• polar coordinate: 극좌표계
• dot product: 내적
• cross product: 외적
• substitution integration: 치환 적분
• first-order linear differential equation: 일차 선형 미분 방정식
• formula: 공식
• equation: 방정식

1. Relations and Functions

- Relation

A relation is a set of ordered pairs $(x,y)$.

• The domain of a relation is the set of all $x$-values from the ordered pairs.
• The range of a relation is the set of all $y$-values from the ordered pairs.

- Function

A function is a special type of relation in which each element of the domain is paired with exactly one element of the range.

• A relation is a function if and only if no vertical line intersects its graph more than once.

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용어 정리

• ordered pair: 순서쌍

2. Slope

The slope $m$ of the line passing throught points $(x_1,y_1)$ and $(x_2,y_2)$ is given by...
$m = \frac{y_2-y_1}{x_2-y_1}$, where $x_1 \neq x_2$.

3. Linear Functions

A function $f$ is a linear function if it can be written in the form $f(x) = mx + b$, where $m$ and $b$ are constants.

• The Graph of a linear function is a nonvertical straight line with slope $m$ and $y$-intercept $b$.

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용어 정리

• intercept: 절편(截片), 끊을 절, 조각 편

4. Equation of Lines

• Slope-Intercept Form: $y = mx + b$, where $m$ is the slope and $b$ is the $y$-intercept
• Point-Slope Form: $y-y_1=m(x-x_1)$, where $m$ is the slope and $(x_1,y_1)$ is a point on the line.
• Standard Form: $Ax + By = C$, where $A$, $B$, and $C$ are integers.
• Vertical Line: $x = a$, where $a$ is the $x$-intercept. the slope is undefined.
• Horizontal Line: $y = b$, where $b$ is the $y$-intercept. The slope is zero.

5. Parallel and Perpendicular Lines

Two different lines with equations $y=m_1x + b_1$ and $y=m_2x+b_2$ are parallel if $m_1=m_2$ (the slopes are equal). They are perpendicular if $m_1 \cdot m_2=-1$ (the slopes are negative reciprocals).

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용어 정리

• perpendicular: 직각의, 수직적인
• reciprocals: 상호의, 호혜적인, 역수의

6. Quadratic Functions

A Quadratic function is function that can be written in the form $f(x) = ax^2 + bx + c$ ($a \neq 0$).

• The graph of a quadratic function is a parabola.
• The line of symmetry for a quadratic function is $x = -\frac{b}{2a}$, and the vertex is $(-\frac{b}{2a}, f(-\frac{b}{2a}))$.

- Solutions, Roots, and Zeros of Functions.

The solutions of a function is the value(s) of $x$ for which $f(x) = 0$. Solutions of functions are also called roots or zeros.

• On a graph, the solution of the function is the $x$-intercept(s).

- Discriminant and Roots of a Quadratic Functions

The solutions of the quadratic equation $ax^2 + bx + c = 0$ are given by the quadratic formula:

$x = \frac{-b \pm \sqrt{b^2-4ac}}{2a}$

The expression $b^2 - 4ac$ is called the discriminant.

1. If the discriminant $b^2-4ac > 0$, then there are two real roots, and the graph crosses the $x$-axis twice.
2. If the discriminant $b^2-4ac = 0$, then there is one real root, and the graph is tangent to the $x$-axis.
3. If $b^2-4ac > 0$, then there are no real roots, and the graph does not cross the $x$-axis.

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용어 정리

• symmetry: 대칭
• vertex: 꼭짓점
• solution: 해(解), 풀 해
• root: 근(根), 뿌리 근
• zero: 영(0)
• discriminant: 판별식(判別式), 판단할 판, 다를·나눌 별, 법 식
• real roots: 실근
• tangent to: ~에 접하다

오탈자?

Example 6: what is one ossible value of $k$

7. Laws of Exponents

Name	Laws
Products Of Powers	$a^m \cdot a^n = a^{m+n}$
Quotient of Powers	$\frac{a^m}{a^n}=a^{m-n}$
Power of a Power	$(a^m)^n = a^{mn}$
Power of a Quotient	$(\frac{a}{b})^n = \frac{a^n}{b^n}$ $(a \neq b)$
Negative Exponent	$a^{-n} = \frac{1}{a^n}$ $(\frac{a}{b})^{-n} = (\frac{b}{a})^n$ $(ab \neq 0)$
Zero Exponent	$a^0=1$ $(a \neq 0)$

용어 정리

• power: 거듭제곱
• exponent: 지수
• quotient: 정수의 나눗셈 -> 몫, 그 외의 상황 -> 비율, 분수꼴

8. Special Products and Factoring Polynomials

Name	Laws
Difference of Two Squres	$a^2 - a^2 = (a + b)(a - b)$
Sum of Two cubes	$a^3 + b^3 = (a + b)(a^2 - ab + b^2)$
Difference of Two Cubes	$a^3 - b^3 = (a - b)(a^2 + ab - b^2)$
Perfect Square Trinomials	$(a^2 + 2ab + b^2) = (a + b)^2$ $(a^2 - 2ab + b^2) = (a - b)^2$
Cubes of Binomials	$(a^3 + 3a^2b + 3ab^2 + b^3) = (a + b)^3$ $(a^3 - 3a^2b + 3ab^2 - b^3) = (a - b)^3$

용어 정리

• Square: 제곱
• Cube: 세제곱
• Trinomials: 삼항식
• Binomials: 이항식

9. Composition of Functions

Given the two functions $f$ and $g$, the composition function, denoted by $(f \circ g)(x) = f(g(x))$.

In order for a value of $x$ to be in the domain of $f \circ g$, two conditions must be satisfied...

1. $x$ must be in the domain of $g$, and
2. $g(x)$ must be in the domain of $f$.

10. Inverse Functions

The functions $f$ and $g$ are inverse functions if...

1. $f(g(x)) = x$ for all $x$-values in the domain of $g$ and
2. $g(f(x)) = x$ for all $x$-values in the domain of $f$.

- The inverse of a function $f$ is usually denoted by $f^{-1}$, which is read "$f$ inverse".
- The graph of a function $f$ and the graph of its inverse $f^{-1}$ are symmetric about the line $y = x$.

- Finding an Inverse Function

1. Replace $f(x)$ with $y$.
2. Interchange $x$ and $y$.
3. Solve for $y$.

11. Even and Odd Functions

 The function $f(x)$ is even if $f(-x) = f(x)$. The graph of an even function is symmetric about they $y$-axis.

 The function $f(x)$ is odd if $f(-x) = -f(x)$. The graph of an odd function is symmetric about the origin.

용어 정리

• Even: 짝수
• Odd: 홀수
• Even function: 짝함수, 우함수
• Odd function: 홀함수, 기함수

12. Special Functions

 The greatest integer function is defined as $f(x) = [x] =$ the greatest integer less than or equal to $x$.

 When the domain of a function is divided into several parts and a different function rule is applied to each part, the function is called a piecewise-defined function.

오탈자?

$f(x)=[x]$$=$

용어 정리

• greatest integer function: 최대 정수 함수
• piecewise-defined function: 조각별-정의된 함수, 하이브리드 함수, 클래스에 의한 정의 함수, etc.

13. Distance Formula

The distance between two points with coordinates $(x_1, y_1)$ and $(x_2, y_2)$ is given by...

$d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2}$.

14. Midpoint Formula

The midpoint of a line segment with endpoints $(x_1, y_1)$ and $(x_2, y_2)$ is the point with coordinates $M(\frac{x_1 + x_2}{2}, \frac{y_1 + y_2}{2})$.

용어 정리

• line segment: 선분
• midpoint: 중점

15. Circles

The equation of a circle with center $(h, k)$ and radius $r$ units if $(x - h)^2 + (y - k)^2 = r^2$.

16. Parabolas

- Open Up-downward Parabola

$y = a(x - h)^2 + k$

• $a > 0$, open upward
• $a < 0$, open downward
• Vertex: $V(h, k)$
• axis of symmetry: $x = h$

- Open Right-left Parabola

$x = a(y - k)^2 + h$

• $a

용어 정리

parabola: 포물선

오탈자?

Figure 1.8 과 상단에서 describe 하는 수식이 서로 다름.

17. Ellipses

- Horizontal major axis

$\frac{(x-h)^2}{a^2} + \frac{(y-k)^2}{b^2} = 1$, where $a^2 > b^2$.

- Center: $(h, k)$
- Length of horizontal axis: $2a$.
- Length of vertical axis: $2b$.
- Area of an ellipse = $\pi ab$.

- Vertical major axis

The Vertical major axis is the opposite of a horizontal major axis.

용어 정리

ellipse: 타원
axis: 축
major axis: 장축
minor axis: 단축

18. Hyperbolas

left-right opened style

$\frac{(x-y)^2}{a^2} - \frac{(y-k)^2}{b^2} = 1$

- Center: $(h, k)$
- Distance between vertices: $a$

upward-downward opened style

upward-downward opened style is the opposite of a left-right opened style.

용어 정리

• hyperbola: 쌍곡선

19. Definition of Logarithm

If $b$ and $x$ are positive and $b \neq 1$, $\log_{b}{x} = y$ if and only if $x = b^y$. An equation of the form $y = \log_{b}{x}$ is called a logarithm function. The domain of a logarithmic function is $(0, \infty)$ and the range is $(-\infty , \infty)$.

20. Properties of Logarithms

1. $\log_{b}{MN} = \log_{b}{M} + \log_{b}{N}$
2. $\log_{b}{\frac{M}{N}} = \log_{b}{M} - \log_{b}{N}$
3. $\log_{b}{M^p} = p \cdot \log_{b}{M}$
4. $b^{\log_{b}{N}} = N$
5. $\log_{b}{b} = 1$
6. $\log_{b}{1} = 0$
7. $\log_{b}{x} = \frac{\log_{a}{M}}{\log_{a}{b}}$

21. Common and Natural Logarithms

Common Logarithms

The function $y = \log_{10}{x}$ is called the common logarithmic function. It is usually written without the subscript $10$, so $\log_{10}{x}$ is written as $\log{x}$.

Natural Logarithms

The function $y = \log_{e}{x}$ is called the natural logarithmic function. It is usually written without the subscript $e$, and is written as $\ln{x}$. The number of $e$ is an irrational number whose value is approximately $2.718$.

용어 정리

irrational number: 무리수

22. Trigonometric Functions of Acute Angles

The trigonometric functions of any angle $0\degree < \theta < 90\degree$ are defined as follows:

$\sin{\theta} = \frac{opposite side}{hypotenuse} = \frac{y}{r}$	$\csc{\theta} = \frac{1}{\sin{\theta}} = \frac{r}{y} (y \neq 0)$
$\cos{\theta} = \frac{adjacent side}{hypotenuse} = \frac{x}{r}$	$\sec{\theta} = \frac{1}{\cos{\theta}} = \frac{r}{x} (x \neq 0)$
$\tan{\theta} = \frac{opposite side}{adjacent side} = \frac{y}{x}$	$\cot{\theta} = \frac{1}{\tan{\theta}} = \frac{x}{y} (y \neq 0)$

용어 정리

• opposite side: 대변
• hypotenuse: 빗변
• adjacent side: 밑변
• acute, right, obtuse angle: 예각, 직각, 둔각

오탈자?

$\sin{\theta} = \frac{opposite side}{hypotenuse} =$ $\frac{x}{r}$ (x) $\frac{y}{r}$ (o)

23. Trigonometric Functions and the Unit Circle

The unit circle is a circle with center $(0, 0)$ and radius $1$, whose equation is $x^2 + y^2 = 1$. Let $P(x, y)$ be a point on the unit circle on the terminal side of $\theta$. Then the six triogonometric functions for any angle $\theta$ are defined as follows...

$\sin{\theta} = \frac{y}{r} = \frac{y}{1} = y$	$\csc{\theta} = \frac{1}{y}$
$\cos{\theta} = \frac{x}{r} = \frac{x}{1} = x$	$\sec{\theta} = \frac{1}{x}$
$\tan{\theta} = \frac{y}{x}$	$\cot{\theta} = \frac{x}{y}$

- Reference angle

The reference angle associated with $\theta$ is the aucte angle formed by the $x$-axis and the terminal side of the angle $\theta$

용어 정리

 암만 initial side, terminal side 의 번역을 찾으려 해도 나오질 않아서, initial side 를 시선과 terminal side 를 종선으로 번역한 분의 블로그를 보고 아래와 같이 정리하였음. 실제 수학계에서 이와 같이 사용하는지는 의문...

• initial sie: 시선(始線), 비로소 시, 줄 선
• terminal side: 종선(終線), 마칠 종, 줄 선
  영어로는 다음과 같이 정의되어 있음: A straight line that has been rotated around a poin on another line to form an angle measured in a clockwise or counterclockwise direction.
• unit circle: 단위원
• reference angle: 참조각? 마땅한 뜻이 안 나옴.

24. Familiar Angles

Angles in standard position whose measure are multiple of $\frac{\pi}{6}$ radians or multiples of $\frac{\pi}{4}$ radians are called familiar angles.

$\theta$	0	$\frac{\pi}{6}$	$\frac{\pi}{4}$	$E\frac{\pi}{3}$	$\frac{\pi}{2}$
$\sin{\theta}$	$\frac{\sqrt{0}}{2} = 0$	$\frac{\sqrt{1}}{2} = \frac{1}{2}$	$\frac{\sqrt{2}}{2} = 0$	$\frac{\sqrt{3}}{2} = 0$	$\frac{\sqrt{4}}{2} = 1$
$\cos{\theta}$	$\frac{\sqrt{4}}{2} = 1$	$\frac{\sqrt{3}}{2}$	$\frac{\sqrt{2}}{2} = 0$	$\frac{\sqrt{1}}{2} = \frac{1}{2}$	$\frac{\sqrt{0}}{2} = 0$

- Standard Position

An angle is in standard position if its vertex is located at the origin and one ray is on the $x$-axis.

25. Basic Trigonometric Identities

- Identities from the Defenitions of Trigonometric Functions

- $\tan{\theta} = \frac{\sin{\theta}}{\cos{\theta}}$
- $\sec{\theta} = \frac{1}{\cos{\theta}}$
- $\csc{\theta} = \frac{1}{\sin{\theta}}$
- $\cot{\theta} = \frac{1}{\tan{\theta}}$

- Pythagorean Identities

- $\sin^2{\theta} + \cos^2{\theta} = 1$
- $\tan^2{\theta} + 1 = \sec^2{\theta}$
- $\cot^2{\theta} + 1 = \csc^2{\theta}$

- Opposite Angle Formulas

- $\sin{-\theta} = -\sin{\theta}$
- $\cos{-\theta} = \cos{\theta}$
- $\tan{-\theta} = -\tan{\theta}$

- Cofunction Identities

- $\sin{\theta} = \cos{(90\degree - \theta)}$
- $\cos{\theta} = \sin{(90\degree - \theta)}$
- $\tan{\theta} = \cot{(90\degree - \theta)}$

용어 정리

• Pythagorean Identities: 피타고라스 삼각항등식, 간단히 피타고라스 항등식
• Identity: 항등식

26. Graphs of $\sin{x}$, $\cos{x}$, and $\tan{x}$.

• For the functions $y = a\sin{bx}$ and $y = a\cos{bx}$, the period is $\frac{2\pi}{\vert b \vert}$ and the amplitude is $\vert a \vert$.
• For the function $y = \tan{bx}$ the period is $\frac{\pi}{\vert b \vert}$. Since the tangent function has no maximum or minimum value, it has no amplitude.

용어 정리

• period: 주기
• amplitude: 진폭

27. Law of Sines

In any triangle $ABC$, $\frac{\sin{A}}{a} = \frac{\sin{B}}{b} = \frac{\sin{C}}{c}$

28. Law of Cosines

In any triangle $ABC$,

• $c^2 = a^2 + b^2 - 2ab\cdot\cos{C}$
• $b^2 = a^2 + c^2 - 2ac\cdot\cos{B}$
• $a^2 = b^2 + c^2 - 2bc\cdot\cos{A}$

29. Trigonometric Identities

- Sum and Difference Formulas

$\sin{(A + B)} = \sin{A}\cos{B} + \cos{A}\sin{B}$	$\sin{(A - B)} = \sin{A}\cos{B} - \cos{A}\sin{B}$
$\cos{(A + B)} = \cos{A}\cos{B} - \sin{A}\sin{B}$	$\cos{(A - B)} = \cos{A}\cos{B} + \sin{A}\sin{B}$
$\tan{(A + B)} = \frac{\tan{A} + \tan{B}}{1 - \tan{A}\tan{B}}$	$\tan{(A - B)} = \frac{\tan{A} - \tan{B}}{1 + \tan{A}\tan{B}}$

- Double-Angle Formulas

- $\sin{2\theta} = 2\sin{\theta}\cos{\theta}$
- $\cos{2\theta} = \cos^2{\theta} - \sin^2{\theta} = 2\cos^2{\theta} - 1 = 1 - 2\sin^2{\theta}$
- $\tan{2\theta} = \frac{2\tan{\theta}}{1 - \tan^2{\theta}}$

30. Trigonometric Equation

A trigonometric equations is any statement involving the conditional equality of two trigonometric expressions (Most trigonometric equations are true for some but not all values of the variable.) The values that satisfy the equation are called solution of the equation.

31. Inverse Trigonometric Functions

- arcsine

The inverse sine function, denoted by $y = \sin^{-1}{x}$ or $y = \arcsin{x}$, is the function with a domain of $[-1, 1]$ and a range of $[-\frac{\pi}{2}, \frac{\pi}{2}]$ that satisfies the relation $\sin{y} = x$.

- arccosine

The inverse cosine function, denoted by $y = cos^{-1}{x}$ or $y = \arccos{x}$, is the function with a domain of $[-1, 1]$ and a range of $[0, \pi]$ that satisfies the relation $\cos{y} = x$.

- arctangent

The inverse tangent function, denoted by $y = tan^{-1}{x}$ or y = $\arctan{x}$, is the function with a domain of $(-\infty, \infty)$ and a range of $(-\frac{\pi}{2}, \frac{\pi}{2})$ that satisfies the relation $\tan{y} = x$.

32. Vectors

A vector is a directed line segment. The symbol $\overrightarrow{PQ}$ (read "vector PQ") denotes the vector extending from point $P(x_1, y_1)$, the initial point, to point $Q(x_2, y_2)$, the terminal point. $\overrightarrow{PQ} = \langle x_2, - x_1, y_2 - y_1 \rangle$.

• Boldface letters such as u and v are often used to denote vectors, and the vector $\overrightarrow{PQ}$ can also be written as u $= \langle a, b \rangle = \langle x_2 - x_1, y_2 - y_1 \rangle$.
• The length or magnitude of the vector u $= \langle a, b \rangle$ is. denoted by $|u|$, and $|u| = \sqrt{a^2 + b^2}$.
• If two vectors u and v have the same length and direction, we say that they are equal and we write u = v.

- Algebraic Approach

• $k$u$=k\langle a, b \rangle = \langle ka, kb \rangle$.
• Let u $= \langle a, b \rangle$ and v $= \langle c, d \rangle$ be two vectors. Then...
  - u $+$ v $= \langle a, b \rangle + \langle c, d \rangle = \langle a + c, b + d \rangle$
  - u $-$ v $= \langle a, b \rangle - \langle c, d \rangle = \langle a - c, b - d \rangle$

용어 정리

• line segment: 선분
• Algebraic: 대수적

33. Polar Coordinates

A polar coordinate system consists of a point $O$ called the pole, and a ray called the polar axis, whose endpoint is the fixed point $O$.

The polar coordinates of a point $P$ are an ordered pair $(r, \theta)$, where $r$ is the distance from the origin $O$ to the point $P$, and $\theta$ is the measure of the angle formed by the polar axis and the segment $\overrightarrow{OP}$.

- Comparing Rectangular and Polar Coordinate Systems

Rectangular Coordinates	Polar Coordinates
Origin	Pole
Positive $x$-axis	Polar axis or $\theta = 0$
$P(x, y)$	$P(r, \theta)$
Positive $y$-axis	$\theta = \pi / 2$

- Coordinate-System Conversion Formulas

• $x = r\cos{\theta}$   $y = r\sin{\theta}$
• $r^2 = x^2 + y^2$   $\tan{\theta} = \frac{y}{x} (x \neq 0)$

용어 정리

• pole: 극
• polar axis: 극축
• polar coordinate: 극좌표계

+ 추가

 모든 파트를 다 정리하고 느낀 점은 mathematics fields 에서 사용하는 terminology 를 번역하는 것이 큰 의미가 없다는 생각이 들었다. 그 이유는 아래와 같다:

1. 번역된 용어가 원용어 의미를 완전하게 전달하지 못함.

 local maximum 과 local minimum 을 극대(極大), 그리고 극소(極小) 로 번역하는 것보단 원문을 그대로 놓고 보는 것이 그 의미가 더 명확하다는 생각이 들었다.

2. 사고 과정이 한 단계 추가되어 의미를 이해하는데 시간이 오래 걸린다.

 달리다 가 무슨 뜻인가? 보통 이런 생각을 하는 사람은 없다. 달리다 가 무슨 뜻인지 생각하기 이전에 달리다 라는 용어가 주는 그 느낌과 의미가 머리를 지배한다. 그러나 원어를 한국어로 번역하기 시작하면서 필자는 아래와 같이 사고하기 시작했다.

1. "run 이 한국어로 무슨 뜻이지?"
2. "아! run 은 한국어로 달리다 구나?"
3. 이제 달리다 라는 단어의 느낌과 그 의미를 머릿 속으로 떠올림.

 따라서 원용어 그 자체가 주는 느낌과 의미를 더 잘 기억할 수 있도록, 원어를 한국어로 번역하지 않을 예정이다. 그것 자체가 주는 느낌과 의미, 감각만을 떠올리기 위해 노력할 것이다.

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따라서 새로운 terms 가 나오면 그 해석을 한국어가 아닌 원어로 풀어쓴다.

terminal side: 종선(終線), 마칠 종, 줄 선

위 구문을 이후의 챕터부턴 아래와 같이 작성할 것이다:

terminal side: A straight line that has been rotated around a poin on another line to form an angle measured in a clockwise or counterclockwise direction.

출처

[책] Man Sik Min · Hyeong Chul Jeong · Hyejung Lee, 『CALCULUS』, 한티미디어, p1-47.
[이미지] https://www.mathopenref.com/trigterminalside.html
[용어] https://blog.naver.com/changkh/30183722678
[용어] https://wildcatprecalc.weebly.com/5-initial-and-terminal-side.
[이미지] https://www.seekpng.com/ima/u2w7e6q8u2q8a9w7/
[용어] https://libraryguides.centennialcollege.ca/c.php?g=716824&p=5112868
[용어] https://suhak.tistory.com/300