미적분학은 자연 계열 및 공학 계열 학생들이 대학 과정을 이수하는 데 있어서 기초가 되는 분야이며, 대부분의 전공서적을 이해하는 데 반드시 필요한 과목이다. 이런 관점에서 이 책은 자연 계열 및 공학 계열 학생을 위한 미적분학 입문서로 개발되었다. 자연 계열 및 공학 계열 학생들에게 논리적인 사고력과 전공을 공부하는 데 필수적인 수학적 기본지식을 갖추도록 하는 것이 이 책을 쓰게 된 목적이다. 이 책은 학생들이 논리적인 사고력을 갖추도록 함과 동시에, 충분한 계산 능력을 갖추도록 다양한 예를 수록하여 집필하였다. 특히, 수학의 기초가 부족한 학생을 위하여, 함수의 기본적인 내용과 함께 여러 가지 함수, 방정식, 행렬 등을 소개하고 있으며, 공학 계열 학생을 위하여 벡터를 소개하고 있다. 또한 예제와 함께 같은 유형의 문제를 제시하여, 학생들이 예제를 복습하고 스스로 문제를 해결할 수 있는 능력을 익힐 수 있도록 구성하려고 노력하였다.

이 책의 내용은 주당 3시간 분량으로 2학기 동안 강의할 수 있는 내용으로 구성되어 있다. 1장에서는 함수에 관한 기본적인 내용과 여러 가지 함수들을 소개하고 있으며, 2장에서는 기본적인 방정식의 해법을 다루고 있다. 그리고 3장에서는 미분을 공부하기 위한 전 단계로서 함수의 극한과 연속을 다루고 있다. 4장과 5장에서는 미분과 미분의 응용을 다루고 있으며, 6장과 7장에서는 적분과 적분의 응용을 다루고 있다. 8장에서는 행렬, 9장에서는 벡터에 관한 내용을 다루고 있다. 그리고 10장에서는 편미분과 중적분에 관한 내용을 다루고 있다.

CHAPTER 01 함수

1.1 함수의 정의 ···········································1

1.2 함수의 종류 ··········································10

1.3 지수함수와 로그함수 ···································23

1.4 삼각함수와 역삼각함수 ·································33

1.5 쌍곡선함수 ···········································47

CHAPTER 02 방정식

2.1 분수방정식 ···········································51

2.2 무리방정식 ···········································54

2.3 지수방정식 ···········································56

2.4 로그방정식 ···········································58

CHAPTER 03 함수의 극한과 연속

3.1 극한의 개념 ··········································63

3.2 극한의 계산 ··········································72

3.3 함수의 연속성 ········································82

3.4 무한극한 ·············································90

CHAPTER 04 미분

4.1 도함수 ·············································105

4.2 미분법 ·············································111

∙ 다항함수의 미분 _ 113

∙ 유리함수의 미분 _ 115

∙ 무리함수의 미분 _ 116

4.3 삼각함수 미분 ·······································120

4.4 지수함수, 로그함수 미분 ······························123

4.5 연쇄법칙 ············································126

4.6 음함수의 미분 ·······································139

∙ 음함수와 역함수 _ 144

4.7 고차도함수 ··········································149

∙ 음함수의 고계도함수 _ 150

4.8 매개변수로 표현된 함수의 미분 ·························153

CHAPTER 05 미분의 응용

5.1 로피탈 정리 ·········································165

∙ 로피탈 정리 _ 169

5.2 그래프 그리기 ·······································174

∙ 그래프 그리기 _ 188

5.3 최적화 문제 ·········································201

CHAPTER 06 적분

6.1 부정적분 ············································209

6.2 정적분 ·············································216

6.3 미적분학의 기본 정리 ·································229

6.4 적분법 ·············································236

∙ 치환적분 _ 236

∙ 유리함수의 적분 _ 245

∙ 부분적분 _ 251

∙ 삼각함수의 적분 _ 254

6.5 이상적분 ············································262

CHAPTER 07 적분의 응용

7.1 넓이 ···············································267

7.2 체적 ···············································276

7.3 곡선의 길이 ·········································292

7.4 회전면의 겉넓이 ·····································298

7.5 매개변수로 표현된 함수의 적분 ·························304

CHAPTER 08 행렬

8.1 행렬의 정의와 기본 성질 ······························311

8.2 행렬식과 역행렬 ·····································317

8.3 크레이머 공식(Cramer’s rule) ························325

8.4 기본 행연산과 가우스-조르단 소거법 ····················328

CHAPTER 09 벡터

9.1 기본 개념 ···········································333

9.2 벡터의 내적 ·········································340

9.3 벡터의 외적 ·········································347

∙ 스칼라 삼중곱 _ 350

9.4 직선과 평면의 방정식 ·································353

∙ 직선의 방정식 _ 353

∙ 평면의 방정식 _ 355

9.5 극좌표 ·············································362

∙ 극방정식과 그래프 _ 365

∙ 대칭성 _ 367

∙ 극방정식의 그래프의 접선 _ 367

∙ 극좌표에서의 넓이와 길이 _ 369

∙ 극좌표에서의 곡선의 길이 _ 372

CHAPTER 10 편도함수와 중적분

10.1 다변수함수의 정의 및 그래프 ··························377

10.2 다변수함수의 편도함수 ·······························386

∙편도함수의 기하학적 의미 _ 390

∙고계편도함수 _ 392

10.3 이변수함수의 극값 ··································394

10.4 중적분 ············································401

∙이중적분의 개념 _ 401

∙반복적분 _ 402

∙극좌표를 이용한 이중적분 _ 408

∙이중적분으로 나타낸 체적 _ 411