Ⅰ권

1 함수와 극한 ··· 1
1.1 함수와 그 표현 ····················································1
1.2 필수적인 함수 목록 ···············································11
1.3 함수의 극한 ·····················································26
1.4 극한 셈 ·························································38
1.5 연속 ····························································49
1.6 무한대와 관련된 극한 ·············································61
1 복습 ···························································75
2 도함수 ··· 79
2.1 미분 계수와 변화율 ···············································79
2.2 도함수 ··························································90
2.3 기본적인 미분법 ·················································102
2.4 곱과 몫의 미분법 ················································114
2.5 연쇄 법칙 ······················································122
2.6 음함수의 미분법 ·················································131
2.7 서로 관련된 변화율 ··············································137
2.8 선형 근사와 미분 ················································144
2 복습 ···························································150
3 역함수: 지수·로그·역삼각 함수 ··· 155
3.1 지수 함수 ······················································155
3.2 역함수와 로그 ···················································161
3.3 로그 함수와 지수 함수의 도함수 ···································174
3.4 지수적 증가와 감소 ··············································182
3.5 역삼각 함수 ·····················································190
3.6 쌍곡선 함수 ·····················································196
3.7 부정형과 로피탈의 법칙 ··········································203
3 복습 ···························································211
4 미분의 활용 ··· 215
4.1 최댓값과 최솟값 ·················································215
4.2 평균값 정리 ····················································223
4.3 도함수와 그래프의 모양 ··········································230
4.4 곡선 그리기 ····················································239
4.5 최적화 문제 ····················································247
4.6 뉴턴의 방법 ····················································259
4.7 역도함수 ·······················································265
4 복습 ··························································272
5 적분 ··· 276
5.1 넓이와 거리 ····················································276
5.2 정 적분 ························································289
5.3 정 적분의 값 찾기 ···············································303
5.4 미적분학의 기본 정리 ············································314
5.5 치환 적분법 ····················································324
5 복습 ··························································332
6 적분의 기법 ··· 336
6.1 부분 적분법 ····················································336
6.2 삼각 적분과 삼각 치환 ···········································343
6.3 부분 분수 ······················································354
6.4 적분표와 컴퓨터 대수 체계 ········································362
6.5 근사 적분 ······················································368
6.6 특이 적분 ······················································382
6 복습 ··························································392
7 적분의 활용 ··· 396
7.1 곡선 사이의 넓이 ················································396
7.2 부피 ···························································402
7.3 기둥 껍질에 의한 부피 ············································413
7.4 호의 길이 ······················································419
7.5 회전면의 넓이 ··················································427
7.6 물리학과 공학에의 활용 ··········································432
7.7 미분 방정식 ····················································447
7 복습 ··························································458
8 급수 ··· 462
8.1 수열 ···························································462
8.2 급수 ···························································473
8.3 적분 판정법과 비교 판정법 ········································485
8.4 다른 수렴 판정법 ················································494
8.5 거듭제곱 급수 ···················································505
8.6 거듭제곱 급수에 의한 함수의 표현 ··································511
8.7 테일러 급수와 매클로린 급수 ······································518
8.8 테일러 다항 함수의 활용 ··········································532
8 복습 ··························································540
9 매개 방정식과 극 좌표 ··· 545
9.1 매개 곡선 ······················································545
9.2 매개 곡선의 미분과 적분 ··········································552
9.3 극 좌표 ························································560
9.4 극 좌표에서의 넓이와 길이 ········································570
9.5 극 좌표에서의 원뿔 곡선 ··········································575
9 복습 ·························································· 581
부록 부록 ··· A1
A 삼각법 ··························································A1
B 합의 기호 ······················································A10
C 적분으로 정의된 로그 함수 ········································A15
D 증명 ···························································A23


Ⅱ권
10 벡터와 공간 기하학 ··· 585
10.1 삼차원 좌표 체계 ···············································585
10.2 벡터 ··························································591
10.3 내적 ··························································600
10.4 외적 ··························································607
10.5 직선과 평면의 방정식 ···········································616
10.6 기둥과 이차 곡면 ···············································625
10.7 벡터 함수와 공간 곡선 ···········································631
10.8 호의 길이와 곡률 ···············································644
10.9 공간 운동: 속도와 가속도 ········································652
10 복습 ··························································662
11 편도함수 ··· 667
11.1 다변수 함수 ···················································667
11.2 극한과 연속 ···················································678
11.3 편도함수 ······················································686
11.4 접평면과 선형 근사 ·············································695
11.5 연쇄 법칙 ·····················································704
11.6 방향 도함수와 물매 벡터 ·········································714
11.7 최댓값과 최솟값 ················································726
11.8 라그랑주 곱수 ··················································735
11 복습 ·························································743
12 다중 적분 ··· 747
12.1 직사각형 위에서의 이중 적분 ·····································747
12.2 일반적인 영역 위에서의 이중 적분 ································760
12.3 극 좌표에서의 이중 적분 ·········································769
12.4 이중 적분의 활용 ···············································775
12.5 삼중 적분 ·····················································782
12.6 원기둥 좌표에서의 삼중 적분 ·····································793
12.7 구면 좌표에서의 삼중 적분 ·······································798
12.8 다중 적분에서의 변수 변환 ·······································804
12 복습 ··························································813
13 벡터 해석 ··· 818
13.1 벡터 마당 ·····················································818
13.2 선 적분 ·······················································825
13.3 선 적분의 기본 정리 ············································838
13.4 그린의 정리 ···················································848
13.5 회전과 발산 ···················································856
13.6 매개 곡면과 그 넓이 ············································864
13.7 면 적분 ·······················································876
13.8 스토크스의 정리 ················································888
13.9 발산 정리 ·····················································894
13 복습 ··························································902
부록 부록 ··· A1
D 증명 ···························································A38