第1章AP微積分簡介 Introduction of AP Calculus001
1.1課程及考試The Courses and Examinations001
1.2AP微積分AB和BC大綱要求 The Examination Outline of AP Calculus AB & BC004
1.3AP微積分參考詞匯錶 Reference Vocabulary of AP Calculus006
1.4圖形計算器的使用 Use of Graphing Calculators013
第2章函數 Functions019
2.1函數的定義 Definition of Functions020
2.2函數的基本性質 Function Basic Properties022
2.3基本初等函數 Basic Elementary Functions023
2.4反函數&復閤函數 InverseFunctions & Composite Functions033
2.5函數變換 Transforming of Functions035
2.6#參數方程&嚮量函數 ParametricEquations & Vector Functions037
2.7#極坐標函數 Polar Functions039
2.8習題 Practice Exercises041
第3章極限 Limit043
3.1極限的定義 Definition of the Limit044
3.2極限存在的判定 The Limit does Exist or Not045
3.3極限的運算 Operations of Limit047
3.4極限的應用 Applications of Limit052
3.5習題 Practice Exercises053
第4章連續 Continuity055
4.1連續性的定義 Definition of the Continuity056
4.2間斷點的分類 Kinds of Discontinuities059
4.3連續函數定理 The Continuous Functions Theorem061
4.4習題 Practice Exercises063
第5章導數和微分 Derivative andDifferential065
5.1導數的定義 Definition of the Derivative066
5.2可導性和連續性 Derivability and Continuity072
5.3導數的基本公式和法則 Basic Differentiation Formulas andRules075
5.4鏈式法則和反函數求導 The Chain Rule & Derivative of anInverse Function077
5.5隱函數求導和二階導數 Implicit Differentiation & SecondDerivatives082
5.6#參數方程求導 Derivatives of Parametric Equations088
5.7#嚮量函數和極坐標函數求導 Derivatives of Vector Functions andPolar Functions090
5.8微分 Differential093
5.9習題 Practice Exercises096
第6章微分的應用 Applications ofDifferential Calculus098
6.1切綫方程和法綫方程Equations of Tangent and Normal099
6.2最值問題The Problems of Maxima and Minima101
6.3運動問題The Problems of Motion112
6.4微分中值定理The Mean Value Theorem for Derivatives118
6.5洛必達法則L’Hpital’s Rule120
6.6估算問題The Problems of Estimate125
6.7#歐拉方法Euler’s Method129
6.8習題Practice Exercises130
第7章不定積分 The IndefiniteIntegral132
7.1不定積分的定義Definition of The Indefinite Integral133
7.2不定積分公式Formulas of The Indefinite Integral135
7.3U-替換法U-Substitution138
7.4#分部積分法Integration by Parts148
7.5#有理函數的積分Integration of Rational Functions153
7.6不定積分的應用Applications of Indefinite Integral156
7.7習題Practice Exercises157
第8章定積分 The Definite Integral159
8.1黎曼和與梯形法則Riemann Sums and Trapezoid Rule160
8.2定積分的定義Definition of the Definite Integral165
8.3微積分基本定理The Fundamental Theorem of Calculus169
8.4定積分的性質Properties of Definite Integral174
8.5積分中值定理The Mean Value Theorem for Integrals176
8.6定積分的計算The Operations of Definite Integrate178
8.7#廣義積分Improper Integrals180
8.8習題Practice Exercises185
第9章積分的應用 Applications ofIntegral186
9.1麵積Area187
9.2體積Volume195
9.3#弧長 Arc Length204
9.4位移和距離Displacement and Distance206
9.5習題 Practice Exercises207
第10章微分方程 DifferentialEquations209
10.1一階微分方程First-Order Differential Equations210
10.2求解可分離變量微分方程Solving Separable D.E.211
10.3斜率場 Slope Fields213
10.4指數增長與衰減 Exponential Growth and Decay216
10.5約束增長與衰減 Restricted Growth and Decay219
10.6#邏輯斯諦微分方程Logistic Differential Equation222
10.7習題 Practice Exercises225
第11章無窮級數 Infinite Series226
11.1數列的極限 The Limit of The Sequence227
11.2無窮級數 Infinite Series228
11.3四類重要級數Four Important Series232
11.4正項級數的四大判彆法Four Tests of Nonnegative Series235
11.5絕對收斂和條件收斂 Absolute and Conditional Convergence240
11.6冪級數 Power Series242
11.7泰勒級數和麥剋勞林級數 Taylor and Maclaurin Series245
11.8冪級數的計算 Computations with Power Series251
11.9習題 Practice Exercises254
習題答案Practice Answer255
附錄Appendix287
A.1常用公式和定理Common Formulas and Theorems287
A.2AP微積分公式總結Summary AP Calculus Formula291
A.3VIP服務及網站298
參考文獻References299
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