미적분학 1에서는 일변수 함수를 다룬다. 어떤 독자들에게는 이 책이 과학적 내용이 지나치게 많아 다소 당혹스럽게 느껴질 수도 있을 것이다. 그러나 뉴턴이 케플러 법칙을 보이기 위하여 만든 것이 미적분학의 시작이 되었고 그것이 이 책 전반부의 핵심 주제이다. 극한, 미분, 적분, 관성 좌표계, 극좌표계, 변수 변환, 미분 방정식은 이 과학적 목표를 달성하기 위한 수학적 요소들이다. 미적분학은 뉴턴 이후에도 많은 발전을 이루어 왔다. 이 책의 후반부에서는 적분과 미분 기법을 배우지만 핵심은 수열과 급수를 이용한 근사 기법이다. 이 책에서 학생들이 주도적으로 생각할 수 있도록 적절한 질문을 제시하려고 노력하였다. 질문은 학습을 이끄는 원동력이자 창의적 사고의 출발점이다. 이 책에 담긴 질문과 문제들에 대해 생각하고 답을 찾아가는 과정 속에서 학생들이 미적분학을 더욱 깊이 이해할 수 있기를 바란다.

Part I Differentiation: Mathematical Description of Motion 1 Limit and Continuity #1 (with Common Language) 3 1.1 Common Language Definitions 4 1.2 ε-δ arguments for quality control 9 2 Limit and Continuity #2 (with ε-δ argument) 15 2.1 Rigorous definitions using ε-δ 15 2.2 Examples 20 2.3 Limits as and 22 3 Differentiation 25 3.1 Rate of Increase 25 3.2 Differentiation Rules 28 3.3 Intermediate and Mean Value Theorem 32 4 Chain Rule and Variable-Centered Notation 37 4.1 Chain Rule 37 4.2 Variable-Centered Notation 42 5 Integration & Fundamental Theorem of Calculus 49 5.1 Antiderivative 49 5.2 Area Function as an Integral 51 5.3 Riemann Sum and Area 53 6 Inverse Functions and Their Derivatives 63 6.1 Graph and Differentiation 63 6.2 Inverse Functions in Variable-Centered Notation 67 6.3 Integration by change of variables 68 7 Logarithm, Exponential, & Implicit Differentiation 73 7.1 Natural logarithm 73 7.2 Exponential function 76 7.3 Implicit differentiation 78 Part II Kepler and Newton’s Laws of Motion 8 Rectangular Coordinate System and Curves in 85 8.1 Projection and coordinate system 85 8.2 Vector Space 90 8.3 Inner Product 91 8.4 Cross product 94 9 Polar coordinates in 99 9.1 Moving particle and trajectory curves in space 99 9.2 Polar coordinates 100 9.3 Motion in polar coordinates 103 9.4 Ellipse in polar coordinates 105 9.5 Curves in polar coordinates (optional) 107 10 First Order Differential Equations 111 10.1 First order differential equations 111 10.2 Separation of variables 116 10.3 First-Order Linear Equation and Integrating Factor 118 11 Second Order Differential Equations 123 11.1 Second Order Linear Equation 123 11.2 Homogeneous Problem and Characteristic Polynomial 127 11.3 Initial value problem 131 12 Newton’s Law on Earth’s Surface 135 12.1 Newton’s law of motion and gravitation 135 12.2 Work and energy 136 12.3 Gravity force and potential energy on Earth 137 12.4 Projectile motion on Earth 139 13 Newton’s Law in Space: Two-Body Problem 145 13.1 Two-body problem 145 13.2 Center of mass (barycenter) 146 13.3 Kepler problem 148 13.4 Orbit of ICBM #1 (Optional) 152 14 Kepler’s Law (Optional) 157 14.1 Kepler’s first law and elliptical orbits 157 14.2 Examples and Applications 161 Part III The Arts of Calculus 15 Curves and Particle Trajectories in 169 15.1 Arclength as a variable 169 15.2 TNB coordinate system 172 15.3 Computation formulas 177 16 Linearization and Differentiation 181 16.1 Linearization 181 16.2 Differentials 183 16.3 Differentials for linear approximation 185 17 Inverse trigonometric and hyperbolic functions 189 17.1 Inverse trigonometric functions 189 17.2 Hyperbolic functions 194 18 L’H?pital’s rule and big-oh / little-oh 199 18.1 L’H?pital’s rule 199 18.2 Calculating limits using inverse functions 203 18.3 Big-oh and little-oh 204 19 Integration by substitution and by parts 207 19.1 Substitution 208 19.2 Integration by parts 210 19.3 Trigonometric substitution 213 20 Rational Functions and Improper Integration 217 20.1 Integration of rational functions 217 20.2 Integral over unbounded domains 221 20.3 Integral of unbounded functions 224 Part IV Approximation Techniques and Series 21 Numerical Integration 231 21.1 Numerical integration and Riemann sum 231 21.2 Convergence order 233 21.3 Numerical integration and Gauss-Legendre quadrature 236 22 Sequences and Series 243 22.1 Sequence of real numbers 243 22.2 Series of real numbers 248 22.3 Power series 250 23 Tests for Absolute Convergence 255 23.1 Integral Test 255 23.2 Comparison Test 257 23.3 Ratio test 258 23.4 Root test 261\ 24 Power Series 267 24.1 Convergence of a power series 267 24.2 Radius of convergence 269 24.3 Alternating series 272 24.4 Rearrangement and conditional convergence 274 25 Taylor Series 279 25.1 Taylor series 279 25.2 Applications and two other versions 285 25.3 Convergence order of Gauss-Legendre 290 Part V Appendix 26 Energy of Planet Orbits 295 26.1 Potential energy in space 296 26.2 Energy of circular orbits 296 26.3 Energy of elliptical orbits 298 26.4 Interstellar and solar system objects 301 27 Elliptic and Hyperbolic Orbits 303 27.1 Eccentricity and focus of an ellipse 303 27.2 Directrices and ellipses 304 27.3 Polar equations of an ellipse 307 27.4 Display solar system orbits 309 27.5 Orbit of ICBM #2 310 28 Calculus for Planet Orbits 315 28.1 Variable centered notation 315 28.2 Coordinate system 316 28.3 Motion in polar coordinates 316 28.4 Newton’s law of motion and gravitation 317 28.5 Two-body problem 318 28.6 Kepler problem 319 Index 321