Calculus Min yan Department of mathematics Hong Kong University of Science and Technology November 4. 2016
Calculus Min Yan Department of Mathematics Hong Kong University of Science and Technology November 4, 2016
Contents 1.1 Limit of Sequence 1.1.1 Arithmetic Rule 9 1.1.2 Sandwich Rule 1.1.3 Some Basic Limits 19 1.2 Rigorous Definition of Sequence Limit 1.2.1 Rigorous Definition 1.2.2 The art of estimation 1.2.3 Rigorous Proof of limits 31 1. 2. 4 Rigorous Proof of Limit Propertie 1.3 Criterion for Convergence 1.3.1 Monotone Sequence 1.3.2 Application of Monotone Sequence 1.3.3 Cauchy Criterion 1.4 Infinity 1.4.1 Divergence to Infinity 1.4.2 Arithmetic Rule for Infinity 1.4.3 Unbounded Monotone Sequence 1.5 Limit of Function 1.5. 1 Properties of Function Limit 53 1.5.2 Limit of Composition Function 1. 5.3 One Sided Limit 5.4 Limit of Trigonometric Function 1.6 Rigorous Definition of Function Limit 1.6.1 Rigorous Proof of Basic Limits 67 1.6.3 Relation to Sequence Lims of Limit 1.6.2 Rigorous Proof of Propertie 1.6.4 More Properties of Function Limit 1.7 Continuity 1.7.1 Meaning of Continuity 1.7.2 Intermediate Value Theorem
Contents 1 Limit 7 1.1 Limit of Sequence . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7 1.1.1 Arithmetic Rule . . . . . . . . . . . . . . . . . . . . . . . . . . 9 1.1.2 Sandwich Rule . . . . . . . . . . . . . . . . . . . . . . . . . . 11 1.1.3 Some Basic Limits . . . . . . . . . . . . . . . . . . . . . . . . 15 1.1.4 Order Rule . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19 1.1.5 Subsequence . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22 1.2 Rigorous Definition of Sequence Limit . . . . . . . . . . . . . . . . . . 24 1.2.1 Rigorous Definition . . . . . . . . . . . . . . . . . . . . . . . . 26 1.2.2 The Art of Estimation . . . . . . . . . . . . . . . . . . . . . . 28 1.2.3 Rigorous Proof of Limits . . . . . . . . . . . . . . . . . . . . . 31 1.2.4 Rigorous Proof of Limit Properties . . . . . . . . . . . . . . . 33 1.3 Criterion for Convergence . . . . . . . . . . . . . . . . . . . . . . . . 37 1.3.1 Monotone Sequence . . . . . . . . . . . . . . . . . . . . . . . . 38 1.3.2 Application of Monotone Sequence . . . . . . . . . . . . . . . 42 1.3.3 Cauchy Criterion . . . . . . . . . . . . . . . . . . . . . . . . . 45 1.4 Infinity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 48 1.4.1 Divergence to Infinity . . . . . . . . . . . . . . . . . . . . . . . 48 1.4.2 Arithmetic Rule for Infinity . . . . . . . . . . . . . . . . . . . 50 1.4.3 Unbounded Monotone Sequence . . . . . . . . . . . . . . . . . 52 1.5 Limit of Function . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 53 1.5.1 Properties of Function Limit . . . . . . . . . . . . . . . . . . . 53 1.5.2 Limit of Composition Function . . . . . . . . . . . . . . . . . 56 1.5.3 One Sided Limit . . . . . . . . . . . . . . . . . . . . . . . . . 61 1.5.4 Limit of Trigonometric Function . . . . . . . . . . . . . . . . . 63 1.6 Rigorous Definition of Function Limit . . . . . . . . . . . . . . . . . . 66 1.6.1 Rigorous Proof of Basic Limits . . . . . . . . . . . . . . . . . 67 1.6.2 Rigorous Proof of Properties of Limit . . . . . . . . . . . . . . 70 1.6.3 Relation to Sequence Limit . . . . . . . . . . . . . . . . . . . 73 1.6.4 More Properties of Function Limit . . . . . . . . . . . . . . . 78 1.7 Continuity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 80 1.7.1 Meaning of Continuity . . . . . . . . . . . . . . . . . . . . . . 81 1.7.2 Intermediate Value Theorem . . . . . . . . . . . . . . . . . . . 82 3
CONTENTS 1.7.3 Continuous Inverse Function 1.7.4 Continuous Change of Variable 2 Differentiation 93 2.1 Linear Approximation 2.1.1 Derivative 2.1.2 Basic derivative 2.1.3 Constant Approximation 4 One Sided Derivative 100 2.2 Property of Derivative 2.2.1 Arithmetic Combination of Linear Approximation 2.2.2 Composition of Linear Approximation 102 2.2.3 Implicit Linear Approximation 109 2.3 Application of Linear Approximation 113 2.3.1 Monotone Property and Extrema 113 2.3.2 Detect the Monotone Property 115 2.3.3 Compare Functions 118 2.3.4 First derivative Test 120 2.3.5 Optimization Problem 122 2.4 Main value theorem 125 2.4.1 Mean value Theorem 125 2.4.2 Criterion for Constant Function 127 2.4.3 L Hospital's rule 129 2.5 High Order Approximation 133 2.5.1 Taylor Expansion 2.5.2 High Order Approximation by Substitution 2.5.3 Combination of High Order Approximations 143 2.5.4 Implicit High Order Differentiation 2.5.5 Two Theoretical Examples 2.6 Application of High Order Approximation 2.6.2 Convex Function 152 ch of Graph 2.7 Numerical Application 2.7.1 Remainder Formula 2.7.2 wton's method 3 Integration 169 3.1 Area and Definite Integral 169 3.1.1 Area below Non-negative Function 3.1.2 Definite Integral of Continuous Function 3.1.3 Property of Area and Definite Integral 175 3.2 Rigorous Definition of Integral 178
4 CONTENTS 1.7.3 Continuous Inverse Function . . . . . . . . . . . . . . . . . . . 84 1.7.4 Continuous Change of Variable . . . . . . . . . . . . . . . . . 88 2 Differentiation 93 2.1 Linear Approximation . . . . . . . . . . . . . . . . . . . . . . . . . . 93 2.1.1 Derivative . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 94 2.1.2 Basic Derivative . . . . . . . . . . . . . . . . . . . . . . . . . . 95 2.1.3 Constant Approximation . . . . . . . . . . . . . . . . . . . . . 98 2.1.4 One Sided Derivative . . . . . . . . . . . . . . . . . . . . . . . 100 2.2 Property of Derivative . . . . . . . . . . . . . . . . . . . . . . . . . . 101 2.2.1 Arithmetic Combination of Linear Approximation . . . . . . . 101 2.2.2 Composition of Linear Approximation . . . . . . . . . . . . . 102 2.2.3 Implicit Linear Approximation . . . . . . . . . . . . . . . . . . 109 2.3 Application of Linear Approximation . . . . . . . . . . . . . . . . . . 113 2.3.1 Monotone Property and Extrema . . . . . . . . . . . . . . . . 113 2.3.2 Detect the Monotone Property . . . . . . . . . . . . . . . . . 115 2.3.3 Compare Functions . . . . . . . . . . . . . . . . . . . . . . . . 118 2.3.4 First Derivative Test . . . . . . . . . . . . . . . . . . . . . . . 120 2.3.5 Optimization Problem . . . . . . . . . . . . . . . . . . . . . . 122 2.4 Main Value Theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . 125 2.4.1 Mean Value Theorem . . . . . . . . . . . . . . . . . . . . . . . 125 2.4.2 Criterion for Constant Function . . . . . . . . . . . . . . . . . 127 2.4.3 L’Hospital’s Rule . . . . . . . . . . . . . . . . . . . . . . . . . 129 2.5 High Order Approximation . . . . . . . . . . . . . . . . . . . . . . . . 133 2.5.1 Taylor Expansion . . . . . . . . . . . . . . . . . . . . . . . . . 136 2.5.2 High Order Approximation by Substitution . . . . . . . . . . . 139 2.5.3 Combination of High Order Approximations . . . . . . . . . . 143 2.5.4 Implicit High Order Differentiation . . . . . . . . . . . . . . . 148 2.5.5 Two Theoretical Examples . . . . . . . . . . . . . . . . . . . . 149 2.6 Application of High Order Approximation . . . . . . . . . . . . . . . 151 2.6.1 High Derivative Test . . . . . . . . . . . . . . . . . . . . . . . 151 2.6.2 Convex Function . . . . . . . . . . . . . . . . . . . . . . . . . 152 2.6.3 Sketch of Graph . . . . . . . . . . . . . . . . . . . . . . . . . . 157 2.7 Numerical Application . . . . . . . . . . . . . . . . . . . . . . . . . . 161 2.7.1 Remainder Formula . . . . . . . . . . . . . . . . . . . . . . . . 162 2.7.2 Newton’s Method . . . . . . . . . . . . . . . . . . . . . . . . . 164 3 Integration 169 3.1 Area and Definite Integral . . . . . . . . . . . . . . . . . . . . . . . . 169 3.1.1 Area below Non-negative Function . . . . . . . . . . . . . . . 169 3.1.2 Definite Integral of Continuous Function . . . . . . . . . . . . 172 3.1.3 Property of Area and Definite Integral . . . . . . . . . . . . . 175 3.2 Rigorous Definition of Integral . . . . . . . . . . . . . . . . . . . . . . 178
CONTENTS 3.2.1 What is Area? 3.2.3 Riemann Sum 183 3.3 Numerical Calculation of Integral 3.3.1 Left and right Rule 3.3.2 Midpoint Rule and Trapezoidal Rule 186 3.3.3 Simpson's rule 189 3.4 Indefinite Integral 3.4.1 Fundamental Theorem of Calculus 3.4.2 Indefinite Integral 3.5 Properties of Integration 198 3.5.1 Linear Property 198 3.5.2 Integration by Part 204 3.5.3 Change of Variable 214 3.6 Integration of Rational Function 228 3.6.1 Rational function r+b 3.6.2 Rational Function of 3.6.3 Rational Function of sin a and cos x 3.7 Improper Integral 240 3.7.1 Definition and Property 240 3.7.2 Comparison Test 245 3.7.3 Conditional Convergence 249 3.8 Application to Geometry 3.8.1 Length of Curve 253 3.8.2 Area of Region 3. 8.3 Surface of Revolution 3.8.4 Solid of Revolution 5 Cavalieri's Principle 3.9 Polar Coordinate 3.9.1 Curves in Polar Coordinate 3.9.2 Geometry in Polar Coordinate 3.10 Application to Physics 3.10.1 Work and Pressure 3.10.2 Center of Mass 4 Series 297 4.1 Series of numbers 4.1.1 Sum of se erles 298 4. 1.2 Property of Converging Series 300 4.2 Comparison Test 4.2.1 Integral test 4.2.2 Comparison Test
CONTENTS 5 3.2.1 What is Area? . . . . . . . . . . . . . . . . . . . . . . . . . . . 178 3.2.2 Darboux Sum . . . . . . . . . . . . . . . . . . . . . . . . . . . 181 3.2.3 Riemann Sum . . . . . . . . . . . . . . . . . . . . . . . . . . . 183 3.3 Numerical Calculation of Integral . . . . . . . . . . . . . . . . . . . . 184 3.3.1 Left and Right Rule . . . . . . . . . . . . . . . . . . . . . . . 184 3.3.2 Midpoint Rule and Trapezoidal Rule . . . . . . . . . . . . . . 186 3.3.3 Simpson’s Rule . . . . . . . . . . . . . . . . . . . . . . . . . . 189 3.4 Indefinite Integral . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 191 3.4.1 Fundamental Theorem of Calculus . . . . . . . . . . . . . . . 191 3.4.2 Indefinite Integral . . . . . . . . . . . . . . . . . . . . . . . . . 195 3.5 Properties of Integration . . . . . . . . . . . . . . . . . . . . . . . . . 198 3.5.1 Linear Property . . . . . . . . . . . . . . . . . . . . . . . . . . 198 3.5.2 Integration by Parts . . . . . . . . . . . . . . . . . . . . . . . 204 3.5.3 Change of Variable . . . . . . . . . . . . . . . . . . . . . . . . 214 3.6 Integration of Rational Function . . . . . . . . . . . . . . . . . . . . . 228 3.6.1 Rational Function . . . . . . . . . . . . . . . . . . . . . . . . . 228 3.6.2 Rational Function of n r ax + b cx + d . . . . . . . . . . . . . . . . . . 235 3.6.3 Rational Function of sin x and cos x . . . . . . . . . . . . . . . 238 3.7 Improper Integral . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 240 3.7.1 Definition and Property . . . . . . . . . . . . . . . . . . . . . 240 3.7.2 Comparison Test . . . . . . . . . . . . . . . . . . . . . . . . . 245 3.7.3 Conditional Convergence . . . . . . . . . . . . . . . . . . . . . 249 3.8 Application to Geometry . . . . . . . . . . . . . . . . . . . . . . . . . 253 3.8.1 Length of Curve . . . . . . . . . . . . . . . . . . . . . . . . . . 253 3.8.2 Area of Region . . . . . . . . . . . . . . . . . . . . . . . . . . 258 3.8.3 Surface of Revolution . . . . . . . . . . . . . . . . . . . . . . . 265 3.8.4 Solid of Revolution . . . . . . . . . . . . . . . . . . . . . . . . 269 3.8.5 Cavalieri’s Principle . . . . . . . . . . . . . . . . . . . . . . . . 276 3.9 Polar Coordinate . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 283 3.9.1 Curves in Polar Coordinate . . . . . . . . . . . . . . . . . . . 284 3.9.2 Geometry in Polar Coordinate . . . . . . . . . . . . . . . . . . 288 3.10 Application to Physics . . . . . . . . . . . . . . . . . . . . . . . . . . 292 3.10.1 Work and Pressure . . . . . . . . . . . . . . . . . . . . . . . . 292 3.10.2 Center of Mass . . . . . . . . . . . . . . . . . . . . . . . . . . 295 4 Series 297 4.1 Series of Numbers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 297 4.1.1 Sum of Series . . . . . . . . . . . . . . . . . . . . . . . . . . . 298 4.1.2 Property of Converging Series . . . . . . . . . . . . . . . . . . 300 4.2 Comparison Test . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 302 4.2.1 Integral Test . . . . . . . . . . . . . . . . . . . . . . . . . . . . 303 4.2.2 Comparison Test . . . . . . . . . . . . . . . . . . . . . . . . . 305
6 CONTENTS 4.2.3 Special Comparison Test 309 4.3 Conditional Convergence 4.3.1 Test for Conditional Convergence 4.3.2 Absolute v.s. Conditional 317 4.4 Power Series 320 4.4.1 Radius of Convergence 322 4.4.2 Function Defined by Power Series 325 4.5 Fourier Series 328 4.5.1 Fourier Coefficient 329 4.5.2 Complex Form of Fourier Series 4.5.3 Derivative and Integration of Fourier Series 336 4.5.4 Sum of Fourier Series 339 4.5.5 Parseval's Identity
6 CONTENTS 4.2.3 Special Comparison Test . . . . . . . . . . . . . . . . . . . . . 309 4.3 Conditional Convergence . . . . . . . . . . . . . . . . . . . . . . . . . 313 4.3.1 Test for Conditional Convergence . . . . . . . . . . . . . . . . 314 4.3.2 Absolute v.s. Conditional . . . . . . . . . . . . . . . . . . . . 317 4.4 Power Series . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 320 4.4.1 Radius of Convergence . . . . . . . . . . . . . . . . . . . . . . 322 4.4.2 Function Defined by Power Series . . . . . . . . . . . . . . . . 325 4.5 Fourier Series . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 328 4.5.1 Fourier Coefficient . . . . . . . . . . . . . . . . . . . . . . . . 329 4.5.2 Complex Form of Fourier Series . . . . . . . . . . . . . . . . . 334 4.5.3 Derivative and Integration of Fourier Series . . . . . . . . . . . 336 4.5.4 Sum of Fourier Series . . . . . . . . . . . . . . . . . . . . . . . 339 4.5.5 Parseval’s Identity . . . . . . . . . . . . . . . . . . . . . . . . 341