DIFFERENTIAL AND INTEGRAL CALCULUS
I The questions 1 1 Limits 3 11 Drawing Tangents and a First Limit 3 12 Another Limit and Computing Velocity 4 13 The Limit of a Function 5 14 Calculating Limits with Limit Laws 7 15 Limits at Infinity 13 16 Continuity 15 2 Derivatives 19 21 Revisiting tangent lines 19 22 Definition of the derivative 22 23 Interpretations of the derivative 26 24 Arithmetic of derivatives - a differentiation toolbox 27 25 Proofs of the arithmetic of derivatives 29 26 Using the arithmetic of derivatives - examples 29 27 Derivatives of Exponential Functions 31 28 Derivatives of trigonometric functions 33 29 One more tool - the chain rule 36 210 The natural logarithm 39 211 Implicit Differentiation 41 212 Inverse Trigonometric Functions 43 213 The Mean Value Theorem 46 214 Higher Order Derivatives 48 3 Applications of derivatives 53 31 Velocity and acceleration 53 32 Related Rates 55 33 Exponential Growth and Decay 63 331 Carbon Dating 63 332 Newton’s Law of Cooling 64 333 Population Growth 66 334 Further problems 67 34 Taylor polynomials 68 341 Zeroeth Approximation 68 342 First Approximation 70 343 Second Approximation 71 344 Still Better Approximations 73 345 Some Examples 74 346 Estimating Changes and Dx, Dy notation 75 347 Further Examples 76 348 The error in Taylor polynomial approximation 78 349 Further problems 80 35 Optimization 82 351 Local and global maxima and minima 82 352 Finding global maxima and minima 84 353 Max/min examples 85 36 Sketching Graphs 88 361 Domain, intercepts and asymptotes 88 362 First derivative - increasing or decreasing 90 363 Second derivative - concavity 92 364 Symmetries 93 365 A checklist for sketching 96 366 Sketching examples 96 37 L’Hˆ opital’s Rule and indeterminate forms 100 4 Towards Integral Calculus 103 41 Introduction to antiderivatives 103 II Hints to problems 107 11 Drawing Tangents and a First Limit 109 12 Another Limit and Computing Velocity 109 13 The Limit of a Function 109 14 Calculating Limits with Limit Laws 110 15 Limits at Infinity 112 16 Continuity 113 21 Revisiting tangent lines 114 22 Definition of the derivative 114 23 Interpretations of the derivative 116 24 Arithmetic of derivatives - a differentiation toolbox 116 25 Proofs of the arithmetic of derivatives 116 26 Using the arithmetic of derivatives - examples 116 27 Derivatives of Exponential Functions 117 28 Derivatives of trigonometric functions 118 29 One more tool - the chain rule 119 210 The natural logarithm 120 211 Implicit Differentiation 122 212 Inverse Trigonometric Functions 123 213 The Mean Value Theorem 124 214 Higher Order Derivatives 125 31 Velocity and acceleration 126 32 Related Rates 127 331 Carbon Dating 129 332 Newton’s Law of Cooling 130 333 Population Growth 130 334 Further problems 131 341 Zeroeth Approximation 131 342 First Approximation 132 343 Second Approximation 132 344 Still Better Approximations 133 345 Some Examples 133 346 Estimating Changes and Dx, Dy notation 134 347 Further Examples 134 348 The error in Taylor polynomial approximation 135 349 Further problems 136 351 Local and global maxima and minima 137 352 Finding global maxima and minima 137 353 Max/min examples 138 361 Domain, intercepts and asymptotes 139 362 First derivative - increasing or decreasing 139 363 Second derivative - concavity 140 364 Symmetries 140 365 A checklist for sketching 141 366 Sketching examples 141 37 L’Hˆ opital’s Rule and indeterminate forms 142 41 Introduction to antiderivatives 143 III Answers to problems 145 11 Drawing Tangents and a First Limit 147 12 Another Limit and Computing Velocity 147 13 The Limit of a Function 148 14 Calculating Limits with Limit Laws 149 15 Limits at Infinity 152 16 Continuity 153 21 Revisiting tangent lines 156 22 Definition of the derivative 157 23 Interpretations of the derivative 159 24 Arithmetic of derivatives - a differentiation toolbox 160 25 Proofs of the arithmetic of derivatives 161 26 Using the arithmetic of derivatives - examples 161 27 Derivatives of Exponential Functions 163 28 Derivatives of trigonometric functions 163 29 One more tool - the chain rule 165 210 The natural logarithm 167 211 Implicit Differentiation 169 212 Inverse Trigonometric Functions 170 213 The Mean Value Theorem 174 214 Higher Order Derivatives 177 31 Velocity and acceleration 180 32 Related Rates 181 331 Carbon Dating 182 332 Newton’s Law of Cooling 183 333 Population Growth 183 334 Further problems 184 341 Zeroeth Approximation 184 342 First Approximation 185 343 Second Approximation 186 344 Still Better Approximations 186 345 Some Examples 187 346 Estimating Changes and Dx, Dy notation 187 347 Further Examples 188 348 The error in Taylor polynomial approximation 188 349 Further problems 190 351 Local and global maxima and minima 193 352 Finding global maxima and minima 195 353 Max/min examples 196 361 Domain, intercepts and asymptotes 197 362 First derivative - increasing or decreasing 197 363 Second derivative - concavity 197 364 Symmetries 200 365 A checklist for sketching 201 366 Sketching examples 202 37 L’Hˆ opital’s Rule and indeterminate forms 210 41 Introduction to antiderivatives 211 IV Solutions to problems 215 11 Drawing Tangents and a First Limit 217 12 Another Limit and Computing Velocity 218 13 The Limit of a Function 220 14 Calculating Limits with Limit Laws 222 15 Limits at Infinity 241 16 Continuity 249 21 Revisiting tangent lines 256 22 Definition of the derivative 258 23 Interpretations of the derivative 270 24 Arithmetic of derivatives - a differentiation toolbox 271 25 Proofs of the arithmetic of derivatives 275 26 Using the arithmetic of derivatives - examples 275 27 Derivatives of Exponential Functions 280 28 Derivatives of trigonometric functions 283 29 One more tool - the chain rule 292 210 The natural logarithm 306 211 Implicit Differentiation 317 212 Inverse Trigonometric Functions 325 213 The Mean Value Theorem 337 214 Higher Order Derivatives 349 31 Velocity and acceleration 359 32 Related Rates 368 331 Carbon Dating 390 332 Newton’s Law of Cooling 396 333 Population Growth 402 334 Further problems 405 341 Zeroeth Approximation 409 342 First Approximation 410 343 Second Approximation 413 344 Still Better Approximations 420 345 Some Examples 423 346 Estimating Changes and Dx, Dy notation 430 347 Further Examples 432 348 The error in Taylor polynomial approximation 437 349 Further problems 452 351 Local and global maxima and minima 460 352 Finding global maxima and minima 466 353 Max/min examples 468 361 Domain, intercepts and asymptotes 478 362 First derivative - increasing or decreasing 481 363 Second derivative - concavity 482 364 Symmetries 486 365 A checklist for sketching 490 366 Sketching examples 490 37 L’Hˆ opital’s Rule and indeterminate forms 519 41 Introduction to antiderivatives 526