The Diver Returns Circular Functions, Vector Components, and Complex Numbers

Transcription

1 Co n t e n t s The Diver Returns Circular Functions, Vector Components, and Complex Numbers Back to the Circus 3 The Circus Act 4 As the Ferris Wheel Turns 6 Graphing the Ferris Wheel 8 Distance with Changing Speed 9 Free Fall 10 Moving Cart, Turning Ferris Wheel 12 POW 1: Which Weights Weigh What? 13 The Standard POW Write-up 16 A Simple Summary and a Complex Beginning 17 A Falling Start 18 Look Out Below! 19 The Diver and the POW 20 Big Push 21 Finding with the Formula 23 Using Your ABCs 24 Imagine a Solution 25 Complex Numbers and Quadratic Equations 27 Complex Components 28 Three O Clock Drop 29 Up, Down, Splat! 30 Falling Time for Vertical Motion 31 Components of Velocity 32 High Noon 33 Leap of Faith 34 Contents ix

2 The Ideal Skateboard 35 Racing the River 36 One O Clock Without Gravity 38 Swimming Pointers 39 Vector Velocities 40 Velocities on the Wheel 41 Release at Any Angle 42 An Expanded Portfolio of Formulas 43 Moving Diver at Two O Clock 44 The Danger of Simplification 45 The Diver Really Returns 46 The Diver s Success 47 A Circus Reflection 48 Beginning Portfolio Selection 49 The Diver Returns Portfolio 50 Supplemental Activities 52 Complex Conjugation 53 Absolutely Complex 54 The Polar Complex 55 Polar Roots 57 Number Research 58 The World of Functions Families of Functions and the Algebra of Functions The What and Why of Functions 61 Brake! 62 POW 2: One Mile at a Time 63 Story Sketches 64 Story Sketches II 65 What Good Are Functions? 66 More Families 67 x Interactive Mathematics Program

3 Tables 68 Linear Tables 69 Story Sketches III 70 Quadratic Tables 71 Back to Basics 72 Quadratic Tables by Algebra 74 A General Quadratic 75 Exponential Tables 76 A Cubic Pattern 78 POW 3: A Spin on Transitivity 79 Mystery Tables 82 Brake! Revisited 83 Bigger Means Smaller 84 Going to the Limit 85 Don t Divide That! 86 Difficult Denominators 88 Return of the Shadow 89 An Average Drive 90 Approaching Infinity 91 The End of the Function 92 Creating the Ending You Want 73 Who s Who? 94 Families Have Many Different Members 95 Fitting Mia s Birdhouses Again 96 Mystery Tables II 97 What Will It Be Worth? 98 The Decision About Dunkalot 99 A Tight Fit 101 Let s Regress 102 Midnight Express 104 Contents xi

4 POW 4: It s Off to College We Go 105 In the Lead 106 Back to Arithmetic 107 The Arithmetic of Functions 108 The Arithmetic of Graphs 110 Back to the Corral 111 Name That Family! 113 Small World, Isn t It? Revisited 114 Composing Functions 115 Rumble, Grumble 116 The Composition of Functions 117 The Cost of Pollution 119 Order Among the Functions 121 Cozying Up to Composition 122 Taking Functions Apart 124 Fish, Ladders, and Bacteria 125 Functions in Verse 126 Linear Functions in Verse 128 An Inventory of Inverses 129 Transforming Functions 131 Double Dose of Functions 132 Slide That Function 134 Transforming Graphs, Tables, and Situations 135 Back to the Beginning 137 Better Braking 138 Beginning Portfolio Selection 139 The World of Functions Portfolio 140 Supplemental Activities 142 From Second Differences to Quadratics 143 Real Domains 144 Absolutely Functions 146 xii Interactive Mathematics Program

5 Odd or Even? 148 Graphing Power 150 Ferris Wheel on a Ramp 151 Freddie on the Ferris Wheel 152 Over, and Over, and Over, and Its Own Inverse 155 A Hyperbolic Approach 156 Small World Again! 157 The Pollster s Dilemma The Binomial Distribution and the Central Limit Theorem What s a Pollster to Think? 161 The Pollster s Dilemma 162 No Bias Allowed! 163 POW 5: The King s Switches 164 Sampling Seniors 166 Pennant Fever Reflection 168 Bags of Marbles and Bowls of Ice Cream 169 Polls and Pennant Fever 171 The Theory of Three-Person Polls 172 Graphs of the Theory 174 The Theory of Polls 175 Civics in Action 177 Normal Distributions Revisited 178 The Central Limit Theorem 179 Deviations of Swinging 183 Means and More in Middletown 184 Graphing Distributions 185 Gifts Aren t Always Free 187 Normal Areas 188 The Normal Table 189 More Middletown Musings 191 Contents xiii

6 Back to the Circus 192 Gaps in the Table 193 A Normal Poll 195 A Plus for the Community 196 Means and Standard Deviations 197 Mean and Standard Deviation for Probability Distributions 198 A Distribution Example 200 The Search Is On! 202 Why Is That Batter Sneezing? 204 Putting Your Formulas to Work 206 From Numbers to Proportions 208 Is Twice as Many Twice as Good? 209 A Matter of Confidence 210 Different p, Different 211 Let s Vote on It! 212 Project Topics and Random Polls 214 Mean, Median, and Mode 215 The Worst-Case Scenario 217 A Teaching Dilemma 218 What Does It Mean? 219 Confidence and Sample Size 220 Polling Puzzles 221 How Big? 222 Putting It Together 223 Roberto and the Coin 224 How Much Better Is Bigger? 225 The Pollster s Dilemma Revisited 226 Final Data Collection 227 The Pollster s Dilemma Portfolio 228 Supplemental Activities 230 What Is Random? 231 xiv Interactive Mathematics Program

7 Random Number Generators 232 The Tack or the Coin? 233 Three-Person Races 234 Generalizing Linear Interpolation 236 Another View of the Central Limit Theorem 237 It s the News 239 How Much? How Fast? Accumulated Change, Rates of Growth, and the Fundamental Theorem of Calculus Adding Up the Parts 243 Building the Pyramid 244 How Far Did You Go? 245 Another Trip 246 POW 6: Advanced Pool Pockets 247 How Fast? How Much? 249 Leaky Faucet 251 Units for Measuring Electricity 252 What s Watt? 253 Electrical Meter 254 Tilted Duct 255 Warming Up 256 Total Heat 257 Rate and Accumulation 258 How Fast Were You Going? 259 A Distance Graph 260 Let It Fall! 261 Basic Derivatives 263 Summer Job 265 Going Up? 266 Down the Drain 268 Zero to Sixty 269 Contents xv

8 Polynomial Derivatives 270 Area and Distance 271 A Fundamental Relationship 272 The Leading Edge 275 Pyramids and Energy 277 Filling the Reservoir 278 A Pyramid of Bright Ideas 279 Trying a New Angle 280 Different Angles 282 A Solar Formula 283 A Sine Derivative 284 A Derivative Proof 285 A Cosine Derivative 287 The Inside Story 288 A Solar Summary 289 How Much? How Fast? Portfolio 290 Supplemental Activities 291 Derivative Power 292 Parabolic Area 293 Ana on the Train 294 Widget Wisdom 295 As the Cube Turns Programming and Transformational Geometry Calculator Pictures 299 Picture This! 300 POW 7: A Sticky Gum Problem Revisited 301 Starting Sticky Gum 304 Programming Without a Calculator 305 Programming Loops 307 Learning the Loops 308 An Animated Shape 310 xvi Interactive Mathematics Program

9 A Flip Book 311 Movin On 312 Some Back and Forth 313 Arrow 314 Sunrise 315 POW 8: A Wider Windshield Wiper, Please 316 Translation in Two Dimensions 317 Move That Line! 318 Double Dotting 319 Memories of Matrices 320 Cornering the Cabbage 323 Rotation in Two Dimensions 324 Goin Round the Origin 325 Double Trouble 326 The Sine of a Sum 328 A Broken Button 329 Oh, Say What You Can See 330 Comin Round Again (and Again...) 332 More Memories of Matrices 333 Taking Steps 335 How Did We Get Here? 337 Swing That Line! 338 Doubles and Differences 339 What s Going On Here? 340 Projecting Pictures 341 POW 9: An Animated POW 342 A Snack in the Middle Revisited 343 Fractional Snacks 344 More Walking for Clyde 345 Monorail Delivery 346 Another Mystery 347 Contents xvii

10 A Return to the Third Dimension 348 Where s Madie? 349 And Fred Brings the Lunch 350 Flipping Points 352 Where s Bonita? 354 Lunch in the Window 355 Further Flips 356 Cube on a Screen 357 Spiders and Cubes 359 Find Those Corners! 360 An Animated Outline 361 Mirrors in Space 362 Where Are We Now? 363 Rotation in Three Dimensions 364 Follow That Point! 365 One Turn of a Cube 366 Rotation Matrix in Three Dimensions 367 The Turning Cube Outline 368 Beginning Portfolio Selection 369 Creating Animations 370 An Animated POW Write-up 371 Continued Portfolio Selection 372 As the Cube Turns Portfolio 373 Supplemental Activities 375 Loopy Arithmetic 376 Sum Tangents 377 Moving to the Second Quadrant 378 Adding Sums for All Quadrants 381 Deriving the Polar Complex 382 Sine and Cosine Derivatives Again 383 xviii Interactive Mathematics Program

11 Bugs in Trees 384 Half a Sine 387 The General Isometry 388 Perspective on Geometry 390 Let the Calculator Do It! 391 Glossary 393 Index of Mathematical Ideas 407 Index of Activity Titles 427 Photographic Credits 431 Contents xix