From 7432f09be9b35c57b606d78924c41abac3ab46f2 Mon Sep 17 00:00:00 2001 From: Jan Meyer Date: Thu, 4 Jun 2026 19:29:51 +0200 Subject: [PATCH] chore: add maths6 --- 11 Homework/Mathe II/Homework 6.pdf | 1156 ++++++++++----------------- 11 Homework/Mathe II/Homework 6.typ | 30 +- 2 files changed, 439 insertions(+), 747 deletions(-) diff --git a/11 Homework/Mathe II/Homework 6.pdf b/11 Homework/Mathe II/Homework 6.pdf index 1181275..61990af 100644 --- a/11 Homework/Mathe II/Homework 6.pdf +++ b/11 Homework/Mathe II/Homework 6.pdf @@ -4,8 +4,8 @@ 1 0 obj << /Type /Pages - /Count 2 - /Kids [93 0 R 95 0 R] + /Count 1 + /Kids [77 0 R] >> endobj @@ -26,7 +26,7 @@ endobj /Last 5 0 R /Count -2 /Title (Problem 2) - /Dest 88 0 R + /Dest 72 0 R >> endobj @@ -35,7 +35,7 @@ endobj /Parent 3 0 R /Next 5 0 R /Title ((a)) - /Dest 86 0 R + /Dest 70 0 R >> endobj @@ -44,7 +44,7 @@ endobj /Parent 3 0 R /Prev 4 0 R /Title ((b)) - /Dest 87 0 R + /Dest 71 0 R >> endobj @@ -56,7 +56,7 @@ endobj /Last 9 0 R /Count -3 /Title (Problem 3) - /Dest 92 0 R + /Dest 76 0 R >> endobj @@ -65,7 +65,7 @@ endobj /Parent 6 0 R /Next 8 0 R /Title ((a)) - /Dest 89 0 R + /Dest 73 0 R >> endobj @@ -75,7 +75,7 @@ endobj /Next 9 0 R /Prev 7 0 R /Title ((b)) - /Dest 90 0 R + /Dest 74 0 R >> endobj @@ -84,7 +84,7 @@ endobj /Parent 6 0 R /Prev 8 0 R /Title ((c)) - /Dest 91 0 R + /Dest 75 0 R >> endobj @@ -98,28 +98,33 @@ endobj /Strong /Span /Em /Span >> - /K [13 0 R] + /K [12 0 R] /ParentTree << - /Nums [0 11 0 R 1 12 0 R] + /Nums [0 11 0 R] >> - /ParentTreeNextKey 2 + /ParentTreeNextKey 1 >> endobj 11 0 obj -[65 0 R 64 0 R 63 0 R 63 0 R 63 0 R 63 0 R 63 0 R 63 0 R 63 0 R 63 0 R 63 0 R 63 0 R 63 0 R 63 0 R 63 0 R 63 0 R 63 0 R 63 0 R 63 0 R 63 0 R 63 0 R 63 0 R 63 0 R 63 0 R 63 0 R 63 0 R 63 0 R 63 0 R 63 0 R 63 0 R 63 0 R 63 0 R 62 0 R 61 0 R 61 0 R 61 0 R 61 0 R 61 0 R 61 0 R 61 0 R 61 0 R 61 0 R 61 0 R 61 0 R 61 0 R 60 0 R 59 0 R 59 0 R 59 0 R 59 0 R 59 0 R 59 0 R 59 0 R 59 0 R 59 0 R 59 0 R 59 0 R 59 0 R 59 0 R 59 0 R 59 0 R 59 0 R 59 0 R 59 0 R 59 0 R 59 0 R 59 0 R 59 0 R 59 0 R 59 0 R 58 0 R 57 0 R 56 0 R 56 0 R 56 0 R 56 0 R 56 0 R 56 0 R 56 0 R 56 0 R 56 0 R 56 0 R 56 0 R 56 0 R 56 0 R 55 0 R 54 0 R 54 0 R 54 0 R 54 0 R 54 0 R 53 0 R 52 0 R 52 0 R 52 0 R 51 0 R 50 0 R 50 0 R 50 0 R 50 0 R 50 0 R 50 0 R 50 0 R 49 0 R 48 0 R 48 0 R 48 0 R 48 0 R 48 0 R 48 0 R 48 0 R 48 0 R 48 0 R 48 0 R 48 0 R 48 0 R 48 0 R 48 0 R 48 0 R 48 0 R 48 0 R 48 0 R 48 0 R 48 0 R 48 0 R 48 0 R 48 0 R 48 0 R 48 0 R 48 0 R 48 0 R 48 0 R 48 0 R 48 0 R 48 0 R 48 0 R 48 0 R 48 0 R 48 0 R 48 0 R 48 0 R 48 0 R 48 0 R 48 0 R 48 0 R 48 0 R 48 0 R 48 0 R 48 0 R 48 0 R 48 0 R 48 0 R 48 0 R 48 0 R 48 0 R 48 0 R 48 0 R 48 0 R 47 0 R 46 0 R 46 0 R 46 0 R 46 0 R 46 0 R 45 0 R 44 0 R 44 0 R 44 0 R 44 0 R 44 0 R 44 0 R 44 0 R 44 0 R 44 0 R 43 0 R 42 0 R 42 0 R 42 0 R 42 0 R 42 0 R 42 0 R 42 0 R 42 0 R 42 0 R 42 0 R 42 0 R 42 0 R 42 0 R 42 0 R 42 0 R 42 0 R 42 0 R 42 0 R 42 0 R 42 0 R 42 0 R 42 0 R 42 0 R 42 0 R 42 0 R 41 0 R 40 0 R 40 0 R 40 0 R 40 0 R 40 0 R 40 0 R 40 0 R 40 0 R 40 0 R 40 0 R 39 0 R] +[49 0 R 48 0 R 47 0 R 47 0 R 47 0 R 47 0 R 47 0 R 47 0 R 47 0 R 47 0 R 47 0 R 47 0 R 47 0 R 47 0 R 47 0 R 47 0 R 47 0 R 47 0 R 47 0 R 47 0 R 47 0 R 47 0 R 47 0 R 47 0 R 47 0 R 47 0 R 47 0 R 47 0 R 47 0 R 47 0 R 47 0 R 47 0 R 44 0 R 46 0 R 46 0 R 46 0 R 46 0 R 46 0 R 46 0 R 46 0 R 46 0 R 46 0 R 46 0 R 46 0 R 46 0 R 44 0 R 45 0 R 45 0 R 45 0 R 45 0 R 45 0 R 45 0 R 45 0 R 45 0 R 45 0 R 45 0 R 45 0 R 45 0 R 45 0 R 45 0 R 45 0 R 45 0 R 45 0 R 45 0 R 45 0 R 45 0 R 45 0 R 45 0 R 45 0 R 45 0 R 43 0 R 33 0 R 42 0 R 42 0 R 42 0 R 42 0 R 42 0 R 42 0 R 42 0 R 42 0 R 42 0 R 42 0 R 42 0 R 42 0 R 42 0 R 33 0 R 41 0 R 41 0 R 41 0 R 41 0 R 41 0 R 33 0 R 40 0 R 40 0 R 40 0 R 33 0 R 39 0 R 39 0 R 39 0 R 39 0 R 39 0 R 39 0 R 39 0 R 33 0 R 38 0 R 38 0 R 38 0 R 38 0 R 38 0 R 38 0 R 38 0 R 38 0 R 38 0 R 38 0 R 38 0 R 38 0 R 38 0 R 38 0 R 38 0 R 38 0 R 38 0 R 38 0 R 38 0 R 38 0 R 38 0 R 38 0 R 38 0 R 38 0 R 38 0 R 38 0 R 38 0 R 38 0 R 38 0 R 38 0 R 38 0 R 38 0 R 38 0 R 38 0 R 38 0 R 38 0 R 38 0 R 38 0 R 38 0 R 38 0 R 38 0 R 38 0 R 38 0 R 38 0 R 38 0 R 38 0 R 38 0 R 38 0 R 38 0 R 38 0 R 38 0 R 38 0 R 38 0 R 38 0 R 33 0 R 37 0 R 37 0 R 37 0 R 37 0 R 37 0 R 33 0 R 36 0 R 36 0 R 36 0 R 36 0 R 36 0 R 36 0 R 36 0 R 36 0 R 36 0 R 33 0 R 33 0 R 35 0 R 35 0 R 35 0 R 35 0 R 35 0 R 35 0 R 35 0 R 35 0 R 35 0 R 35 0 R 35 0 R 35 0 R 35 0 R 35 0 R 35 0 R 35 0 R 35 0 R 35 0 R 35 0 R 35 0 R 35 0 R 35 0 R 35 0 R 35 0 R 35 0 R 33 0 R 34 0 R 34 0 R 34 0 R 34 0 R 34 0 R 34 0 R 34 0 R 34 0 R 34 0 R 34 0 R 33 0 R 33 0 R 32 0 R 23 0 R 31 0 R 31 0 R 31 0 R 31 0 R 31 0 R 31 0 R 31 0 R 31 0 R 31 0 R 23 0 R 30 0 R 23 0 R 29 0 R 23 0 R 28 0 R 23 0 R 27 0 R 27 0 R 27 0 R 27 0 R 27 0 R 27 0 R 27 0 R 27 0 R 27 0 R 27 0 R 27 0 R 27 0 R 27 0 R 27 0 R 27 0 R 27 0 R 27 0 R 27 0 R 27 0 R 23 0 R 26 0 R 26 0 R 26 0 R 26 0 R 26 0 R 26 0 R 26 0 R 23 0 R 25 0 R 23 0 R 24 0 R 24 0 R 24 0 R 24 0 R 24 0 R 24 0 R 24 0 R 23 0 R 22 0 R 21 0 R 21 0 R 21 0 R 21 0 R 21 0 R 21 0 R 21 0 R 21 0 R 21 0 R 21 0 R 21 0 R 21 0 R 21 0 R 21 0 R 21 0 R 21 0 R 21 0 R 21 0 R 21 0 R 21 0 R 21 0 R 21 0 R 21 0 R 21 0 R 21 0 R 21 0 R 21 0 R 21 0 R 21 0 R 21 0 R 21 0 R 21 0 R 21 0 R 21 0 R 21 0 R 21 0 R 21 0 R 21 0 R 21 0 R 21 0 R 21 0 R 21 0 R 21 0 R 21 0 R 21 0 R 21 0 R 21 0 R 19 0 R 18 0 R 18 0 R 18 0 R 18 0 R 18 0 R 18 0 R 18 0 R 18 0 R 18 0 R 18 0 R 18 0 R 18 0 R 18 0 R 18 0 R 18 0 R 18 0 R 18 0 R 18 0 R 18 0 R 18 0 R 18 0 R 18 0 R 18 0 R 18 0 R 18 0 R 18 0 R 18 0 R 18 0 R 18 0 R 18 0 R 18 0 R 18 0 R 18 0 R 18 0 R 18 0 R 18 0 R 18 0 R 18 0 R 18 0 R 18 0 R 18 0 R 18 0 R 18 0 R 18 0 R 18 0 R 18 0 R 18 0 R 18 0 R 18 0 R 18 0 R 18 0 R 16 0 R 17 0 R 17 0 R 17 0 R 17 0 R 17 0 R 17 0 R 17 0 R 17 0 R 17 0 R 17 0 R 17 0 R 17 0 R 17 0 R 17 0 R 17 0 R 17 0 R 17 0 R 17 0 R 17 0 R 17 0 R 17 0 R 17 0 R 17 0 R 17 0 R 17 0 R 17 0 R 17 0 R 17 0 R 17 0 R 17 0 R 17 0 R 17 0 R 17 0 R 17 0 R 17 0 R 17 0 R 17 0 R 17 0 R 17 0 R 17 0 R 17 0 R 17 0 R 17 0 R 17 0 R 17 0 R 17 0 R 17 0 R 17 0 R 17 0 R 17 0 R 17 0 R 15 0 R 14 0 R 14 0 R 14 0 R 14 0 R 14 0 R 14 0 R 14 0 R 14 0 R 14 0 R 14 0 R 14 0 R 14 0 R 14 0 R 14 0 R 14 0 R 14 0 R 14 0 R 14 0 R 14 0 R 14 0 R 14 0 R 14 0 R 14 0 R 14 0 R 14 0 R 14 0 R 14 0 R] endobj 12 0 obj -[38 0 R 37 0 R 36 0 R 36 0 R 36 0 R 36 0 R 36 0 R 36 0 R 36 0 R 36 0 R 36 0 R 35 0 R 34 0 R 33 0 R 32 0 R 31 0 R 30 0 R 29 0 R 28 0 R 28 0 R 28 0 R 28 0 R 28 0 R 28 0 R 28 0 R 28 0 R 28 0 R 28 0 R 28 0 R 28 0 R 28 0 R 28 0 R 28 0 R 28 0 R 28 0 R 28 0 R 28 0 R 27 0 R 26 0 R 26 0 R 26 0 R 26 0 R 26 0 R 26 0 R 26 0 R 25 0 R 24 0 R 23 0 R 22 0 R 22 0 R 22 0 R 22 0 R 22 0 R 22 0 R 22 0 R 21 0 R 20 0 R 19 0 R 19 0 R 19 0 R 19 0 R 19 0 R 19 0 R 19 0 R 19 0 R 19 0 R 19 0 R 19 0 R 19 0 R 19 0 R 19 0 R 19 0 R 19 0 R 19 0 R 19 0 R 19 0 R 19 0 R 19 0 R 19 0 R 19 0 R 19 0 R 19 0 R 19 0 R 19 0 R 19 0 R 19 0 R 19 0 R 19 0 R 19 0 R 19 0 R 19 0 R 19 0 R 19 0 R 19 0 R 19 0 R 19 0 R 19 0 R 19 0 R 19 0 R 19 0 R 19 0 R 19 0 R 19 0 R 19 0 R 18 0 R 17 0 R 17 0 R 17 0 R 17 0 R 17 0 R 17 0 R 17 0 R 17 0 R 17 0 R 17 0 R 17 0 R 17 0 R 17 0 R 17 0 R 17 0 R 17 0 R 17 0 R 17 0 R 17 0 R 17 0 R 17 0 R 17 0 R 17 0 R 17 0 R 17 0 R 17 0 R 17 0 R 17 0 R 17 0 R 17 0 R 17 0 R 17 0 R 17 0 R 17 0 R 17 0 R 17 0 R 17 0 R 17 0 R 17 0 R 17 0 R 17 0 R 17 0 R 17 0 R 17 0 R 17 0 R 17 0 R 17 0 R 17 0 R 17 0 R 17 0 R 17 0 R 16 0 R 16 0 R 16 0 R 16 0 R 16 0 R 16 0 R 16 0 R 16 0 R 16 0 R 16 0 R 16 0 R 16 0 R 16 0 R 16 0 R 16 0 R 16 0 R 16 0 R 16 0 R 16 0 R 16 0 R 16 0 R 16 0 R 16 0 R 16 0 R 16 0 R 16 0 R 16 0 R 16 0 R 16 0 R 16 0 R 16 0 R 16 0 R 16 0 R 16 0 R 16 0 R 16 0 R 16 0 R 16 0 R 16 0 R 16 0 R 16 0 R 16 0 R 16 0 R 16 0 R 16 0 R 16 0 R 16 0 R 16 0 R 16 0 R 16 0 R 16 0 R 15 0 R 14 0 R 14 0 R 14 0 R 14 0 R 14 0 R 14 0 R 14 0 R 14 0 R 14 0 R 14 0 R 14 0 R 14 0 R 14 0 R 14 0 R 14 0 R 14 0 R 14 0 R 14 0 R 14 0 R 14 0 R 14 0 R 14 0 R 14 0 R 14 0 R 14 0 R 14 0 R 14 0 R] +<< + /Type /StructElem + /S /Document + /P 10 0 R + /K [49 0 R 48 0 R 44 0 R 43 0 R 33 0 R 32 0 R 23 0 R 22 0 R 20 0 R 19 0 R 16 0 R 15 0 R 13 0 R] +>> endobj 13 0 obj << /Type /StructElem - /S /Document - /P 10 0 R - /K [65 0 R 64 0 R 63 0 R 62 0 R 61 0 R 60 0 R 59 0 R 58 0 R 57 0 R 56 0 R 55 0 R 54 0 R 53 0 R 52 0 R 51 0 R 50 0 R 49 0 R 48 0 R 47 0 R 46 0 R 45 0 R 44 0 R 43 0 R 42 0 R 41 0 R 40 0 R 39 0 R 38 0 R 37 0 R 36 0 R 35 0 R 34 0 R 33 0 R 32 0 R 31 0 R 30 0 R 29 0 R 28 0 R 27 0 R 26 0 R 25 0 R 24 0 R 23 0 R 22 0 R 21 0 R 20 0 R 19 0 R 18 0 R 17 0 R 16 0 R 15 0 R 14 0 R] + /S /P + /P 12 0 R + /K [14 0 R] >> endobj @@ -128,12 +133,8 @@ endobj /Type /StructElem /S /Formula /P 13 0 R - /A [<< - /O /Layout - /Placement /Block - >>] - /K [208 209 210 211 212 213 214 215 216 217 218 219 220 221 222 223 224 225 226 227 228 229 230 231 232 233 234] - /Pg 95 0 R + /K [423 424 425 426 427 428 429 430 431 432 433 434 435 436 437 438 439 440 441 442 443 444 445 446 447 448 449] + /Pg 77 0 R >> endobj @@ -141,24 +142,20 @@ endobj << /Type /StructElem /S /H2 - /P 13 0 R + /P 12 0 R /T ((c)) - /K [207] - /Pg 95 0 R + /K [422] + /Pg 77 0 R >> endobj 16 0 obj << /Type /StructElem - /S /Formula - /P 13 0 R - /A [<< - /O /Layout - /Placement /Block - >>] - /K [156 157 158 159 160 161 162 163 164 165 166 167 168 169 170 171 172 173 174 175 176 177 178 179 180 181 182 183 184 185 186 187 188 189 190 191 192 193 194 195 196 197 198 199 200 201 202 203 204 205 206] - /Pg 95 0 R + /S /P + /P 12 0 R + /K [18 0 R 370 17 0 R] + /Pg 77 0 R >> endobj @@ -166,73 +163,60 @@ endobj << /Type /StructElem /S /Formula - /P 13 0 R - /A [<< - /O /Layout - /Placement /Block - >>] - /K [105 106 107 108 109 110 111 112 113 114 115 116 117 118 119 120 121 122 123 124 125 126 127 128 129 130 131 132 133 134 135 136 137 138 139 140 141 142 143 144 145 146 147 148 149 150 151 152 153 154 155] - /Pg 95 0 R + /P 16 0 R + /K [371 372 373 374 375 376 377 378 379 380 381 382 383 384 385 386 387 388 389 390 391 392 393 394 395 396 397 398 399 400 401 402 403 404 405 406 407 408 409 410 411 412 413 414 415 416 417 418 419 420 421] + /Pg 77 0 R >> endobj 18 0 obj << /Type /StructElem - /S /H2 - /P 13 0 R - /T ((b)) - /K [104] - /Pg 95 0 R + /S /Formula + /P 16 0 R + /K [319 320 321 322 323 324 325 326 327 328 329 330 331 332 333 334 335 336 337 338 339 340 341 342 343 344 345 346 347 348 349 350 351 352 353 354 355 356 357 358 359 360 361 362 363 364 365 366 367 368 369] + /Pg 77 0 R >> endobj 19 0 obj << /Type /StructElem - /S /Formula - /P 13 0 R - /A [<< - /O /Layout - /Placement /Block - >>] - /K [57 58 59 60 61 62 63 64 65 66 67 68 69 70 71 72 73 74 75 76 77 78 79 80 81 82 83 84 85 86 87 88 89 90 91 92 93 94 95 96 97 98 99 100 101 102 103] - /Pg 95 0 R + /S /H2 + /P 12 0 R + /T ((b)) + /K [318] + /Pg 77 0 R >> endobj 20 0 obj << /Type /StructElem - /S /H2 - /P 13 0 R - /T ((a)) - /K [56] - /Pg 95 0 R + /S /P + /P 12 0 R + /K [21 0 R] >> endobj 21 0 obj << /Type /StructElem - /S /P - /P 13 0 R - /K [55] - /Pg 95 0 R + /S /Formula + /P 20 0 R + /K [271 272 273 274 275 276 277 278 279 280 281 282 283 284 285 286 287 288 289 290 291 292 293 294 295 296 297 298 299 300 301 302 303 304 305 306 307 308 309 310 311 312 313 314 315 316 317] + /Pg 77 0 R >> endobj 22 0 obj << /Type /StructElem - /S /Formula - /P 13 0 R - /A [<< - /O /Layout - /Placement /Block - >>] - /K [48 49 50 51 52 53 54] - /Pg 95 0 R + /S /H2 + /P 12 0 R + /T ((a)) + /K [270] + /Pg 77 0 R >> endobj @@ -240,9 +224,9 @@ endobj << /Type /StructElem /S /P - /P 13 0 R - /K [47] - /Pg 95 0 R + /P 12 0 R + /K [215 31 0 R 225 30 0 R 227 29 0 R 229 28 0 R 231 27 0 R 251 26 0 R 259 25 0 R 261 24 0 R 269] + /Pg 77 0 R >> endobj @@ -250,23 +234,19 @@ endobj << /Type /StructElem /S /Formula - /P 13 0 R - /A [<< - /O /Layout - /Placement /Block - >>] - /K [46] - /Pg 95 0 R + /P 23 0 R + /K [262 263 264 265 266 267 268] + /Pg 77 0 R >> endobj 25 0 obj << /Type /StructElem - /S /P - /P 13 0 R - /K [45] - /Pg 95 0 R + /S /Formula + /P 23 0 R + /K [260] + /Pg 77 0 R >> endobj @@ -274,23 +254,19 @@ endobj << /Type /StructElem /S /Formula - /P 13 0 R - /A [<< - /O /Layout - /Placement /Block - >>] - /K [38 39 40 41 42 43 44] - /Pg 95 0 R + /P 23 0 R + /K [252 253 254 255 256 257 258] + /Pg 77 0 R >> endobj 27 0 obj << /Type /StructElem - /S /P - /P 13 0 R - /K [37] - /Pg 95 0 R + /S /Formula + /P 23 0 R + /K [232 233 234 235 236 237 238 239 240 241 242 243 244 245 246 247 248 249 250] + /Pg 77 0 R >> endobj @@ -298,23 +274,19 @@ endobj << /Type /StructElem /S /Formula - /P 13 0 R - /A [<< - /O /Layout - /Placement /Block - >>] - /K [18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36] - /Pg 95 0 R + /P 23 0 R + /K [230] + /Pg 77 0 R >> endobj 29 0 obj << /Type /StructElem - /S /P - /P 13 0 R - /K [17] - /Pg 95 0 R + /S /Formula + /P 23 0 R + /K [228] + /Pg 77 0 R >> endobj @@ -322,37 +294,30 @@ endobj << /Type /StructElem /S /Formula - /P 13 0 R - /A [<< - /O /Layout - /Placement /Block - >>] - /K [16] - /Pg 95 0 R + /P 23 0 R + /K [226] + /Pg 77 0 R >> endobj 31 0 obj << /Type /StructElem - /S /P - /P 13 0 R - /K [15] - /Pg 95 0 R + /S /Formula + /P 23 0 R + /K [216 217 218 219 220 221 222 223 224] + /Pg 77 0 R >> endobj 32 0 obj << /Type /StructElem - /S /Formula - /P 13 0 R - /A [<< - /O /Layout - /Placement /Block - >>] - /K [14] - /Pg 95 0 R + /S /H1 + /P 12 0 R + /T (Problem 3) + /K [214] + /Pg 77 0 R >> endobj @@ -360,9 +325,9 @@ endobj << /Type /StructElem /S /P - /P 13 0 R - /K [13] - /Pg 95 0 R + /P 12 0 R + /K [71 42 0 R 85 41 0 R 91 40 0 R 95 39 0 R 103 38 0 R 158 37 0 R 164 36 0 R 174 175 35 0 R 201 34 0 R 212 213] + /Pg 77 0 R >> endobj @@ -370,23 +335,19 @@ endobj << /Type /StructElem /S /Formula - /P 13 0 R - /A [<< - /O /Layout - /Placement /Block - >>] - /K [12] - /Pg 95 0 R + /P 33 0 R + /K [202 203 204 205 206 207 208 209 210 211] + /Pg 77 0 R >> endobj 35 0 obj << /Type /StructElem - /S /P - /P 13 0 R - /K [11] - /Pg 95 0 R + /S /Formula + /P 33 0 R + /K [176 177 178 179 180 181 182 183 184 185 186 187 188 189 190 191 192 193 194 195 196 197 198 199 200] + /Pg 77 0 R >> endobj @@ -394,44 +355,39 @@ endobj << /Type /StructElem /S /Formula - /P 13 0 R - /A [<< - /O /Layout - /Placement /Block - >>] - /K [2 3 4 5 6 7 8 9 10] - /Pg 95 0 R + /P 33 0 R + /K [165 166 167 168 169 170 171 172 173] + /Pg 77 0 R >> endobj 37 0 obj << /Type /StructElem - /S /P - /P 13 0 R - /K [1] - /Pg 95 0 R + /S /Formula + /P 33 0 R + /K [159 160 161 162 163] + /Pg 77 0 R >> endobj 38 0 obj << /Type /StructElem - /S /H1 - /P 13 0 R - /T (Problem 3) - /K [0] - /Pg 95 0 R + /S /Formula + /P 33 0 R + /K [104 105 106 107 108 109 110 111 112 113 114 115 116 117 118 119 120 121 122 123 124 125 126 127 128 129 130 131 132 133 134 135 136 137 138 139 140 141 142 143 144 145 146 147 148 149 150 151 152 153 154 155 156 157] + /Pg 77 0 R >> endobj 39 0 obj << /Type /StructElem - /S /P - /P 13 0 R - /K [211] - /Pg 93 0 R + /S /Formula + /P 33 0 R + /K [96 97 98 99 100 101 102] + /Pg 77 0 R >> endobj @@ -439,23 +395,19 @@ endobj << /Type /StructElem /S /Formula - /P 13 0 R - /A [<< - /O /Layout - /Placement /Block - >>] - /K [201 202 203 204 205 206 207 208 209 210] - /Pg 93 0 R + /P 33 0 R + /K [92 93 94] + /Pg 77 0 R >> endobj 41 0 obj << /Type /StructElem - /S /P - /P 13 0 R - /K [200] - /Pg 93 0 R + /S /Formula + /P 33 0 R + /K [86 87 88 89 90] + /Pg 77 0 R >> endobj @@ -463,47 +415,40 @@ endobj << /Type /StructElem /S /Formula - /P 13 0 R - /A [<< - /O /Layout - /Placement /Block - >>] - /K [175 176 177 178 179 180 181 182 183 184 185 186 187 188 189 190 191 192 193 194 195 196 197 198 199] - /Pg 93 0 R + /P 33 0 R + /K [72 73 74 75 76 77 78 79 80 81 82 83 84] + /Pg 77 0 R >> endobj 43 0 obj << /Type /StructElem - /S /P - /P 13 0 R - /K [174] - /Pg 93 0 R + /S /H2 + /P 12 0 R + /T ((b)) + /K [70] + /Pg 77 0 R >> endobj 44 0 obj << /Type /StructElem - /S /Formula - /P 13 0 R - /A [<< - /O /Layout - /Placement /Block - >>] - /K [165 166 167 168 169 170 171 172 173] - /Pg 93 0 R + /S /P + /P 12 0 R + /K [47 0 R 32 46 0 R 45 45 0 R] + /Pg 77 0 R >> endobj 45 0 obj << /Type /StructElem - /S /P - /P 13 0 R - /K [164] - /Pg 93 0 R + /S /Formula + /P 44 0 R + /K [46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65 66 67 68 69] + /Pg 77 0 R >> endobj @@ -511,253 +456,56 @@ endobj << /Type /StructElem /S /Formula - /P 13 0 R - /A [<< - /O /Layout - /Placement /Block - >>] - /K [159 160 161 162 163] - /Pg 93 0 R + /P 44 0 R + /K [33 34 35 36 37 38 39 40 41 42 43 44] + /Pg 77 0 R >> endobj 47 0 obj << /Type /StructElem - /S /P - /P 13 0 R - /K [158] - /Pg 93 0 R + /S /Formula + /P 44 0 R + /K [2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31] + /Pg 77 0 R >> endobj 48 0 obj << /Type /StructElem - /S /Formula - /P 13 0 R - /A [<< - /O /Layout - /Placement /Block - >>] - /K [104 105 106 107 108 109 110 111 112 113 114 115 116 117 118 119 120 121 122 123 124 125 126 127 128 129 130 131 132 133 134 135 136 137 138 139 140 141 142 143 144 145 146 147 148 149 150 151 152 153 154 155 156 157] - /Pg 93 0 R + /S /H2 + /P 12 0 R + /T ((a)) + /K [1] + /Pg 77 0 R >> endobj 49 0 obj << /Type /StructElem - /S /P - /P 13 0 R - /K [103] - /Pg 93 0 R + /S /H1 + /P 12 0 R + /T (Problem 2) + /K [0] + /Pg 77 0 R >> endobj 50 0 obj -<< - /Type /StructElem - /S /Formula - /P 13 0 R - /A [<< - /O /Layout - /Placement /Block - >>] - /K [96 97 98 99 100 101 102] - /Pg 93 0 R ->> -endobj - -51 0 obj -<< - /Type /StructElem - /S /P - /P 13 0 R - /K [95] - /Pg 93 0 R ->> -endobj - -52 0 obj -<< - /Type /StructElem - /S /Formula - /P 13 0 R - /A [<< - /O /Layout - /Placement /Block - >>] - /K [92 93 94] - /Pg 93 0 R ->> -endobj - -53 0 obj -<< - /Type /StructElem - /S /P - /P 13 0 R - /K [91] - /Pg 93 0 R ->> -endobj - -54 0 obj -<< - /Type /StructElem - /S /Formula - /P 13 0 R - /A [<< - /O /Layout - /Placement /Block - >>] - /K [86 87 88 89 90] - /Pg 93 0 R ->> -endobj - -55 0 obj -<< - /Type /StructElem - /S /P - /P 13 0 R - /K [85] - /Pg 93 0 R ->> -endobj - -56 0 obj -<< - /Type /StructElem - /S /Formula - /P 13 0 R - /A [<< - /O /Layout - /Placement /Block - >>] - /K [72 73 74 75 76 77 78 79 80 81 82 83 84] - /Pg 93 0 R ->> -endobj - -57 0 obj -<< - /Type /StructElem - /S /P - /P 13 0 R - /K [71] - /Pg 93 0 R ->> -endobj - -58 0 obj -<< - /Type /StructElem - /S /H2 - /P 13 0 R - /T ((b)) - /K [70] - /Pg 93 0 R ->> -endobj - -59 0 obj -<< - /Type /StructElem - /S /Formula - /P 13 0 R - /A [<< - /O /Layout - /Placement /Block - >>] - /K [46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65 66 67 68 69] - /Pg 93 0 R ->> -endobj - -60 0 obj -<< - /Type /StructElem - /S /P - /P 13 0 R - /K [45] - /Pg 93 0 R ->> -endobj - -61 0 obj -<< - /Type /StructElem - /S /Formula - /P 13 0 R - /A [<< - /O /Layout - /Placement /Block - >>] - /K [33 34 35 36 37 38 39 40 41 42 43 44] - /Pg 93 0 R ->> -endobj - -62 0 obj -<< - /Type /StructElem - /S /P - /P 13 0 R - /K [32] - /Pg 93 0 R ->> -endobj - -63 0 obj -<< - /Type /StructElem - /S /Formula - /P 13 0 R - /A [<< - /O /Layout - /Placement /Block - >>] - /K [2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31] - /Pg 93 0 R ->> -endobj - -64 0 obj -<< - /Type /StructElem - /S /H2 - /P 13 0 R - /T ((a)) - /K [1] - /Pg 93 0 R ->> -endobj - -65 0 obj -<< - /Type /StructElem - /S /H1 - /P 13 0 R - /T (Problem 2) - /K [0] - /Pg 93 0 R ->> -endobj - -66 0 obj << /Type /Font /Subtype /Type0 /BaseFont /FQBMTL+LibertinusSerif-Regular-Identity-H /Encoding /Identity-H - /DescendantFonts [67 0 R] - /ToUnicode 70 0 R + /DescendantFonts [51 0 R] + /ToUnicode 54 0 R >> endobj -67 0 obj +51 0 obj << /Type /Font /Subtype /CIDFontType0 @@ -767,13 +515,13 @@ endobj /Ordering (Identity) /Supplement 0 >> - /FontDescriptor 69 0 R + /FontDescriptor 53 0 R /DW 0 /W [0 0 500 1 1 839 2 2 457 3 3 316 4 4 538 5 5 447 6 6 250 7 7 297 8 8 730 9 9 531 10 10 390 11 11 310 12 12 500 13 13 493 14 15 465 16 16 322 17 17 542 18 18 515 19 19 372 20 23 465 24 24 701 25 25 220 26 26 271 27 27 505.99997 28 28 519 29 29 512 30 30 747 31 31 685 32 32 424 33 33 236 34 34 604 35 35 504 36 36 497 37 37 790 38 38 264 39 39 428 40 40 457 41 41 541 42 42 220 43 43 328 44 44 531 45 45 485 46 46 951 47 47 485 48 48 588 49 49 637] >> endobj -68 0 obj +52 0 obj << /Length 13 /Filter /FlateDecode @@ -783,7 +531,7 @@ x endstream endobj -69 0 obj +53 0 obj << /Type /FontDescriptor /FontName /FQBMTL+LibertinusSerif-Regular @@ -794,12 +542,12 @@ endobj /Descent -246 /CapHeight 658 /StemV 95.4 - /CIDSet 68 0 R - /FontFile3 71 0 R + /CIDSet 52 0 R + /FontFile3 55 0 R >> endobj -70 0 obj +54 0 obj << /Length 1292 /Type /CMap @@ -888,7 +636,7 @@ end endstream endobj -71 0 obj +55 0 obj << /Length 5831 /Filter /FlateDecode @@ -915,18 +663,18 @@ p endstream endobj -72 0 obj +56 0 obj << /Type /Font /Subtype /Type0 /BaseFont /KTNANF+LibertinusSerif-Bold-Identity-H /Encoding /Identity-H - /DescendantFonts [73 0 R] - /ToUnicode 76 0 R + /DescendantFonts [57 0 R] + /ToUnicode 60 0 R >> endobj -73 0 obj +57 0 obj << /Type /Font /Subtype /CIDFontType0 @@ -936,13 +684,13 @@ endobj /Ordering (Identity) /Supplement 0 >> - /FontDescriptor 75 0 R + /FontDescriptor 59 0 R /DW 0 /W [0 0 500 1 1 614 2 2 428 3 3 551 4 4 542 5 5 325 6 6 489 7 7 905 8 8 250 9 9 514 10 10 315 11 11 505.99997 12 12 315 13 13 514 14 14 456] >> endobj -74 0 obj +58 0 obj << /Length 10 /Filter /FlateDecode @@ -952,7 +700,7 @@ x endstream endobj -75 0 obj +59 0 obj << /Type /FontDescriptor /FontName /KTNANF+LibertinusSerif-Bold @@ -963,12 +711,12 @@ endobj /Descent -246 /CapHeight 645 /StemV 168.6 - /CIDSet 74 0 R - /FontFile3 77 0 R + /CIDSet 58 0 R + /FontFile3 61 0 R >> endobj -76 0 obj +60 0 obj << /Length 802 /Type /CMap @@ -1022,7 +770,7 @@ end endstream endobj -77 0 obj +61 0 obj << /Length 1734 /Filter /FlateDecode @@ -1037,62 +785,62 @@ Ty endstream endobj -78 0 obj +62 0 obj << /Type /Font /Subtype /Type0 - /BaseFont /RJSFJP+NewCMMath-Book-Identity-H + /BaseFont /XLWTGD+NewCMMath-Book-Identity-H /Encoding /Identity-H - /DescendantFonts [79 0 R] - /ToUnicode 82 0 R + /DescendantFonts [63 0 R] + /ToUnicode 66 0 R >> endobj -79 0 obj +63 0 obj << /Type /Font /Subtype /CIDFontType0 - /BaseFont /RJSFJP+NewCMMath-Book + /BaseFont /XLWTGD+NewCMMath-Book /CIDSystemInfo << /Registry (Adobe) /Ordering (Identity) /Supplement 0 >> - /FontDescriptor 81 0 R + /FontDescriptor 65 0 R /DW 0 - /W [0 0 500 1 1 490 2 2 706 3 3 389 4 4 572 5 5 389 6 6 778 7 7 394 8 8 278 9 9 556 10 10 600 11 11 778 12 13 500 14 14 648 15 15 569 16 16 278 17 17 833 18 18 1000 19 19 1139 20 20 500 21 21 569 22 22 278 23 23 451 24 24 278 25 25 778 26 26 278 27 27 778 28 29 422 30 30 569 31 31 778 32 33 556 34 34 667 35 35 530 36 36 339 37 37 778 38 38 613 39 39 1032 40 40 569 41 41 778 42 42 654 43 43 970 44 44 850 45 45 736 46 46 500 47 47 736 48 48 716 49 49 500 50 50 601 51 51 666 52 52 778 53 53 500 54 55 875 56 56 584 57 57 1000 58 58 569] + /W [0 0 500 1 1 490 2 2 706 3 3 389 4 4 572 5 5 389 6 6 778 7 7 394 8 8 278 9 9 556 10 10 648 11 11 778 12 13 569 14 14 881 15 15 792 16 16 681 17 17 278 18 18 833 19 19 1000 20 20 1139 21 21 500 22 22 569 23 23 681 24 24 278 25 25 451 26 26 278 27 27 500 28 28 778 29 29 278 30 31 778 32 33 422 34 34 681 35 35 778 36 36 600 37 37 778 38 39 556 40 40 667 41 41 530 42 42 339 43 43 778 44 44 613 45 45 1032 46 46 778 47 47 654 48 48 970 49 49 850 50 50 597 51 51 569 52 52 597 53 53 716 54 54 569 55 55 601 56 56 666 57 57 569 58 59 663 60 60 584 61 61 1000 62 62 681] >> endobj -80 0 obj +64 0 obj << /Length 13 /Filter /FlateDecode >> stream -x # +x # endstream endobj -81 0 obj +65 0 obj << /Type /FontDescriptor - /FontName /RJSFJP+NewCMMath-Book + /FontName /XLWTGD+NewCMMath-Book /Flags 131076 - /FontBBox [-15 -1245 1060 1745] + /FontBBox [-15 -796 1060 1296] /ItalicAngle 0 /Ascent 806 /Descent -194 /CapHeight 683 /StemV 95.4 - /CIDSet 80 0 R - /FontFile3 83 0 R + /CIDSet 64 0 R + /FontFile3 67 0 R >> endobj -82 0 obj +66 0 obj << - /Length 1494 + /Length 1558 /Type /CMap /WMode 0 >> @@ -1119,7 +867,7 @@ end def 1 begincodespacerange <0000> endcodespacerange -58 beginbfchar +62 beginbfchar <0001> <0002> <0003> <0028> @@ -1129,55 +877,59 @@ endcodespacerange <0007> <0073> <0008> <0069> <0009> <006E> -<000A> +<000A> <000B> <002B> <000C> <0033> <000D> <0031> -<000E> -<000F> <0031> -<0010> <006C> -<0011> <006D> -<0012> <2192> -<0013> <221E> -<0014> <0030> -<0015> <0032> -<0016> <005B> -<0017> -<0018> <002C> -<0019> <003E> -<001A> <007C> -<001B> <2212> -<001C> <0028> -<001D> <0029> -<001E> <0033> -<001F> <2265> -<0020> <0075> -<0021> <0070> -<0022> <2208> -<0023> -<0024> <002C> -<0025> <2264> -<0026> <2130FE00> -<0027> -<0028> <0030> -<0029> <212CFE00> -<002A> -<002B> -<002C> <212CFE00> -<002D> <0028> -<002E> <0032> -<002F> <0029> -<0030> -<0031> <0034> -<0032> -<0033> <2130FE00> -<0034> <2212> -<0035> <0036> -<0036> <0028> -<0037> <0029> -<0038> -<0039> <2190> -<003A> <0034> +<000E> +<000F> +<0010> <0031> +<0011> <006C> +<0012> <006D> +<0013> <2192> +<0014> <221E> +<0015> <0030> +<0016> <0030> +<0017> <0032> +<0018> <005B> +<0019> +<001A> <002C> +<001B> <0031> +<001C> <003E> +<001D> <007C> +<001E> <2212> +<001F> <2212> +<0020> <0028> +<0021> <0029> +<0022> <0033> +<0023> <2265> +<0024> +<0025> <002B> +<0026> <0075> +<0027> <0070> +<0028> <2208> +<0029> +<002A> <002C> +<002B> <2264> +<002C> <2130FE00> +<002D> +<002E> <212CFE00> +<002F> +<0030> +<0031> <212CFE00> +<0032> <0028> +<0033> <0032> +<0034> <0029> +<0035> +<0036> <0034> +<0037> +<0038> <2130FE00> +<0039> <0036> +<003A> <0028> +<003B> <0029> +<003C> +<003D> <2190> +<003E> <0034> endbfchar endcmap CMapName currentdict /CMap defineresource pop @@ -1188,54 +940,63 @@ end endstream endobj -83 0 obj +67 0 obj << - /Length 7624 + /Length 8143 /Filter /FlateDecode /Subtype /CIDFontType0C >> stream -xyy|SevcH(1*#&q6Pe,{.i{4M6;ٷM閦(- -# 41sD9mY`nC1EAA0%+% SOƈSbVoߺ-fUn(f 9% %fΜt(yDissiRssD_S0/2lΪٰ͛v[5W$Ť -c)$AVܨ(Q? 눕~f:318Mٖ;A`Ԟ\6f6gjj[v(JTTTk >j0ETTTԒHLz-Ry.u7eAZ}.]Ę0D JVNO:gI33n=ӏx_D_eU=:~뱭ϟjfOgxꙧpމI95F#<6Byxhx>2!ekZdI-o~2>88>=ؼEK]ҿuBQ,wkka4y"I[Z")R(6_ZQ̛'($s Զ5 Wt]]|ЂnSХqEW][鮆*<,PW^aQttDc\pE}mu/"63 XZ&sr wh*!&.pfsA͗'dM/# Jgk]sZŰlw6V4}N.DKVE|m1a0 CW}Ӂ` *@mTk -d4~ PlB_OC I[Gm [`bG, &u8@.Iz/fe*/7}hb\Z7 ^͛ ,Yܽw,!Pt56.ănN"DC]ɢ<~YH@iӺvم}'_O'W&;#@fF'!CoXtN(``SweWZƯZPR5{Q;~J=ZĺCʯ5qZB\&%ǩIޛ7jǑp}\twsCzd\ 3,侵iI4e-ɚvg^gЁ>.0d1#&9gQ7T/tJUZT/tYU!ȸ)/-24yx)ȶv,o:OOue{G w -u,O-5l:7 Uк9+TB+ ^7TD9mSrĺpF֖&!MWPk.fHlJ76QS髫Cye}yW\&y1UQbZuL}<ˋw߱[ y-pCckOze@YcE^jƺ@RhmPwhZK`D_֩u4@|30*EcGxGA>=LKJ~ڐ -R0Âzeؤ(,:T 6}@q">6.! uzƩ"cDgMYmU[u@x 9LdP._;>n)hIF2dZ@=X-y;j cleLԑm2l.@\e 2mb"^-yu%܎G_`:kCHUF +E, i @GCH^Re9(u|nb]S9RJ!.FdOIdEv#{>]Fr9=~2pX[t~h_),2 -?k8y ->~q)-ȫ/O2DuVIS̍G8ÛJ/ۓ5~:kB~kG+Y˗qcc#va_T@\ ;oHvށk(y1gC7Q}Pq\6/ixI$%GCʕZVOΧ7u8ll0tMeV@5"1 |>u_a]qlΉ܄m -|)?'CZ¼EՇ: x S dېM/JRqUAKlÜ!_4P]x@]b -uW@< Rd%;vm _l3LK VX{G(//|T(7siOx(z^P :hBQiٚmxtʊW렟mmx V`4lYI!~Ei S#H!#Ic_PwhkM&<&XpѣXYb{aG̸_~xi8&gr?w2+dn);z6t~+Ĥ 8LcʽEXvXvpCgW[ۓFs;$aWeT%\4UvY2 J"0 n}|I'7'/wޛ`ѺށށpN%$ -nUpwbZVVF~beeu C;{7վaKf{OBO¡p 4;!+ Iv$U;3Cqح(nMkM X+k*֜ok;HgvfvJ ‡_r [Q ͅlb:paHz[$/ᗹx -v[\[N D ryA8 -#a9akshp:wx f|E4GSpя>Ʉ̈́P^ )UR/Ra&<2Stx+ʝ\&T|,z;97J?'\$t FaC9 -^];6,FkMY'auidO_N~FfI5-+۟AUE%g[zŶ,&1ӊzػ?0~p!~oinNOaso#.qH ER`}aiu-3a,EQ}[g(@*wk1:-spWku t6aC}-w(`Q8c [h r e |근p>&fD!`O\l&&1i&|}3N*P\4+f?fp_9~醠ή-BH@ [G/amh>`.Z*ʮ sd?'INJƒVpqn<;DL*l|ɺ'an NQoxwp3삄>c4mȼi 2 og';"MߟuYXWī^h5 ]W\ZҊ&N)z#l)ZAnS:ygڊ)=7|@%MGmm] #zsJaOSBdH$+C1H!QP)iK<..c6kE?:Zy@\+| V{6t v^T@Vk0TTC-xb. 0+G]{k1eN?5̺{?Ǯ,%Ԡe -)AP AwnzpZZD xYUSjjwİ\F5nT eA eUXf3lo@5 - -ȃT,Y*YZL_6ٓ}0QrfhzāH+EF(OP?]UARe67+zEb%h<Wܶi=a_1hG׈EƺK|H<63;\u1EZMuyUI0rВlzV^A[([/QHR%.Ue׹ df5Լ@+Zjv Or͆یiC,w:IWwVCq.u-IA -\_^ -P.t -hhYMJ!{,[^&#LNHH٬[\F m$UpKm+5% `59,#O|Eؕnp"b&9k)TJĘH0Ll-?-u7YwgcΘeZu/ *7:xӺHveО!~ozN4 ᾋx($ZAؐPG[JAm1;Il}{G"16#+}+vk|M> ;{ԅ/YM>@(YeDZksbW9U\pZC>p -\ tGptFq߱6%'W&2h+Z F>Cى& k& 7%g\{mMo5DlSnvG`*z;E`\.H,NMKM$^ у Sp ބa`eBd& 4^4.uFf"_Ξ\(5@yi&~X5_/9w `~y?5spxZtf]='_Ʀ;Ē#16{?RJF琬dI i!_$Z^j(u"l &xTG"hPZ[P%d]0fpQ{!σr}tM "x D q% ->`x0Y*^$5wg2ȧ"YijO8wﻢmW&>p7[D.SJPi5JDCXL}jZ+1d!$!cȀ 6M1}<p2l'R`Ø!{J(1Өu3- ;ҔkA#FJ

?΢5k~Cګ.>9ᩤs3 Ƽbr~:`#F>F%דVUa lf׿g"r<c?!b;.eŷǧAoTGTrv32Kܡryk]3*6!w'd6`;O}4KeX8]2tA|ʨ-mO;GM fETgbJ+hzAcԁ"b9U5\h kGu=&7/9 &82Pi2^ /7=ig`TE\m89=sȜ/ _$_f0Jn$[1eM=iЏD_)S3 -[ -Jȏ',A~E iE+l+.aukz&#M # +xy|Se[K[06fd; KY,BMӴM4i\ٓi鞦ntE6YEA|SOy?uy}??wsW~-XhgJ;+7fRm&Ka!X̹' 0\ٷ¥ݤ- + +f-/\Ash#ԟȅ + + 3'&lMN%o|byy峫}f9^ȂܬlW{"yˋry)Y) ) sS)ܻ'rp^RJNA+,O^ =O Ke͟SF]r󮝑+÷nܴsϦł婹)^VAAw<@|.'6q8ZdgAd»- So7LUf…[Mfd,/$(8(((h=z*h˂P1YZA޵;!C\ \0L O0Pk﮹=ݴ𡅶-<(lQEG}ߟvP_跬|~-a_=% K>YZ"u޶+D+`:<>COBxjv+I2Ȅ6Dr/^-x|}C$X* :R fI#d 婢B(rnvy5Kyl^@( *=ړ^k/zh \]G ?=ZڡRK ;%-mmYqW( (>d/;W"L-K9N ZZumK UK>2Ӕ&@2Z,^{V5jm6XlUMccmGj !7'?Xwd!4BocXÀH@S2Ӂtu8(E%x\ӋXp^g9,q{P@L2!CR϶v.&)?Ὤy]K3\{F&NsU(yu˰Ȁ5u/X51FQ̐/͖v7KD#q7{$'xHz~$Afl^rYCYU v vyg3.!$dm:іx(QjF/+<܌Qa t6xb0 -}>P/ +P4Hydinu Njpj*ĺW!)<_-i7ڹL2 rԲFOc*\}|=Ɓwh nn%AmL& nB-k:*\Z5*I֪ACmz0#3;t\^ZjtVS%uHSZ)24x%Hd'vda}gp1j쑺Cl!v]Ѯn=)uzY] arexTvG +%29ߝeOg$xx9>AWNΟt@,D!nƩV*e\(6WTr`6ڬ.֗]6eWUE^L <om즼6?5e@AiBIZDuc9:*ca@\V uihŸ/Y {! .YT@FL@/-zQfT[ +] uh<5v!ƺD>&Z7klJ(ŀboΚ +B¤0Չ9<ƾǟ~muB'rb!07`(>R!YC$'8j1;Qᓧj $ +$?*RRcArkPc;0?Ƹ#p;릯 m{j*er.)$r@yfNOkGy&ɢ`Jٙ1 ۄ/Fy0btRHB h~zo78Ӡ}ݐiVV%N5rJ|( +AX=#PLB=  H<^cUZ&`&.wh^|`)0~x=)qȈ+2ѥц78sרiX5ٙG_J?;uIpB, %,a[& X(] ]"$E\t4OT97DmFii"qbtG=]Ýc.$;á\r\dwN񢩤 ;a3.<XYYt_N&qq}v돛(<,ttftO 6;ӗ4?\Iɦ4W"D3^+k*oa0W͠/l(d?t\2D@ÛH)&x`[ӍiS y Q9Mn>Fpxa?RZS_屸4tsF̴nnN| Vy= rT|)yWLa3dvn|?c>ۑj >mwVCA7W2vo? PrAļxqmͅo? +5!Pz~QeLU1(@: mNs4nWP͒x(є{ AWSӸȮ2K5&c!3;q Žt&*5%{ٸS#*܋{9h@n\W, WREaC"c@1*:auz~4E] 4O 21ɘ5 6p>G[z"6~ɺ'$0U OI8^4<8za쇸dh[Լ=P̱IB4]tjz!X[bȵTwĿ+.̦jwMi(>=Q Aeun w_!lWauܥ/8 jb^H ?+GQ:1nbqwhCwuG~]] +rPZA앉A2/4T6U6!-W7kk +7{ zT6ټR!WXOՀ`` ]o Dj,x]9> +]?IVfr6}o0--AhFD@.E/Mnk:maV倱`6w}Qͅ8uGQNxi;';3 g?a!>" /S5=Չ*kkD {-$[L^i jɆx^^,Ak(KMr~z#4K{ۊ64ћ{y? (fn^8=Xćowըڵ.t +Yw*%f1L+Քȧ@2Nb=3,C)E8EV!@h$҃Ԍ]5ՠ3C޷mMӾK>W_ g p UCpI)x~ztyM(u&N-ʒ! z]4~I|^njFI +ՕB5#&H.8!ՆC 8ǀ| [֒K7Ɛ0;2fR }w}xUqŖZTN^d5诪=(^)3 LTO`5:^L4Jl1},. =@+)F$1;@RELԡ \"CgV"+ B)\8ix6p!+̥:LzсOgEJjw"!AP&)U)ź,@BVebnE_m3S^]ܐX7QYV>䕜C9)e!5ɭJށ| pz[[|Y]dIP0Xą#G+E^'f|_eC\aJlA^h hXOmj.# W)bxM4e@k>?Ro_06]~%.S߼BvR֬6> ӣxY!^x1x0^6ӯ1 f@RZT׿B޳|&!WK|cc箒|ެ3 @,"'\} 6XnollBw'U_?:{l(&=V$+*)#ir +h~;h},fiw";1ult_ׂXj ' +GwwuoH8$ŇO/sr|r?-f{*~~̇W^2%9xnv+/AI}}$}| %@.BrI,,|~I_E֪@I*0Qo#&"񾿄8GAAZ[ JK,Iī{5ю#ɃrMxޝ!(,<<֢8A^\`g0}由'|E2Qx>Ooh ɒT)5\̆޹2r_pu̼L%!C@)WBMx: TC QaV5UFpu &56g7M~CL5os:>IM4_ Ny/"_E߯^ !3pS]/T8ОvR@Hatzd +PYZ-RX}9\i0e6VU5ĉb <@ Nð.@r K-NjRP4:F{}n<84P (AbFR}` +Gé'@`O,w;Sհ15[k C) + $©@B ~'섘܌-C=|ǡZ*%E*D2ܢNsqGϫr6Wr,!֝4s,"~/Cec> !vZ9ة̏^|/ލwq`~ MP+xf +TՐ ^#n$w E 92|=r9rWRƤ3bC0? 6^p+<^ہ_6dX~q+^K4F^EYbu0뮐#w &2 ȕK>'&wJ.~tB$VsS}B/)eF*&T]QWHy]ʌRɝs}Uf>7 + fAk KAh7MF^SQ)hU<"rT 5++xэZk4Yo dgvZ&74ZN 2j34on4{+;rm'q ' YuzFJPqbnzBT,XQ+A +ҪPk'~()\63&wv$9y~&]DJQ(f&f^f tR4{lDhPQ,&JOM55L U_ޭNj&"b#Uml'lodZ endstream endobj -84 0 obj -[/ICCBased 85 0 R] +68 0 obj +[/ICCBased 69 0 R] endobj -85 0 obj +69 0 obj << /Length 258 /N 1 @@ -1250,157 +1011,106 @@ F endstream endobj -86 0 obj -[93 0 R /XYZ 70.86614 755.2506 0] +70 0 obj +[77 0 R /XYZ 70.86614 755.2506 0] endobj -87 0 obj -[93 0 R /XYZ 70.86614 577.2266 0] +71 0 obj +[77 0 R /XYZ 70.86614 696.3214 0] endobj -88 0 obj -[93 0 R /XYZ 70.86614 781.0236 0] +72 0 obj +[77 0 R /XYZ 70.86614 781.0236 0] endobj -89 0 obj -[95 0 R /XYZ 70.86614 366.25763 0] +73 0 obj +[77 0 R /XYZ 70.86614 549.3075 0] endobj -90 0 obj -[95 0 R /XYZ 70.86614 300.7636 0] +74 0 obj +[77 0 R /XYZ 70.86614 503.70154 0] endobj -91 0 obj -[95 0 R /XYZ 70.86614 189.17963 0] +75 0 obj +[77 0 R /XYZ 70.86614 437.94354 0] endobj -92 0 obj -[95 0 R /XYZ 70.86614 781.0236 0] +76 0 obj +[77 0 R /XYZ 70.86614 607.69556 0] endobj -93 0 obj +77 0 obj << /Type /Page /Resources << /ProcSet [/PDF /Text /ImageC /ImageB] /ColorSpace << - /c0 84 0 R + /c0 68 0 R >> /Font << - /f0 66 0 R - /f1 72 0 R - /f2 78 0 R + /f0 50 0 R + /f1 56 0 R + /f2 62 0 R >> >> /MediaBox [0 0 595.2756 841.8898] /StructParents 0 /Parent 1 0 R - /Contents 94 0 R + /Contents 78 0 R >> endobj -94 0 obj +78 0 obj << - /Length 3624 + /Length 6820 /Filter /FlateDecode >> stream -x]]q}_1-Tb}W7`hI xl?д(hfkU;yTW7>~ۏ˯~uX_?^>-/7C~yo߿?7}u-5-TQ}i(ٗ?^_77\n>^å/7;7?~? wO{~?.7t_sD7S'>}_| 'bme>t<>͡4f|c3>563^,q{,g`{ŽDPt -|OMo_^;?v›.K ebiֺ̩o-h-$]BWs`q c u)x!@=Xr++08.^b h )^h,s@TԠ[SZx`0O^ j[\T-Z>Or@i..] TRb F3-*;`40|( 1<' E?wK: 2K2lMc d@=3L$z}> |{_`^}Êbp"W4+B s :\(!mtiyE'̀0J"_&BS 1J5c9V 3<3BІSs M -`N\^ҏU -8 -,K0kk |h^!\:/њ_/r N0)Y Dg 0 ^ { -z]V&'_ϖ`i1]( UX<70L8C!nLA/iKyWv2 !u TY\ UL`!:G`Ea9uOUpafA|@74;m ˏBl0*՚%p_? bMTs7?;nBrڊ7Q.u1 -MՍQu,#>-lNܢL69f1u=,>(c٦@Օn2^M@0"((JcZVVѤGn\e1)!w站ѹ$ؠL1 6(;ZhuEic (n>}]vYuߡ5>1&40|mݓ~r|)xPEVWI[-n;VIRS1¹L6hL6 8YU!>}l- WG[Ee7ݲfȱ@ -*s0}zevR{;:7^-/^?ى λ TeP8s&S8(\`BdՔ ]ئ@j9lm -Q|&.Yj $;],cn;Qur$\; Z{P*hǃNrjW;Y! v$5Wg[^Ja^A \bX -DcTz)vKanpq#O`sJc^2džk -&ܼRH)\a՝o}Yr!i S -GJ[}X[ {w1CtL515uxZ)5C8ji.36PAVh خgJ5@v_LL0tauAŹ]~TOj<Tq pI ?>;~}2>;_<I@MB[zɽj˅x{5ejS'\*Yxi,luwL0TPc~>M98 ɒG&@\srr.)LK 6RR9ǟh-,{/CTe-ga^X֚co2xaIp0sqF& ]F]˒VŜۡIdúN'KZvHp W{dX+g|\ɺ}/o-/O`T/5dy(=ƏJO۸c,%dy{IoGpԐe{vOAg<{hPhv~+E#x [+s*om%WrA+=<Ɖ-ɥ-;N1R k1'ci cR&Gq@~Jۉ1PoK;Ը K - !.En$#?^"?<-ZO=mf|4t. +x][%m~_q|약\Y8l@@AYl793sw EH~_yog?;^ۯopͫ߿߿|sz|o/^_?~?_}~;: O/]@O z8?ݽO_O_7wS??ǿE"O~)?=ߝ^ݯ_pnЃUO Ĕ +~ O~͋KB@Dӏw?}Oshl=!Y?~n6ӳ}38޻> +ށ{Oрyj}amz~@ϗ[/>{_A~uB _:[^ގn~_݋;{o¢vK +DTazyc}w`_^? n-4&S6@[bТYG>!`VO Cc$Q|f&H.Dm[RR-I؅F*HbPi.UAQ2mBI#$ ~$&dv42_EgX Jgͧ@B'%; ox|򋫺l>O_~/w?wz.֟_w_^R4jG".47IA"n=d`lj8&@k8ZR*FsvbT͠;nKEf%#lI\w"@W:@%R۷?Do>7Z.}Wh ތ>u`Y].qt  >n(r,}30е/O\e4_+BS{vn]lhƝʎ0"ޟкhZB,\>2R}<qte" @p{e7,3 WաwSvGL[07@>$:!a cZFbZn ԉ! -w0|Y`=R',xrt-͔g,':y<2.M&6Z2c2=.g%__tĹZ@+󣆀]09QsHmVF!lken4T)[0VY q"۩Kd Kl{jiFzAUteaj~bat= +!I8*)+#6d:q(C6LUh> ٰ3/o PFmH'mH_zMa09s?EzPݶx <虗asg4(z7-zaty +8y 30'W{w@5\#0׼TAA\R7j>_wqo^KUnr,h幊oXE7]dwM`_߿|޿JOm`iYxCsQ 6ǐӛrALz6vC$#g.Jˡ 4 °8!9HGԽCDϲ}N έKaPJ4WuZSg@3a u%J=1 lJ 1(%aA VM,%V_ŭԾuvpux1??Ca(T@+3.`WF|{L0rEZj algBƕf@]ҍ ݀^g<T'OݮWA0[8DX{e Xl0r{d,3 3X +ERm/z ˗"P|Q>.|­2_ Ly2_Y<9 PY C\WS/A GLgNCMVNxa|6Oy`槞7UV:?3N<@U)g,J#afsb8 rEXnن1}lǸY 4JCs|¡qi H: PKe#09X([@`,n X;,ހ{ PS;4ΪCv `3KlW\]sS>ӫː2 FlyjDCMoJ/ݎa uMA10x1edk~h:2zvY 8}TLjra*f7BtI_᫬6#Tk*XsAzIGYۚ= -15w2($EdPv51%0IBahbJr!ugn6xl1%zE_'6&p!~11R;ro%5ɛ:8 GR4i]6CpJ' #fa*+9{ 2n߂\TrDx݌`XE%iO V"O +j?к?*~ ވm۰wޒsm;őg4d>E~m>_)OI8C)fW:@s 8e%sM9hI}H_PT njƻyHvIO??}uqI;$}mO?ǽwA'DΤ|8jr n9ԩtM1s@{V\!) ~BʼnҲ?xqzc{nOyū}yhEƭך,9+#كyD) ];DdNaBV hWI"÷Ft$pluyFd.bmiw!i`eC+2 n8;S݇jec,~@BMl}iM* +\~:-yJ lT)r:/ɧj>݃.K#w#ǒ[y`Vj>c6?y,ɕ9]2A[b푃&AQ8qA=};%M1TR51a2=(u_g,vuOB=(Ѥs#k%E=pޓ50[Ėa&"BS +5ِslHG@!%;ujٽ I͒ )ɀs%d_CJVPϼcKFa'+ZRd2A 21sKvc%RZzy=0\!)Iwgd͍?vg~6ڙt1Mͽo +?d5 a=isRAZe`% w9oJͷ*oǙA$9?z֣8;N1LY{!jNvelV2(m22j"vĝ'eVʏ$~@6̻K*cǺV8K!p淘%sj-g=%{gk\Ϻ/;9zJa^0|dܳHZδpkn2_Z/gLyА`V\ӀXK)?6`%- ҉w/v+- nˌc0 jIp˞͚&8Tc|A,#Azǁgk˴$-Dojn@0ӬjI>#ϧl;j=:?ˉWXZ-gT1VKYVQ'C, K Yڋ5ZZ-)n9սEc;jq[zK`6jq˾ +hft-9nKW_V֝ 0[o}Isǜ[y]PݸS8kP*y sueδKU2X ^KR%bg]2rAKi^KR/sg"uA(pU͇aEuJ-Iug`HyOKFc%[3̅t@P[߾kloɦ-7rm0JiMC#VH0цJosv$T: +Ȏ6(5Z[ŚI'oeoAPftmn&Iڊa D7G@1Ӛ!g;Fj[ Ea;ݷMk*t4]XzV{P8aܫ$->m•g%.CNbGy^[XӦ{=]!vsrҦuyz[GAˮZp 2́d|zp&yi+Ck+ u& +Qh7AHǺ#Zk7xJe? endstream endobj -95 0 obj -<< - /Type /Page - /Resources << - /ProcSet [/PDF /Text /ImageC /ImageB] - /ColorSpace << - /c0 84 0 R - >> - /Font << - /f0 66 0 R - /f1 72 0 R - /f2 78 0 R - >> - >> - /MediaBox [0 0 595.2756 841.8898] - /StructParents 1 - /Parent 1 0 R - /Contents 96 0 R ->> -endobj - -96 0 obj -<< - /Length 3679 - /Filter /FlateDecode ->> -stream -x]o]GA|ՅLswJ4 R(~khZ ھv朽x9/gf{g7ӯ}y}e|忿z￞W]|_훷'L_pMTQ(j25N8뷇O8];<Sv|\{/ D:}SNzg9}ϫ/?~w}A:f lOf?6 om(_~έ 8[ Ta֧ 񡰙 k-WSZ+ḛpkvu BZ}h=w8<'~;abK!j!eI4^fWI2wU*Ԡ e$_ó< -.nէ -TuaϮ,x(VD: hƀpim ҡo)>N%!.=tOV= -u~u)ޜW`g z3\- dkH?2T -$"-s1fO_p=2<='oP;~l!/Pyn)Az{,i^ M*5O(t& ><Įz&KVTȷEt%朦>q0\dhQ a - e|+2#.E&kf%AX-Dx軏BʏRA6V/zɒK|2U0}2k]!gA8z¼L]J}"}E\ХZWZ?"O A{#EGnĺ jsNs4 _z ܎_oW(Um'YHFۇ?ŸBze#k,:F%?I;`Ӌ}\R-TGJd|܆=dxe]MwԚtl7sgB;;Z{@ؤ?}=}.c,/i3\1h^IR(A^ -o4V]Cj+WAiDkŽVq\o󯋛5BuͿ.xԺ$npݮ1 -0:ɤдgAR ٠>/D;dIH"j6 /"H5-,J^˃V:)60j"]& H+ Ed n$0^gR*8қH^+T]?M$K]ɫ=ЋMEKRouFŶMF2> +pFZfu\G Z)Yl$WҎ*MѲzFj\@dё\CTA7’EGjlm{7,iKx7% q'7L±I?1zq`=!K>02 1T ʗ|.DV` C#h("!kg񑅻~6#u@=/8|ƴ`A[XA?/gטnصf W|DZ5<2h=3*P2g,?e zVS9'P\M# . -H#ma "5 wΟ PW1^pZ?.t/Ndt' 5A^\ Xs%NTc3\NX!)kDʖ[,Dsg /*vΈ6(|&bGAwXx:K:`1yz:ʺw -2{*꺚niH/VPtߣ[ԫ~kSkiT H4IK#7 }I҈jԓmE- Ү7߸hYK9KZe3D޾ J i9u=akU *Yk*,ުT+UKvֶ0d~s~t':Y0&1hcoܻ &Ƥ (ƀ{NXS -u9ZZD0lh1q3 )P -BZk6Cb/4@,lRR[KR"qƍ6S\n};H[ 1GD1Le'w'\jM'. laN91,k -fgB_ S*EZ -hW@f]8fW9u`.#~Y#%~#fU*>j[*z -W Q~ @p?̧[ WtRx=~g "$xrat]0WA;2 +h]g -,Cؙ̎2V9ꎇ(?Ӂl kr48cE.U)K,v68 ()(꼗e&3km)*ʱn`Aj]%`jqPH+`{79ꞃj%oZG&W1}}b57l^ऐ*Sर*~02ªF\Ͻ)ꌅY\J)ꞇ2ٲM)xK5`}ě·TTt[:NZU'FJOO;TˁZqjı`P-50U8r Jt9T0(a+$p8Ĉ Nت 0R&GA: )bZqrxCPVp 9d @ -z|7mR| .?rQ+m42pbS*#nQ"i+*WJu xU\nq0?Y3nS|ZH`D԰@56xzoi!U8ЗԄ(EyF<'9r=H^;mg8OlԎwP㾍#;v\5GY\G ^gc= xrGzd^4E'-\9Xu,]gjVe:WALvH35C<^Mk7[|nyfhaycY>:D]|!nE=U@e?y[Z 4(m.s=b1l)lMLg6R^qlk7?Zx<[)0N(fj+[_i 8c{OqS6LD$S|ZQlCf37Aoy;R8m%ۻ5>96Gd5S9*Y koPf䗓)ox?`> endobj -98 0 obj +80 0 obj << /Length 996 /Type /Metadata /Subtype /XML >> stream -Typst 0.14.2en2026-05-31T17:54:00+02:002026-05-31T17:54:00+02:002application/pdf8fFSuRIZAEIHYQCqPu8a5A==8fFSuRIZAEIHYQCqPu8a5A==proof1.7 +Typst 0.14.2en2026-05-31T17:56:47+02:002026-05-31T17:56:47+02:001application/pdfJNdKxi/u/31IOoDqFlLHYg==JNdKxi/u/31IOoDqFlLHYg==proof1.7 endstream endobj -99 0 obj +81 0 obj << /Type /Catalog /Pages 1 0 R - /Metadata 98 0 R + /Metadata 80 0 R /Lang (en) /StructTreeRoot 10 0 R /MarkInfo << @@ -1415,114 +1125,96 @@ endobj endobj xref -0 100 +0 82 0000000000 65535 f 0000000016 00000 n -0000000088 00000 n -0000000168 00000 n -0000000297 00000 n -0000000379 00000 n -0000000461 00000 n -0000000590 00000 n -0000000672 00000 n -0000000768 00000 n -0000000850 00000 n -0000001088 00000 n -0000002591 00000 n -0000004255 00000 n -0000004696 00000 n -0000004945 00000 n -0000005044 00000 n -0000005389 00000 n -0000005734 00000 n -0000005833 00000 n -0000006119 00000 n -0000006217 00000 n -0000006303 00000 n -0000006465 00000 n -0000006551 00000 n -0000006695 00000 n +0000000081 00000 n +0000000161 00000 n +0000000290 00000 n +0000000372 00000 n +0000000454 00000 n +0000000583 00000 n +0000000665 00000 n +0000000761 00000 n +0000000843 00000 n +0000001072 00000 n +0000004241 00000 n +0000004409 00000 n +0000004486 00000 n +0000004683 00000 n +0000004782 00000 n +0000004883 00000 n +0000005176 00000 n +0000005469 00000 n +0000005568 00000 n +0000005645 00000 n +0000005922 00000 n +0000006021 00000 n +0000006196 00000 n +0000006313 00000 n +0000006406 00000 n +0000006523 00000 n +0000006688 00000 n 0000006781 00000 n -0000006943 00000 n -0000007029 00000 n -0000007227 00000 n -0000007313 00000 n -0000007457 00000 n -0000007543 00000 n -0000007687 00000 n -0000007773 00000 n -0000007917 00000 n -0000008003 00000 n -0000008163 00000 n -0000008248 00000 n -0000008351 00000 n -0000008438 00000 n -0000008619 00000 n -0000008706 00000 n -0000008947 00000 n -0000009034 00000 n -0000009211 00000 n -0000009298 00000 n -0000009459 00000 n -0000009546 00000 n -0000009903 00000 n -0000009990 00000 n -0000010155 00000 n -0000010241 00000 n -0000010391 00000 n -0000010477 00000 n -0000010633 00000 n -0000010719 00000 n -0000010899 00000 n -0000010985 00000 n -0000011083 00000 n -0000011296 00000 n -0000011382 00000 n -0000011559 00000 n -0000011645 00000 n -0000011868 00000 n -0000011965 00000 n -0000012068 00000 n -0000012250 00000 n -0000012932 00000 n -0000013022 00000 n -0000013275 00000 n -0000014648 00000 n -0000020584 00000 n -0000020763 00000 n -0000021132 00000 n -0000021219 00000 n -0000021468 00000 n -0000022350 00000 n -0000024189 00000 n -0000024362 00000 n -0000025123 00000 n -0000025213 00000 n -0000025460 00000 n -0000027035 00000 n -0000034764 00000 n -0000034800 00000 n -0000035158 00000 n -0000035209 00000 n -0000035260 00000 n -0000035311 00000 n -0000035363 00000 n -0000035414 00000 n -0000035466 00000 n -0000035517 00000 n -0000035821 00000 n -0000039524 00000 n -0000039828 00000 n -0000043586 00000 n -0000043712 00000 n -0000044797 00000 n +0000006874 00000 n +0000006967 00000 n +0000007092 00000 n +0000007197 00000 n +0000007387 00000 n +0000007516 00000 n +0000007705 00000 n +0000007830 00000 n +0000007939 00000 n +0000008244 00000 n +0000008357 00000 n +0000008455 00000 n +0000008559 00000 n +0000008687 00000 n +0000008785 00000 n +0000008895 00000 n +0000009056 00000 n +0000009181 00000 n +0000009352 00000 n +0000009449 00000 n +0000009552 00000 n +0000009734 00000 n +0000010416 00000 n +0000010506 00000 n +0000010759 00000 n +0000012132 00000 n +0000018068 00000 n +0000018247 00000 n +0000018616 00000 n +0000018703 00000 n +0000018952 00000 n +0000019834 00000 n +0000021673 00000 n +0000021846 00000 n +0000022637 00000 n +0000022727 00000 n +0000022973 00000 n +0000024612 00000 n +0000032860 00000 n +0000032896 00000 n +0000033254 00000 n +0000033305 00000 n +0000033356 00000 n +0000033407 00000 n +0000033458 00000 n +0000033510 00000 n +0000033562 00000 n +0000033614 00000 n +0000033918 00000 n +0000040817 00000 n +0000040943 00000 n +0000042028 00000 n trailer << - /Size 100 - /Root 99 0 R - /Info 97 0 R - /ID [(8fFSuRIZAEIHYQCqPu8a5A==) (8fFSuRIZAEIHYQCqPu8a5A==)] + /Size 82 + /Root 81 0 R + /Info 79 0 R + /ID [(JNdKxi/u/31IOoDqFlLHYg==) (JNdKxi/u/31IOoDqFlLHYg==)] >> startxref -45033 +42264 %%EOF \ No newline at end of file diff --git a/11 Homework/Mathe II/Homework 6.typ b/11 Homework/Mathe II/Homework 6.typ index 531a4dc..35c50d7 100644 --- a/11 Homework/Mathe II/Homework 6.typ +++ b/11 Homework/Mathe II/Homework 6.typ @@ -14,29 +14,29 @@ = Problem 2 == (a) -$ f_n (x) = (sin(n x) + n) / (3n + 1) = (sin(n x) / n + 1) / (3 + 1/n) $ -Da $ lim_(n -> oo) sin(n x) / n = 0 $, ist der punktweise Grenzwert: -$ f(x) = lim_(n -> oo) f_n (x) = (0 + 1) / (3 + 0) = 1/3 $ +$f_n (x) = (sin(n x) + n) / (3n + 1) = (sin(n x) / n + 1) / (3 + 1/n)$ +Da $lim_(n -> oo) sin(n x) / n = 0$, ist der punktweise Grenzwert: +$f(x) = lim_(n -> oo) f_n (x) = (0 + 1) / (3 + 0) = 1/3$ == (b) -Zu zeigen: $ f_n (x) = (n x) / (n x^2 + 1) $ konvergiert auf $ [r, 1) $ mit $ r > 0 $ gleichmäßig gegen $ f(x) = 1/x $. -$ |f_n (x) - f(x)| = |(n x) / (n x^2 + 1) - 1/x| = |(n x^2 - (n x^2 + 1)) / (x(n x^2 + 1))| = 1 / (n x^3 + x) $ -Da $ x >= r > 0 $, gilt $ n x^3 + x > n r^3 $. Daraus folgt für das Supremum: -$ sup_(x in [r, 1)) |f_n (x) - f(x)| <= 1 / (n r^3) $ -Wegen $ lim_(n -> oo) 1 / (n r^3) = 0 $ konvergiert die Funktionenfolge gleichmäßig. +Zu zeigen: $f_n (x) = (n x) / (n x^2 + 1)$ konvergiert auf $[r, 1)$ mit $r > 0$ gleichmäßig gegen $f(x) = 1/x$. +$|f_n (x) - f(x)| = |(n x) / (n x^2 + 1) - 1/x| = |(n x^2 - (n x^2 + 1)) / (x(n x^2 + 1))| = 1 / (n x^3 + x)$ +Da $x >= r > 0$, gilt $n x^3 + x > n r^3$. Daraus folgt für das Supremum: +$sup_(x in [r, 1)) |f_n (x) - f(x)| <= 1 / (n r^3)$ +Wegen $lim_(n -> oo) 1 / (n r^3) = 0$ konvergiert die Funktionenfolge gleichmäßig. = Problem 3 -Sei $ cal(E) = (bold(m)_0, bold(m)_1) $ die Standardbasis. Die Basiswechselmatrizen von $ cal(B) $ und $ cal(C) $ in $ cal(E) $ sind: -$ M_cal(B) = mat(3, 2; 3, 1), quad M_cal(C) = mat(4, 1; 0, 1) $ -Der Koordinatenvektor von $ bold(p) = bold(m)_0 + bold(m)_1 $ in $ cal(E) $ ist $ bold(p)^cal(E) = vec(1, 1) $. +Sei $cal(E) = (bold(m)_0, bold(m)_1)$ die Standardbasis. Die Basiswechselmatrizen von $cal(B)$ und $cal(C)$ in $cal(E)$ sind: +$M_cal(B) = mat(3, 2; 3, 1), quad M_cal(C) = mat(4, 1; 0, 1)$ +Der Koordinatenvektor von $bold(p) = bold(m)_0 + bold(m)_1$ in $cal(E)$ ist $bold(p)^cal(E) = vec(1, 1)$. == (a) -$ bold(p)^cal(B) = M_cal(B)^(-1) bold(p)^cal(E) = 1/(3-6) mat(1, -2; -3, 3) vec(1, 1) = mat(-1/3, 2/3; 1, -1) vec(1, 1) = vec(1/3, 0) $ +$bold(p)^cal(B) = M_cal(B)^(-1) bold(p)^cal(E) = 1/(3-6) mat(1, -2; -3, 3) vec(1, 1) = mat(-1/3, 2/3; 1, -1) vec(1, 1) = vec(1/3, 0)$ == (b) -$ T_(cal(C) <- cal(B)) = M_cal(C)^(-1) M_cal(B) = 1/4 mat(1, -1; 0, 4) mat(3, 2; 3, 1) = mat(1/4, -1/4; 0, 1) mat(3, 2; 3, 1) = mat(0, 1/4; 3, 1) $ -$ T_(cal(B) <- cal(C)) = (T_(cal(C) <- cal(B)))^(-1) = 1/(0 - 3/4) mat(1, -1/4; -3, 0) = -4/3 mat(1, -1/4; -3, 0) = mat(-4/3, 1/3; 4, 0) $ +$T_(cal(C) <- cal(B)) = M_cal(C)^(-1) M_cal(B) = 1/4 mat(1, -1; 0, 4) mat(3, 2; 3, 1) = mat(1/4, -1/4; 0, 1) mat(3, 2; 3, 1) = mat(0, 1/4; 3, 1)$ +$T_(cal(B) <- cal(C)) = (T_(cal(C) <- cal(B)))^(-1) = 1/(0 - 3/4) mat(1, -1/4; -3, 0) = -4/3 mat(1, -1/4; -3, 0) = mat(-4/3, 1/3; 4, 0)$ == (c) -$ bold(p)^cal(C) = T_(cal(C) <- cal(B)) bold(p)^cal(B) = mat(0, 1/4; 3, 1) vec(1/3, 0) = vec(0, 1) $ \ No newline at end of file +$bold(p)^cal(C) = T_(cal(C) <- cal(B)) bold(p)^cal(B) = mat(0, 1/4; 3, 1) vec(1/3, 0) = vec(0, 1)$ \ No newline at end of file