图 灵 数 学 统 计 学 丛 书 The Calculus Lifesaver:All the Tools You Need to Excel at Calculus 普 林 斯 顿 微 积 分 读 本 [ 美 ] Adrian Banher 著 杨 爽 赵 晓 婷 高 璞 译 北 京
图 书 在 版 编 目 (CIP) 数 据 普 林 斯 顿 微 积 分 读 本 / ( 美 ) 班 纳 (Banner, A.) 著 ; 杨 爽, 赵 晓 婷, 高 璞 译. 北 京 : 人 民 邮 电 出 版 社, 2010.8 ( 图 灵 数 学 统 计 学 丛 书 ) ISBN 978-7-115-23130-7 Ⅰ. 1 普 Ⅱ. 1 班 2 杨 3 赵 4 高 Ⅲ. 1 微 积 分 Ⅳ. 10172 中 国 版 本 图 书 馆 CIP 数 据 核 字 (2010) 第 118833 号 内 容 提 要 本 书 阐 述 了 求 解 微 积 分 的 技 巧, 详 细 讲 解 了 微 积 分 基 础 极 限 连 续 微 分 导 数 的 应 用 积 分 无 穷 级 数 泰 勒 级 数 与 幂 级 数 等 内 容, 旨 在 教 会 读 者 如 何 思 考 问 题 从 而 找 到 解 题 所 需 的 知 识 点, 着 重 训 练 大 家 自 己 解 答 问 题 的 能 力. 本 书 适 用 于 大 学 低 年 级 学 生 高 中 高 年 级 学 生 想 学 习 微 积 分 的 数 学 爱 好 者 以 及 广 大 数 学 教 师. 即 可 作 为 教 材 习 题 集, 也 可 作 为 学 习 指 南, 同 时 还 有 利 于 教 师 备 课. 图 灵 数 学 统 计 学 丛 书 普 林 斯 顿 微 积 分 读 本 著 [ 美 ] Adrian Banner 译 杨 爽 赵 晓 婷 高 璞 责 任 编 辑 傅 志 红 执 行 编 辑 卢 秀 丽 毛 倩 倩 人 民 邮 电 出 版 社 出 版 发 行 北 京 市 崇 文 区 夕 照 寺 街 14 号 邮 编 100061 电 子 函 件 315@ptpress.com.cn 网 址 :http://www.ptpress.com.cn 北 京 ****** 印 刷 开 本 :700 1000 1/16 印 张 :42 字 数 :895 千 字 2010 年 8 月 第 1 版 印 数 :1 3 000 册 2010 年 8 月 北 京 第 1 次 印 刷 著 作 权 合 同 登 记 号 图 字 : 01-2009-3812 号 ISBN 978-7-115-23130-7 定 价 :95.00 元 读 者 服 务 热 线 :(010)51095186 印 装 质 量 热 线 :(010) 67129223 反 盗 版 热 线 :(010)67171154
Original edition, entitled The Calculus Lifesaver : All the Tools You Need to Excel at Calculus by Adrian Banner, ISBN: 978-0-691-13088-0, published by Princeton University Press. Copyright 2007 by Princeton University Press. All rights reserved. No part of this book may be reproduced or transmitted in any form or by any means, electronic or mechanical, including photocopying, recording or by any information storage and retrieval system, without permission in writing from Princeton University Press. Simplified Chinese translation copyright 2010 by Posts & Telecom Press..,. ( ).,.
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3 ( ) [0, π/2] 1.5 1.6 2.1 2.2 2.3 2.4 9.1 3.6 4 6.5 7.1( 7.1.5) 9.4 14.1 14.2 5.1 5.1.4 6.1 (, / / ) 6.2 6.3 6.6 6.7 7.2, 7.2.1 8.1 9.3 9.5 9.7 10.1 10.2 10.3 17.5
4 ( ) / 8.2 9.6 11.1.3 11.2, 11.3 11.5, 12.1.1 11.4, 12.1.2 / 12.2, 12.3 13.1 13.2 13.3 / 16.2( 16.2.1) 16.3 16.4 16.5, B 16.6 17.4, 17.6 18.1 18.2 18.3 19.1, 19.2 19.3( 19.3.6) 19.4 6.4 6.4 7.2.2 16.1 20.1, 20.2 21 22.1.2, 22.2 23 25 / 26
5 ( ) 30.2 30.3 30.4 30.5 27.1 27.2 28.1 28.5 29.1, 29.2 29.3 29.4,., 6.2 6.2.1 6.2.7.
.,. Vickie Kearn Linny Schenck Lorraine Doneker,, Gerald Folland,., Ed Nelson Maria Klawe Christine Miranda Lior Braunstein Emily Sands Jamaal Clue Alison Ralph Marcher Thompson Ioannis Avramides Kristen Molloy Dave Uppal Nwanneka Onvekwusi Ellen Zuckerman Charles MacCluer Gary Slezak,. Eli Stein Simon Kochen Matthew Ferszt Cott Kenny. INTECH, Bob Fernholz Camm Maguire Marie D Albero Vassilios Papathanakos,. William Pender,..,, +C C., Freda Michael, Carly, Rena, Marianna Michael., Amy, ( ).
1............... 1 1.1............................ 1 1.1.1............... 3 1.1.2................. 3 1.1.3.......... 4 1.1.4................. 5 1.2.......................... 6 1.2.1............... 7 1.2.2..................... 8 1.2.3............... 8 1.2.4.......... 9 1.3.................... 10 1.4................ 12 1.5................ 14 1.6.............. 16 2.................... 21 2.1....................... 21 2.2......... 23 2.2.1 ASTC............. 25 2.2.2 [0, 2π].................... 27 2.3................ 29 2.4.................... 32 3...................... 34 3.1................ 34 3.2................ 36 3.3................ 37 3.4......... 38 3.5...................... 41 3.6.................... 43 3.7........... 45 4.......................... 47 4.1 x a......................... 47 4.2 x a........................... 50 4.3 x......................... 51 4.4 x......................... 56 4.5 x........................... 59 4.6.............. 61 5............... 63 5.1......................... 63 5.1.1........... 63 5.1.2....... 64 5.1.3......... 65 5.1.4................ 67 5.1.5 IVT... 69 5.1.6.................. 70 5.2......................... 71 5.2.1................ 71 5.2.2............. 72 5.2.3................ 73 5.2.4......... 74 5.2.5.................... 75 5.2.6.................. 76 5.2.7....... 78 5.2.8......... 80 5.2.9
2.................... 80 5.2.10........ 81 5.2.11........ 82 6............. 84 6.1.................. 84 6.2 ( ).................. 87 6.2.1........... 88 6.2.2......... 88 6.2.3............. 88 6.2.4............. 90 6.2.5......... 91 6.2.6.... 94 6.2.7............. 96 6.3.................... 98 6.4.................. 99 6.5............... 102 6.6............... 104 6.7........ 107 7........ 111 7.1.......... 111 7.1.1.............. 111 7.1.2................. 113 7.1.3............ 117 7.1.4......... 120 7.1.5... 121 7.2.......... 124 7.2.1................. 127 7.2.2.............. 128 7.2.3........ 129 8..... 132 8.1................... 132 8.1.1............ 133 8.1.2........ 137 8.2................... 138 8.2.1........ 140 8.2.2........ 141 8.2.3........ 142 8.2.4..... 144 9.......... 148 9.1...................... 148 9.1.1........ 148 9.1.2........ 149 9.1.3................. 150 9.1.4.............. 151 9.2 e...................... 153 9.2.1... 153 9.2.2..... 154 9.2.3 e.............. 156 9.3...... 158 9.4................... 161 9.4.1 e... 161 9.4.2 0................... 162 9.4.3 1................... 164 9.4.4............ 165 9.4.5................... 167 9.4.6 0................... 169 9.5................. 170 9.6............. 174 9.6.1.............. 175 9.6.2.............. 176 9.7...................... 178 10........ 182 10.1................ 182
3 10.1.1................. 182 10.1.2.......... 183 10.1.3...... 184 10.1.4...... 186 10.2.................. 188 10.2.1.......... 188 10.2.2.......... 191 10.2.3.......... 193 10.2.4.......... 195 10.2.5............... 196 10.2.6...... 197 10.3.................. 199 11.................. 203 11.1.............. 203 11.1.1................. 203 11.1.2............. 204 11.1.3.......... 205 11.2.................... 208 11.3.................... 210 11.4.............. 213 11.5....... 215 11.5.1...... 216 11.5.2...... 217 12........... 220 12.1........... 220 12.1.1................. 222 12.1.2................. 223 12.2..... 225 12.3......................... 226 12.3.1................. 226 12.3.2 : 1............ 229 12.3.3 2................. 230 12.3.4 3................. 233 12.3.5 4................. 236 13............. 240 13.1.................. 240 13.1.1................. 240 13.1.2................. 241 13.1.3.... 242 13.1.4................. 244 13.1.5...... 247 13.1.6................. 247 13.2....................... 250 13.2.1........ 251 13.2.2................. 253 13.2.3................. 255 13.2.4................. 256 13.3.................... 258 14........................ 264 14.1.................. 264 14.1.1 A 0/0......... 264 14.1.2 A : ± / ±... 267 14.1.3 B1( )..... 268 14.1.4 B2(0 ± ).... 270 14.1.5 C(1 ±, 0 0 0 )............... 271 14.1.6................. 273 14.2.............. 274
4 15........................ 277 15.1.................... 277 15.1.1...... 280 15.1.2.......... 281 15.2.................. 284 15.2.1...... 284 15.2.2.... 286 15.2.3........ 288 15.2.4.......... 289 15.2.5...... 292 16...................... 295 16.1.................... 295 16.2................ 299 16.3................ 303 16.4....................... 307 16.4.1...... 308 16.4.2................. 310 16.4.3 y............... 312 16.5.................... 315 16.6..... 318 16.7................ 321 17............. 323 17.1..... 323 17.2....... 326 17.3....... 330 17.4.................... 331 17.5..................... 333 17.5.1 1................. 334 17.5.2 2............. 334 17.5.3 3............. 336 17.5.4 4................. 337 17.6..................... 337 17.6.1........ 338 17.6.2.......... 340 17.6.3............... 343 17.7................ 346 17.8........................ 347 18....... 349 18.1....................... 349 18.1.1...... 352 18.1.2.... 355 18.1.3.... 357 18.2.................. 358 18.3.................... 363 18.3.1................. 363 18.3.2...... 367 18.3.3................. 369 19....... 374 19.1..... 374 19.2..... 377 19.2.1 sin cos....... 377 19.2.2 tan............. 379 19.2.3 sec............. 380 19.2.4 cot............. 382 19.2.5 csc............. 383 19.2.6............. 383 19.3....... 385 19.3.1 1 a 2 x 2.... 385 19.3.2 2 x 2 + a 2.... 387 19.3.3 3 x 2 a 2.... 388 19.3.4.... 389 19.3.5................. 390 19.3.6
5............... 390 19.4................ 392 20......... 394 20.1.................. 394 20.1.1................. 396 20.1.2........ 398 20.2......... 399 20.3 ( )........... 401 20.4 ( )....... 403 20.4.1...... 403 20.4.2.... 405 20.5 P ( )............. 406 20.6.............. 408 21......... 411 21.1.................... 411 21.1.1............. 411 21.1.2.... 412 21.2.............. 414 21.3... 415 21.3.1.... 416 21.3.2................. 418 21.3.3................. 420 21.3.4....... 423 21.4 0.... 427 21.4.1 0............. 427 21.4.2 0.... 428 21.4.3 0.... 429 21.4.4 0.... 431 21.4.5 0................. 432 21.5 0......................... 433 22....... 435 22.1........... 435 22.1.1.... 436 22.1.2........ 438 22.2........... 439 22.3 n ( )........ 443 22.4......................... 444 22.4.1 ( )... 444 22.4.2 ( )............... 445 22.4.3 p ( )...... 446 22.4.4...... 447 22.5.............. 448 22.5.1 ( )... 448 22.5.2 ( )... 450 22.5.3 ( )... 451 22.5.4 ( )............... 454 23........... 457 23.1......... 457 23.2 n...... 459 23.3......... 460 23.4......... 463 23.5......... 464 23.6 p....... 466 23.7....... 470 24................. 475 24.1......... 475 24.1.1.......... 476 24.1.2............ 476 24.1.3............ 477 24.1.4............ 478 24.2........... 481 24.2.1.......... 482 24.2.2................. 484 24.2.3.... 485
6 24.3................ 488 25........... 490 25.1......................... 490 25.2..... 491 25.3........... 494 25.3.1.......... 495 25.3.2.......... 497 25.3.3.......... 498 25.3.4.......... 499 25.3.5.......... 501 25.3.6................. 502 25.4....... 502 26........................ 505 26.1.............. 505 26.1.1............. 505 26.1.2................. 507 26.2......................... 511 26.2.1...... 512 26.2.2........ 514 26.2.3...... 515 26.2.4................. 517 26.2.5........ 518 26.2.6........ 519 26.3......................... 520 26.4..... 522 27........... 526 27.1.................... 526 27.2....................... 531 27.2.1................. 532 27.2.2.... 534 27.2.3................. 537 27.2.4................. 538 28........................ 541 28.1......................... 541 28.2....................... 544 28.3................ 547 28.4 z n = w................... 548 28.5 e z = w.................... 553 28.6................ 555 28.7........... 557 29......... 559 29.1................ 559 29.1.1............... 560 29.1.2................. 561 29.1.3.......... 563 29.1.4 1 y............. 563 29.1.5 2................. 565 29.1.6 3.......... 567 29.2................ 569 29.3......................... 573 29.4.............. 577 30.................... 581 30.1................ 581 30.2......................... 582 30.3................ 584 30.4.............. 588 30.4.1...... 589 30.4.2...... 589 30.4.3................. 590 30.4.4.... 591 30.4.5............... 592
7 30.4.6........ 593 30.4.7 y P y H................. 596 30.4.8 IVP.................. 596 30.5................ 598 A................. 601 A.1.............. 601 A.1.1............... 601 A.1.2........... 603 A.1.3...... 604 A.2.......... 605 A.2.1.................... 605 A.2.2.................... 606 A.2.3.................... 607 A.2.4.................... 609 A.3.............. 609 A.3.1............. 610 A.3.2...... 611 A.3.3................. 611 A.3.4................. 613 A.4................... 615 A.4.1...... 615 A.4.2...... 617 A.4.3................. 618 A.5..... 619 A.6................... 621 A.6.1......... 622 A.6.2......... 622 A.6.3...... 622 A.6.4......... 623 A.6.5.................. 624 A.6.6...... 624 A.6.7...... 625 A.6.8...... 625 A.6.9......... 626 A.6.10..... 627 A.6.11... 628 A.7.......... 630 B..................... 633 B.1............ 633 B.2..................... 636 B.3................... 638 B.4................... 640 B.4.1...... 641 B.4.2.................. 642............................... 644.................................... 647
1.,,.,.. ; ; ; ;, ;..,,. 1.1.,.,.. f (x) = x 2, f,., R,.,,., f 2 4 1/2 1/4, 1 1.,,. f (2) = 4, f 2 4., f, f (x) x., f (x), f., g (x) = x 2, ( ). f,,., f ( 1/2) = 1/4, g ( 1/2). g. g f, g f, g f.
2 1 f (x) = x 2, f( ),., h (x) = x, h., h ( ) = 4, h ( ) = 6, h ( ) = 0.,., h (2),, 2. 2. h ( ) = 4,,,, h., h ( )., Junkster. Junkster.,,,,. Junkster j (x) = Junkster x, j Junkster.., Junkster, ( ).,,...., ;.,.,.,.. f (x) = x 2, R,.,,.,., 2( 2) 2. g (x) = x 2,, g, R,.,. h (x) x,. 0 2 4 6 8,., 1 3 5 7.,.,., j (x) Junkster x,.,
1.1 3. 1.1.1, R, R ( ).,, {x : 2 x < 5}., 2 5, 2 5.. [a, b] a b, a b. [a, b] a x b x., [2, 5] 2 5 ( 2 5). ( 2 3 4 5, 2 5, 5/2 7 π. ) [a, b].,. (a, b) a b a b., x (a, b), a < x < b. (2, 5) 2 5 2 5. (a, b). [a, b) a b a b ; (a, b] b, a.,.,. {x : 2 x < 5}, [2, 5). (a, ), a a ; [a, ), a. 3.,. 1.1.2,. (, 1.1 g.),,.,
4 1. k (x) = x, R,. [0, ), 0.,., 3. 9 ), (1). (2) (,, ). (3). (, tan(90 ),. tan(90 ) = sin (90 ) cos (90 ) = 1 0,, tan(90 ),. f (x) = log 10 (x + 8) 26 2x, (x 2) (x + 19), f, f (x),. (26 2x),,. 26 2x 0. x 13., (x + 8),,. ( 0, 0.), x + 8 > 0, x > 8., 8 < x 13,, ( 8, 13]. 0, (x 2) 0 (x + 19) 0., x 2 x 19., x ( 8, 13], x 19. 2., 2 ( 8, 13]. ( 8, 13] \ {2},. 1.1.3 F, [ 2, 1],, F (x) = x 2. (,.), x, f (x) = x 2, F f, ( ). 1.1 g, F f.
1.1 5, F 2 1 ( 2 1),,., y. y, y, y. ;, y,. F 1-1. 1-1 1-1 y 0 4( 0 4), [0, 4];, 0 1( 0 1), [0, 1]., [0, 4]. F. 1.1.4,.. 12,,,.,.,, f (x, f (x)),, x f. x. x, (x, f (x)),, x x, f (x). x,., x,,. x., x., x, f (x),,., ;,,.,, 3, 1-2.
6 1 1-2,,., 3 3 ( ),. 3 3,,. x ( 3, 3). x, (x, 0),. f (x).,. x 2 + y 2 = 9,, y = 9 x 2, y = 9 x 2., [ 3, 3].,, (,.), 1-3, [ 3, 3],. 1-3 1.2 f. x; x f,, f (x).,, y, f y y, f x f (x) = y, y f.,, x f (x) = y. f x f (x) = y,., y, f (x) = y. f (x) = x 2 ( R), x 64., x 8 8., g (x) = x 3,, x, 4. g,, ( )., f, f y., x f (x) = y. y
1.2 7,,. y, x f (x) = y. f, f 1.. (1) f, f y, x f (x)=y.,., f 1. (2) f 1 f. (3) f 1 f. (4) f 1 (y) f(x) = y x., f (x) = y, f 1 (y) = x. f 1 f x, f y, y f 1, x. x f (x) = y,?, 3. 1.2.1 f y, x f (x) = y,. f y, x f (x) = y. (0, y), (x, y). x., x,.,,., y,.,,.,., 1-4 f (x) = x 3 g (x) = x 2. 1-4
8 1 y = f (x), f., y = g (x), g. y = x 2 x, y, x = y x = y.. 1.2.2, f y = f (x), x. f (x) = x 3, y = x 3, x = 3 y., f 1 (y) = 3 y y, x,, f 1 (x) = 3 x., x,,.,,. y = x,.. f (x) = x 3, f 1-5 1 1-5. f y = x,. f f 1. 1.2.3,,, y x. x,., x,. 1.1,.,,. g (x) = x 2, 1-6, 1-6. ( ) [0, ),,., [0, ) h, h (x) = x 2.., y = x 2 x.,
1.2 9 x = y x = y,, [0, ),, x = y., h 1 (y) = y.,, (, 0]., (, 0] j, j (x) = x 2,.,, 1-7 j 1 (y) = y, 1-7. 1-8, g g (x) = x 2, (, ),, y = x, 1-8.,,., y = x. 1.2.4 f, f x, f 1 (f (x)) = x ;, f y, f ( f 1 (y) ) = y. (, f f 1, f 1 (y) f y,. ) f (x) = x 3, f f 1 (x) = 3 x, x, f 1 (f (x)) = 3 x 3 = x.,. x f, f 1 ; x., f ( f 1 (y) ) = ( 3 y ) 3., f 1 f, f f 1.,.,. g (x) = x 2,,. [0, ),, g h,. g 1 (x) = x. g ( g 1 (x) ), ( x) 2, x 0, x. (,. ), g 1 (g (x)), x 2, x., x = 2, x 2 = 4, x 2 = 4 = 2.,, g 1 (g (x)) = x.
10 1 2 g.,, g ( 2), 2 g. h, g,.,,,. f, f f 1, f y, f ( f 1 (y) ) = y; f 1 (f (x)) x;, f 1 (f (x)) = x x. 10.2.6,,. 1.3 g(x) = x 2 g, x, g(y) = y 2 g(x + 5) = (x + 5) 2., g(x + 5) = x + 5 2, x + 25 (x + 5) 2.,., f(x) f( ), x ( ),., h(x) = 3 x, h(x 2 + 6) = 3 x2 +6, x 2 + 6. f(x) = cos(x 2 ) f. x, f(x) x,, f(x),, g(x) = x 2, h(x) = cos(x). f x, x g, g g h, h, g x. f, f(x) = h(g(x)), f = h g,, f g h., f g h, h g ( ), g.,.., g(x) = 2 x, h(x) = 5x 4, j(x) = 2x 1, f = g h j j, h, g, f(x) = g(h(j(x))) = g(h(2x 1)) = g(5(2x 1) 4 ) = 2 5(2x 1)4.,.,
1.3 11 f(x) = 1 tan(5 log 2 (x + 3)). f x, 3, g(x) = x + 3; 2, h(x) = log 2 (x); 5, j(x) = 5x;, k(x) = tan(x);, m(x) = 1/x.,, f(x) = m(k(j(h(g(x))))). f = m k j h g. f., h j n, n(x) = 5 log 2 (x), n = j h f = m k n g. ( j h), f, ( n), n(x) = 5 log 2 (x) x.,. f(x) = x 2 sin(x), f, x, f(x) x 2 sin(x)(, ),. g(x) = x 2, h(x) = sin(x), f(x) = g(x)h(x) f = gh. j = g h( ) j(x) = g(h(x)) = g(sin(x)) = (sin(x)) 2 j(x) = sin 2 (x). j x 2 sin(x), k = h g, k g h, k(x) = h(g(x)) = h(x 2 ) = sin(x 2 ). k.,,,. f g(x) = x a(a ),. h(x) = f(x a), y = h(x) y = f(x), y = h(x) a. a, (, 3 3 )., y = (x 1) 2 y = x 2, x 1 x. y = x 2 1, 1-9. 1-9
12 1, y = (x+2) 2 y = x 2 2, (x+2) (x ( 2)). 1.4,. f(x) = x 2 f, ( 3) f( 9)., 3, f( 9).,,. x, f( x) = f(x)., x f, x. g(x) = x 4 h(x) = x 6., n (n ), j(x) = x n., f x f( x) = f(x), f. x, x., f(x)=x 3. ( 3) f( 27),, 3, 27,. f( x) = f(x)., n (n ), j(x) = x n., f x f( x) = f(x), f.,,,.,.,, x f(x) = 0( ).. f, x f( x) = f(x);, x f( x) = f(x),, 0 = 2f(x), f(x) = 0, x, f.,, 0, f(0) = 0. x, f f( x) = f(x). 0, f( 0) = f(0), 0 0, f(0) = f(0), 2f(0) = 0, f(0) = 0., f,.,.,,. f. f(x) = f( x), y = f(x) x x, x, 1-10. y.,
1.4 13, y. y = x 2., f. f( x) = f(x), y = f(x) x x. (, f(x),.) 1-11. 1-10 1-11, 180.,.,, ( 0,,. (,, )., y = x 3. f f(x) = log 5 (2x 6 6x 2 + 3), f x ( x) f( x), x,. f(x), f ; f( x), f ; f(x) f( x), f ( )., f( x) = log 5 (2( x) 6 6( x) 2 + 3) = log 5 (2x 6 6x 2 + 3), f(x), f. g(x) = 2x3 + x 3x 2 + 5 h(x) = 2x3 + x 1 3x 2? + 5 g,, g( x) = 2( x)3 + ( x) 3( x) 2 + 5 g( x) = 2x3 + x 3x 2 + 5, = 2x3 x 3x 2 + 5.
14 1 g(x),,, g. h, h( x) = 2( x)3 + ( x) 1 3( x) 2 + 5 = 2x3 x 1 3x 2. + 5 h( x) = 2x3 + x + 1 3x 2. + 5,, +1,, h..,, f g,, h, h(x) = f(x)g(x), h., h( x) = h(x). f g, f( x) = f(x), g( x) = g(x). h( x), h f g, h( x) = f( x)g( x), f g ( f(x))( g(x)),, f(x)g(x), h(x). ( ) h( x) = f( x)g( x) = ( f(x))( g(x)) = f(x)g(x) = h(x)., h( x) = h(x) h.,. 1.5 f(x) = mx + b. 1-12. m.,,,. 1-12., m,. m,., m,. m ( ),. 0,,,,.,,.,., y. x = 0, y = m 0 + b = b., y b,
1.5 15 (0, b). x, y 0, x.,. b = 0, y = mx., x y., x = 1, y = m., y = mx (1, m)., y = 2x (1, 2), 1-13. m = 0, y = b, (0, b). 1-13 1-14, y = 1 2 x 1., y 1, 1/2., x, y = 0 0= 1 x 1, x=2. 1-14. 2,.,.,. --, (x 0, y 0 ), m, y y 0 = m(x x 0 )., ( 2, 5), 3, y 5= 3(x ( 2)), y = 3x 1.,.,. (x 1, y 1 ) (x 2, y 2 ), y 2 y 1 x 2 x 1. ( 3, 4) (2, 6)., = 6 4 2 ( 3) = 10 = 2. 5 ( 3, 4) 2, y 4 = 2(x ( 3)), y = 2x 2., (2, 6) 2, y ( 6) = 2(x 2), y = 2x 2.,,.
16 1 1.6. (1) x. 1 x x 2 x 3,,., f(x) = 5x 4 4x 3 + 10 x 4 5 x 3 4 10. x 2 x,, x 2 x. x n x n., x 4 x 3 x 2 x 5 4 0 0 10. (, x I?,, x = x 1, 1 = x 0.) n( ). 4, 4 x. n p(x) = a n x n + a n 1 x n 1 + + a 2 x 2 + a 1 x + a 0, a n x n, a n 1 x n 1,, a 0 1. x n,.,. 1-15 x 0 x 7. 1-15., x..,. a n.,
1.6 17 5x 4 4x 3 + 10, 5.,. 4, 1-16. 1-16.. 5x 4 4x 3 + 10, n = 4, a n = 5. 2,. p(x) = a 2 x 2 + a 1 x + a 0, p(x) = ax 2 + bx + c.. = b 2 4ac.. > 0, ; = 0,, ; < 0,. b ± b 2 4ac. 2a... 2x 2 3x + 10. 2 (x 2 32 ) x+5. 1., x 3 2, 2 3 4, 9 16. 9 5, 16 x 2 3 2 x + 5 = x2 3 2 x + 9 16 + 5 9 16. 9 16, 9 (x 16, 3 ) 2. 4, x 2 3 ( 2 x + 5 = x 2 3 2 x + 9 ) + 5 9 ( 16 16 = x 3 ) 2 + 5 9 4 16., 5 9 16 = 71. 2, 16
18 1 2x 2 3x + 10 = 2 (x 2 32 ) x + 5 = 2 ( ( x 3 4 ) ) 2 + 71 ( = 2 x 3 ) 2 + 71 16 4 8.,., 18 19. (2) p(x), p q,. q(x), p q., q(x) 1. 1/x n, n. 1-17. 1-17,.. (3)., y = 2 x. y = b x (b > 1).., ;, y 1 ;, x., x, x,. ( 3,.) y = 2 x y = 2 x y, 1-18. 1-18 1,, y = ( ) x 1. 2 ( ) x 1 = 2
1.6 19 ( ) x 1 1/2 x = 2 x, x, 2 x, 1-18 y = 2 x ( ) 2 x 1 y =. y = b x (0 < b < 1). 2 y = 2 x,. 2 y = log 2 (x). y = x, y = log 2 (x) 1-19., 0. (0, + ),, y. log b (x) (b > 1)., 1-19. 9. (4),. (5) f(x) = x. { x x 0, x = x x < 0. 0 x., x y x y., x 1 3. x 1 3., 1 3. 1, 1-20 1-20 3 2 4, 1-21 1-21 x 1 3 [ 2, 4].
20 1 x = x 2., x 0, x 2 = x; x < 0, x2 = x,,. x 2 = x,,. x, x = x 2., x,..,, x, x, x., x, y = x x, y = x 1-22. 1-22 y = log 2 (x), 1-23. 1-23,..,.
2.,,,.,. ; ( 0 90 ); ;. 2.1., 2π 360.,, 1 2π.,, 2-1. 2-1,.,, 90 π/2., 180 π, 270 3π/2., 2-2
22 2 2-2,, = π 180., 5π/12, 5π 12 = π 180 5π/12 (180/π) (5π/12) = 75., 2-3, sin (θ) =,,. π 180.,,.,.,, θ, 2-3. cos (θ) =, θ,, 2-4., θ, θ.,,.,, tan (θ) =. 2-4 csc(x) = 1 sin(x), sec(x) = 1 cos(x) cot(x) = 1 tan(x).
2.2 23 ( ), 0, π/6, π/4, π/3, π/2.,, sin (π/3) tan (π/4),,,.,,. 0 sin 0 cos 1 tan 0 π 6 1 2 3 π 4 2 1 1 3 π 3 1 3 2 2 1 1 2 tan (π/2)., π/2 (, 2.3 ).,,. (1) sin (π/3) (, 3/2. ) (2) 0 π/2, 3/2 (, π/3. ),.,,,.,,,,,. 2 3 π 2 1 0 2.2 ( ) 0 π/2.,,.,., tan (π/2).,,. 0 2π(, 2π 360 )., sin (θ)( cos (θ) tan (θ)),, θ 0 π/2.,, 2-5. 2-5,
24 2, 1 4( ),.,,. ( ). θ.,, x. θ,,.., 2-2., π/2, y. π, x. 3π/2, y., 2π,, x. 0 2π., 0 2π., θ., 2-6., θ,.,, 2-6 x. x y ( x y ),, r., x y (, 2-6 )., r,., ( ), x y, r = x 2 + y 2. (, 2-7.) 2-7, sin(θ) = y r, cos(θ) = x r tan(θ) = y x. x, y r,, 2.1.,,,,., r = 1, (x, y) (, 1 ).
2.2 25,., sin (7π/6). 7π/6 7π/6 0, π/2, π, 3π/2, 2π., 7/6 1 3/2, 7π/6 π 3π/2., 2-8., 7π/6., r = 1, x., sin (θ) = y/r = y ( r = 1),, y.,, 7π/6 x ( π), π/6.., θ 2-8 θ x, 0 π., x 2, 2-9. 2-9,, r = 1, π/6. y = sin (π/6) = 1/2, x, y., y = 1/2. sin (θ) = y, sin (7π/6) = 1/2.,, x = cos (π/6) = 3/2., (x, y) y, x. cos (7π/6) = 3/2, (x, y) ( 3/2, 1/2 ). 2.2.1 ASTC sin (7π/6) sin (π/6),, π/6 7π/6.,, x, y r.,, 7π/6, π/6; sin (7π/6) sin (π/6) sin (π/6),., y.,, y.,, x.
26 2 y/x, ( x y, ),.., 2-10 (I). (II), ;. (III), ;., (IV), ;. 2-10., ASTC.. A,.,,., 7π/6,.,, sin (7π/6) 1/2 1/2,,. sin (7π/6) = 1/2. ASTC 0, π/2, π 3π/2,., ASTC, y = sin (x)( cos (x), tan (x)),. 2.3., ASTC, 0 2π (1),,,. (2) x y ( ),,, (2.3 ). (3), x ;. (4),.,. (5) ASTC.. cos (7π/4) tan (9π/13). cos (7π/4), 7/4 3/2 2,, 2-11. 2-11
2.2 27, 2π( 0. ),, 2π 7π/4, (2π 7π/4), π/4., cos (7π/4) cos (π/4), cos (π/4) 1/ 2. ASTC,, cos (7π/4) = 1/ 2. tan (9π/13). 9/13 1/2 1, 9π/13, 2-12., π x, π 9π/13, π 9π/13, 4π/13., tan (9π/13) tan (4π/13)., 4π/13, tan (4π/13).., ASTC, 2-12,, tan (9π/13) = tan (4π/13)..,,., tan (4π/13),., tan(4π/13) = 0.768 438 861,., tan (4π/13),,.,, ( ) tan(4π/13) = 0.768.,.,,. 2.2.2 [0, 2π], 2π 0.,, 2π, 0 2π., 2π..,, 450,,, 90,.., 90,,., 450 90,., 5π/2 π/2. 9π/2 2π (
28 2 4π), π/2., π/2,., 9π/2 π/2., π/2 π 2, 5π 2, 9π 2, 13π 2, 17π 2,., 2π..,,..,, 270, 270 ( 3π/2).,,, 270 90., 360 270 90.,, 3π/2 π/2., ( )., π/2, 15π 2, 11π 2, 7π 2, 3π 2, π 2, 5π 2, 9π 2, 13π 2, 17π 2,. ;,,. {π/2 + 2πn}, n,.. sec (15π/4), cos (15π/4), sec (15π/4)., cos (15π/4). 15/4 2, 2., 15/4 2 = 7/4, 0 2,. π, cos (15π/4) cos (7π/4), 1/ 2., cos (15π/4) = 1/ 2., sec (15π/4) 2. 2-13, sin ( 5π/6), 2π 5π/6 0 2π., 2π 5π/6 7π/6,, sin ( 5π/6) = sin (7π/6), 1/2., 2-13.,, π/6,.
2.3 29 2.3.,,., y = sin (x). 0 2π 2-14. 2-14, 0, π/2, π, 3π/2 2π. sin (x) 2π ( sin (x) x, 2π),,, 2-15. 2-15, sin (3π/2) = 1 sin ( π) = 0., π/2 ;., 180,, sin (x) x.( 1.4.) y = cos (x) y = sin (x). x 0 2π, 2-16. 2-16, cos (x) 2π,
30 2, 2-17. 2-17 2-18, cos (π),, 1.,, y., cos (x) x., y = tan (x)., x π/2 π/2, 2-18.,., π, 2π., y = tan (x), 2-19. 2-19, x π/2, y = tan (x) ( ).,, tan (x) x. y = sec (x), y = csc (x) y = cot (x), 2-20 2-21 2-22.
2.3 31 2-20 2-21 2-22
32 2,,. sin (x), tan (x), cot (x) csc (x) x. cos (x) sin (x) x., x, sin ( x) = sin (x), tan ( x) = tan (x) cos ( x) = cos (x). 2.4.,, tan(x) = sin(x) cos(x), cos(x) cos(x) = sin(x). (,,,,.) ( ), cos 2 (x) + sin 2 (x) = 1. x. ( 1, x, cos (x) sin (x).), cos 2 (x). 1 + tan 2 (x) = sec 2 (x).. sin 2 (x),, cot 2 (x) + 1 = csc 2 (x)..., co.. π/2( 90 )..,, ( π ) (x) = 2 x., sin(x) = cos ( π ) ( π ) 2 x, tan(x) = cot 2 x ( π ) sec(x) = csc 2 x.
2.4 33, ;,, co co sin sin, co co tan tan. ( π ) ( π ) cos(x) = sin 2 x, cot(x) = tan 2 x ( π ) csc(x) = sec 2 x.,.., sin(a + B) = sin(a) cos(b) + cos(a) sin(b) cos(a + B) = cos(a) cos(b) sin(a) sin(b).,, sin(a B) = sin(a) cos(b) cos(a) sin(b) cos(a B) = cos(a) cos(b) + sin(a) sin(b). sin (A + B) cos (A + B), A = B = x,., sin (2x) = 2 sin (x) cos (x).,. cos (2x) = cos 2 (x) sin 2 (x);, sin 2 (x) + cos 2 (x) = 1 cos (2x) 2 cos 2 (x) 1 1 2 sin 2 (x)( )., sin(2x) = 2 sin(x) cos(x) cos(2x) = 2 cos 2 (x) 1 = 1 2 sin 2 (x)., sin (x) cos (x) sin (4x), 4x 2x,, sin (4x) = 2 sin (2x) cos (2x).,, sin(4x) = 2(2 sin(x) cos(x))(2 cos 2 (x) 1) = 8 sin(x) cos 3 (x) 4 sin(x) cos(x)., cos(4x) = 2 cos 2 (2x) 1 = 2(2 cos 2 (x) 1) 2 1 = 8 cos 4 (x) 8 cos 2 (x) + 1. ;,.,,.,.,.
3,.,.,,.., ; A., ;, ; ; ( ). 3.1. f x, α. x α, α, f (x),.. f R\ {2}( 2 ), f (x) = x 1. f(x) = x 1 x 2. 3-1., 2, f ( 4.1 )., f, 3-1., f (2) f (2) = 1,, 2 f. f (2)., x 2, f (x),., f (2.01) = 1.01, f (1.999) = 0.999., x 2, f (x) 1.
3.1 35, x 2, 1, 1., f (x) 1 ± 0.000 1, 1.999 9 2.000 1 x ( x = 2, ). f (x) 1 ± 0.000 007, x,., 1.999 993 2.000 007 ( 2)., A A.1., f(x) = 1. x 2, x 2, f (x) 1.,, x 2( 2), f (x) 1.. f(x) 1 x 2., x 2, f (x) 1( )., f., g, 3-2. g,, g (x) { x 1 x 2, g(x) = 3 x = 2 3-2 g (x) g(2) x 2 x 2 g(x), 2,. x = 2, g f.,, g(2) = 3, g (x) = 1. x 2,, f(x) = 1, x 2 x, x 2, f(x) 1., x,., q 2, f(q) 1, f(q) = 1. q 2 f(b) = 1, f(z) = 1, f(α) = 1, b 2 z 2 α 2,!,
36 3 f(x) = 1, x 2, x., ( ) 2., ;,,.,. 3.2,., h (x) 3-3 x = 3, 3-3.,, h (3) = 2., x = 3,,. h (x), x,.,,, 3, 1., x = 3( ),. x = 3, x = 3,., h (x) x = 3 1.,,, x = 3, 2., h (x) x = 3 2. x = 3 ( x = 3 ). x 3 x 3 h(x) = 1 h(x) = 2. + 3, 3. 3,,, h(x), x 3, h (x) x = 3, h (x) x = 3.., x 3 3 x., 3.,, x 3 +, 3.
3.3 37,. x = a, x = a, (, )., f(x) = L f(x) = L x a x a + f(x) = L. x a., h,,. h(x) x 3 DNE. 3.3,.. f (x)=1/x f (x) x 0.,, f (x). 3-4, x x 0 + 0, f (x)., x 0, ;., 3-5 3-4 1 x 0 + x =.,, x 0, f (x). 1 x 0 x =.,., g, g (x) = 1/x 2. 3-5. x = 0, x 0 1/x 2 =.
38 3, f x = a, f(x) f(x), x a + x a.,, g, g (x) = sin (1/x)., x. sin (x) x = π, 2π, 3π, 0,, sin (1/x) 1/x = π, 2π, 3π, 0., sin (1/x) x = 1 π, 1 2π, 1, 0. sin (1/x) x 3π., 3-6. 3-6, 0,., x, sin (x) 1 1,, sin (1/x)., 3-7 3-7, x 0 + sin (1/x) x = 0. 1 1, x = 0,.,,. x x = 0,.,, sin (1/x) (DNE). y = sin (1/x) x 0 +. 3.4. x = a A A.3.4.
3.4 39.,, x,.,, x. f(x) = L x, x, f (x) L,. ( A A.3.3. ), f (x) = L x f y = L., x, f(x) = L, x x (, x ), f (x) L, L. y = f (x).,, f y = L f (x) = L. x f y = M f (x) = M. x, y = x 2, x, y., x x2 =.., sin (x), sin (x) ( x ) 1 1,,., ; sin (x) (DNE). A A.3.4. x f, f (x) = sin (1/x). x, x, 1/x 0. sin (0) = 0, sin (1/x) 0. x, sin (1/x) 0., sin(1/x) = 0. x, sin (1/x) y = 0. y = sin (1/x),. x < 0., f. ( ) ( 1 f( x) = sin = sin 1 ) ( ) 1 = sin = f(x). x x x,, sin (x) x, sin ( 1/x) sin (1/x)., A A.4.1!
40 3,, ( 1.4 ), y = sin (1/x), 3-8 3-8, x 0. x 0,,, x = 0.,,. 1 000 000 000 000., 1 000 000 000 000,,., 0.000 000 001,, 0.000 000 001 (, )., 0.,,. 0( 0),.,. 0, f(x) = L. x, x, f (x) L., f (x) L,, y = f (x) y = L, x., f, 3-9 3-9, f (10) L., x 100, f (x)
3.5 41 L,, 100., f (100) L,, 100., 200., 1 000 000 000 000,,,,, 5 000 000 000 000.,., 1 000 000 000 000, 10 100 ( ), 10 100 10 1 000 000.,. (,,., x.),, x.,., f(x) = L, f(x) = L f(x) = L. x 0 x 0 + x 0,, x 0, f (x) L. (, x,, x. ), x 0 f., ( 0 ),,.,., A A.1 A.3.3. 3.5,.,. 3.3 f (x) = 1/x, y = 0. 1 x x = 0 1 x x = 0., 3-10 y = tan 1 (x) ( y = arctan (x), ) 3-10
42 3 y = π/2, y = π/2,. x tan 1 (x) = π 2 x tan 1 (x) = π 2,, (, ).,., y = 2 x, ( 1.6 ). (, y = tan (x) )..,,.,., f (x) = sin (x) /x f,, x. sin (x) 1 1,, sin (x) /x y = 1/x y = 1/x., sin (x) /x sin (x), π, 2π, 3π,., 3-11. 3-11 y = 1/x y = 1/x.,,, sin(x) = 0. x x, y = f (x), x f
3.6 43.,,.. 3.6,, f g h, x a, g h L,, x a, f L.., a x, g (x) f (x) h (x). f (x) ( )g (x) h (x)., g (x) = L h (x) = L., f (x) = L; x a x a x a x a,., 3-12. 3-12 f g h ; x a, f (x) L. ( A A.2.4. ), g (x) f (x) h (x) a,.,? ( ) 1 x sin? x 0 + x y = x sin (1/x) y = sin (1/x),, x y =x y = x. 3-13 x 0 0.3. 3-13
44 3, x 0,,.,. g y = x, h y = x. x > 0, g (x) f (x) h (x). f (x) x = 0, x < 0. (, x,, x < 0, g (x) h (x),, ), x > 0, g(x) f(x) h(x)? ( 1/x) 1 1 1 sin ( ) 1 1. x x,, x > 0, ( ) 1 x x sin x. x g (x) f (x) h (x)., g(x) = x 0 + x 0 +( x) = 0 h(x) = x = 0. x 0 + x 0 +, x 0 +, g (x) h (x) 0, f (x)., ( ) 1 x sin = 0. x 0 + x, x, ; 3.3, x 0 +, sin (1/x). sin(x) = 0. x x,,., x, g (x) f (x) h (x) ;, g (x) = L h (x) = L., f (x) = L. x x x.,, x, 1 sin (x) 1,, x > 0, x 1 x sin(x) 1 x x, x,, 1/x 1/x 0, sin (x) /x 0., 1 x x = 0 1 x x = 0,
3.7 45 sin(x) = 0. x x, a x g(x) f(x) h(x), g(x) = h(x) = L, f(x) = L. x a x a x a ;, a x. a ;, ( )x,. 3.7.,. (1) x = a. x = a x = a f (x). (, x a, f (x),., x a, f (x). ) 3-14. 3-14 (2) x = a. x = a x = a f (x). 3-15. 3-15
46 3 (3) x = a.,,,.,,,. f (a). 3-16. 3-16 (4) x. 3-17. 3-17 (5) x. 3-18. 3-18
4,.,., ;,,.,.,.,., x x a( a)., x a ; x a ; x ; x ( ) ; x / ;. 4.1 x a x a p(x) q(x),, p q, a. (, p (x) /q (x).) a x. 0,,., x 2 3x + 2 x 1 x 2? x = 1 ( x 2 3x + 2 ) / (x 2), ( 1) 2 3( 1) + 2 = 6 1 2 3 = 2.,, 2. (,,, x = 1, ;,,
48 4,.), x 2 3x + 2, x 2 x 2, x = 2 (4 6 + 2) / (2 2), 0/0..,,,,,.., x 2 3x + 2 (x 2) (x 1),, x 2 3x + 2 x 2 x 2 (x 2)(x 1) = x 2 x 2 = x 2 (x 1), x = 2 (x 1) ; 2 1, 1.. f g, f(x) = x2 3x + 2 x 2 g(x) = x 1 f(x) = x2 3x + 2 (x 2)(x 1) = = x 1 = g(x)? x 2 x 2, x = 2,, x 2 0,., f g 2 f, g. (, f, 3. ),,., f (x) g (x) x = 2, x = 2 f (x) g (x).,..,., a 3 b 3 = (a b)(a 2 + ab + b 2 )., x 3 27 x 3 x 4 5x 3 + 6x 2. x = 3, 0/0( ).,. x 3 3 3,,. x 2, x 2 ( x 2 5x + 6 ). x 2 5x + 6
4.1 x a 49 ;,, x 3 x 3 27 x 4 5x 3 + 6x 2 = x 3 (x 3)(x 2 + 3x + 9) x 2. (x 3)(x 2) x = 3, (x 3).,, x 3 ;, (x 3)( 0),.,, x 3 x 3 27 x 4 5x 3 + 6x 2 = (x 3)(x 2 + 3x + 9) x 2 + 3x + 9 x 3 x 2 = (x 3)(x 2) x 3 x 2 (x 2) = 32 + 3 3 + 9 3 2 (3 2) 0 0, ;, x.,., f, 4-1 x = a = 3. 4-1, f (x) x = a.,,,., 2x 2 x 6 x 1 x(x 1) 3?, x = 1 5/0( ).,. f (x) = ( 2x 2 x 6 ) ( / x (x 1) 3), x 1., x = 1, ( 2x 2 x 6 ) 5,, a x,. x = 1, 1,.
50 4, a x,. (x 1) 3, x > 1, x < 1., ( (+) ( ),, ( ) ( ) = (+),. ) ( ) x > 1 : (+) (+) = ( ); x < 1 : ( ) (+) ( ) = (+)., x 1, f (x), x 1, f (x). ( )., 2x 2 x 6 x 1 x(x 1) 3, ( );, 2x 2 x 6 2x x 1 + x(x 1) 3 = 2 x 6 x 1 x(x 1) 3 =.,, 2x 2 x 6 x 1 x(x 1) 2. x 1,, x,, (x 1) 2, x 1 1,., ( ) x > 1 : (+) (+) = ( ); x < 1 : ( ) (+) (+) = ( )., x = 1, 2x 2 x 6 x 1 x(x 1) 2 =.,. 4.2 x a x2 9 4. x 5 x 5 x = 5, 0/0 ( ). x 2 9 (x 3) (x + 3),, 4., x 2 9 + 4 ; x 2 9 4. (,,., a b a + b,. ),
4.3 x 51 x2 9 4 x2 9 4 x2 9 + 4 = x 5 x 5 x 5 x 5 x2 9 + 4., (a b) (a + b) = a 2 b 2, ( x 2 9 ) 2 4 2, x 2 25., x 2 25 x 5 (x 5)( x 2 9 + 4). x 2 25 (x 5) (x + 5), (x 5)(x + 5) x 5 (x 5)( x 2 9 + 4) = x + 5 x 5 x2 9 + 4., x = 5, 10/8, 5/4.,,, 4.3 x,, x., p(x) x q(x), p q., x,., p,, x, p (x)., p (x) = 3x 3 1 000x 2 + 5x 7. p L (x) = 3x 3, p. x, p (x) p L (x)., p(x) x p L (x) = 1.,., p(x) p L (x) = 1, p (x) = p L (x). ( x ), x,., p(x) = p L(x)? x x,,.,, 1 p (x) p L (x). x, 1, 1.
52 4, ;,,, x, p (x) = 3x 3 1 000x 2 + 5x 7. x = 100., 3x 3 3, 1 000x 2 1. 5x 500, 7., p (100) 7., p L (100) 3, p (100) p L (100).., 100., x 1 000 000., 3x 3 3 000 000 000 000 000 000,, 1000x 2, ( 1 000 000 000 000 000), 5x 5,. 7., p (1 000 000), 3 ( 5 )., 3 3,,, 3., p (1 000 000) 3, p L (1 000 000)?, x,., 1 000 000, x 3 x 2 x.. p(x) x p L (x) = 1.., x p(x) p L (x) = x 3x 3 1000x 2 + 5x 7 3x 3 ( 3x 3 x 3x 3 1000x2 3x 3 + 5x 3x 3 7 ) ( 3x 3 = 1 1000 x 3x + 5 3x 2 7 ) 3x 3.,.,, x, 1, 1 000/3x, 5/3x 2 7/3x 3,,., ;,.,. 1,,, (x + (1 x)). x, (x+ x (1 x)) = 1,, 1., (x) (1 x) (x) x (1 x).,, + ( ) = 1., x + ( ).
4.3 x 53 x, 1. 1 000/3x. x, 1000 x 3x? 1 000/3., 1000 1 x 3 x., 1000/3. x,., ( A A.2.2 ). 1000 1 x 3 x = 1000 1 3 x x., (, )., 1/x = 0,, 1000/3 0. x 1000 x 3x = 0.,., n > 0, C, C x x n = 0, x, 5/3x 2 7/3x 3 0., 3x 3 1000x 2 ( + 5x 7 x 3x 3 = 1 1000 x 3x + 5 3x 2 7 ) 3x 3 = 1 0 + 0 + 0 = 1. x p(x) p(x) = 1 p (x) = 3x 3 1000x 2 + 5x 7.,,,. p, p (x), p(x) p(x) (p(x) ).
54 4,,, p (x)., x, 1,., x 8x 4 x 7x 4 + 5x 3 + 2000x 2 6?,., 8x 4 (, ), x 8x 4 8x 4 ( 8x 4 )., 7x 4,, 7x 4 + 5x 3 + 2000x 2 6 7x 4 (7x 4 )., x x 8x 4 7x 4 + 5x 3 + 2000x 2 6 = x, x 8x 4 8x 4 ( 8x 4 ) 7x 4 + 5x 3 + 2000x 2. 6 7x 4 (7x 4 ) 8x 4 7x 4,,. 1,,., 8/7,., 1,.,,, 1 8x 3 + 1 8x 4 x 1 + 5 7x + 2000 7x 2 6 7x 4. 7x 4 ;, x, C/x n 0( C, n > 0)., x 4, 0 + 1 1 + 0 + 0 0 8 7 = 1 1 8 7 = 8 7. x (x 4 + 3x 99)(2 x 5 ) (18x 7 + 9x 6 3x 2 1)(x + 1).
4.3 x 55, x 4, x 5, 18x 7 x.,,., (x 4 + 3x 99)(2 x 5 ) x (18x 7 + 9x 6 3x 2 1)(x + 1) ( x 4 ) ( ) + 3x 99 2 x x = 4 (x 4 5 ) x 5 ( x 5 ) ( x 18x 7 + 9x 6 3x 2 ) ( ) 1 x + 1 18x 7 (18x 7 ) (x) ( x 1 + 3 x 3 99 ) ( 2x ) x 4 5 + 1 = x = ( 1 + 9 18x 3 18x 5 1 18x 7 (1 + 0 0)(0 + 1) (1 + 0 0 0)(1 + 0) x, (x 4 )( x 5 ) (18x 7 )(x), ) ( 1 + 1 ) (x4 )( x 5 ) (18x 7 )(x) x x 18 = x x 18 =. x/18., x, x/18,,., ( 8/7) ( ).., 4., 4 5,,, 9., 7 1,, 8.,., 2x + 3 x x 2 7. 2x + 3 x x 2 7 = x 2x + 3 (2x) 1 + 3 2x x 2 = 2x 7 x 2 (x 2 x ) 1 7 2x x 2 x 2 = 1 + 0 1 0 2 x x = 0., 2, ( 1).,,, 0., x p(x) q(x)
56 4 p q, (1) p q,. (2) p q,. (3) p q, 0. ( x,, p(x) x q(x) ; 4.5. ),., ;,,. 4.4 x f, g h, 3 f(x) = x 3 + 4x 2 5x 2/3 + 1, g(x) = x 9 7x 2 + 2, h(x) = x 4 x 3 + 5 x 2 2x + 3., n,.,.,.,,. (, )., x 16x4 + 8 + 3x 2x 2 + 6x + 1. 2x 2,, 2x 2 + 6x + 1 2x 2 (2x 2 ). 16x 4 + 8, 16x 4., 4x 2., 4x 2 +3x. 4x 2,., 16x4 + 8 + 3x 4x 2 (4x 2 ). 4x 2, 16x 4 16x4 + 8 + 3x 16x4 + 8 4x 2 = 4x 2 + 3x 16x 4 4x 2 = + 8 16x 4,, + 3x 4x 2.
4.4 x 57 1 + 8 16x 4 + 3 4x. x, x,, 1 + 0 + 0 = 1., 16x4 + 8 + 3x 16x4 + 8 + 3x x 2x 2 + 6x + 1 = 4x 2 (4x 2 ) x 2x 2 + 6x + 1 2x 2 (2x 2 ) 16x 4 +8 = 16x 4 + 3x 1+ 8 4x 2 x 2x 2 4x2 +6x+1 2x 2 = 16x 4 + 3 4x 4 x 1+ 6 2x 2 2x + 1 2 2x 2 1 + 0 + 0 = 1 + 0 + 0 2 = 2.,,.,. 16x4 + 8 + 3x 3 x 2x 2. + 6x + 1, 3x 3x 3., x, 16x 4 + 8 4x 2,, 3x 3., 16x4 + 8 + 3x 3 3x 3 (3x 3 );, 3x 3, 9x 9.,, x 16x4 + 8 + 3x 3 2x 2 + 6x + 1 16x4 + 8 + 3x 3 3x 3 (3x 3 ) 2x 2 + 6x + 1 2x 2 (2x 2 ) 16x 4 +8 9x 6 + 3x3 3x 3 2x 2 3x3 +6x+1 2x 2 = x 2x 2 0 + 0 + 1 = x = x 3x 2 =. 16 x 2 + 8 9x 6 +1 1+ 6 2x + 1 2x 2 = 1 + 0 + 0 x., 16x 4 ;, 4x 2 (3x)., (3x 3 )., 3x 2
58 4, 4x6 5x 5 2x 3 x 3? 27x6 + 8x,,., 4x 6 5x 5, x, 4x 6., 4x 6 5x 5 4x 6, 2x 3 ( x )., 2x 3,, 4.2., 4x6 5x 5 2x 3 4x6 5x x 3 = 5 2x 3 4x6 5x 27x6 + 8x x 3 5 2x 3 27x6 + 8x 4x6 5x 5 + 2x. 3, (a b) (a + b) = a 2 b 2 x x (4x 6 5x 5 ) (2x 3 ) 2 3 27x6 + 8x( 4x 6 5x 5 + 2x 3 )., 5x 5 3 27x6 + 8x( 4x 6 5x 5 + 2x 3 ). ;. 3 27x6 + 8x,, 27x 6, 3 27x6 + 8x 3 3 27x 6, 27x 6 3 27x6 + 8x 3 27x 6 (3x 2 ).,,, 3 27x 6 + 8x 27x 6 (3x 2 ) = 3 1 + 8 27x 5 (3x2 )., x, 1. 4x 6 5x 5 + 2x 3,., 4x 6 5x 5, 4x 6. 2x 3. 2x 3,, 2x 3 + 2x 3 4x 3.. 4x6 5x 5 + 2x 3 4x 3 (4x 3 ),, 4x 3, 16x 6 ; ( 4x 6 5x 5 16x 6 + 2x3 4x 3 ) (4x 3 ) = ( 1 4 5 16x + 1 2 ) (4x 3 ).
4.5 x 59, x, 1 4 + 0 + 1 2 = 1 2 + 1 2 = 1, ( 1 4 1 2.),., x 5x 5 3 27x6 + 8x( 4x 6 5x 5 + 2x 3 ).,, 5x ( ) 5 x 3 ( ). 27x6 + 8x 4x6 3 (3x 2 5x ) 5 + 2x 3 27x 6 4x 3 (4x 3 ) 5x 5, 3x 2 4x 3, 1 ( ) x 3 ( ) 5x5 27x6 + 8x 4x6 5x 3 5 2x 3 (3x 2 )(4x 3 ). 27x 6 4x 3, x 5,, 5/12.,. 4.5 x, x p(x) q(x),., p q.. x,., x, C, n, C/x n 0. ( ),., 4.3.1, x x 8x 4 7x 4 + 5x 3 + 2000x 2 6 x (x 4 + 3x 99)(2 x 5 ) (18x 7 + 9x 6 3x 2 1)(x + 1).,, x,. x ; :
60 4 x x 8x 4 7x 4 + 5x 3 + 2000x 2 6 = x = x x 8x 4 8x 4 ( 8x 4 ) 7x 4 + 5x 3 + 2000x 2 6 7x 4 (7x 4 ) 1 8x 3 + 1 1 + 5 7x + 2000 7x 2 6 7x 4 8 7 = 8 7., n, x, C/x n 0, x., ; (x 4 + 3x 99)(2 x 5 ) x (18x 7 + 9x 6 3x 2 1)(x + 1) ( x 4 ) ( ) + 3x 99 2 x x = 4 (x 4 5 ) x 5 ( x 5 ) ( x 18x 7 + 9x 6 3x 2 ) ( ) 1 x + 1 18x 7 (18x 7 ) (x) x ( 1 + 3 x 3 99 ) ( 2x ) x 4 5 + 1 = x = ( 1 + 9 18x 3 18x 5 1 18x 7 ) ( 1 + 1 ) (x4 )( x 5 ) (18x 7 )(x) x (1 + 0 0)( 0 + 1) (1 + 0 0 0)(1 + 0) x x 18 = x x 18 =.,, x x., x/18.,., x 2. x? x,., 2,, 2.,, x, x 2 = x. x,,. x 4x6 + 8 2x 3 + 6x + 1. 2x 3,,? 4x 6 +8, 4x 6,, 4x 6 + 8 4x 6. 2x 3,! x,, x., 2x 3, 4x 6, 4x 6 2x 3.
4.6 61 x = x = 4x6 + 8 2x 3 + 6x + 1 = 4x 6 + 8 4x 6 2x 3 + 6x + 1 2x 3 x 1 + 0 ( 1) = 1. 1 + 0 + 0 4x 6 4x6 + 8 4x 6 2x 3 = x 4x 6 2x 3 + 6x + 1 2x 3 (2x 3 ) 1 + 8 4x 6 1 + 6x 2x 3 + 1 2x 3,., 4 x4 = x x. 2x 3 2x 3 4,., 4,., 3 x3 = x x(, ), x < 0, x4 = x 2!?,, x 2,, x 4,,! x < 0, n x = x m, x m, n m. 4.6,. x x 0 x., f (x) = x /x,., 0 f, 0, 0.,., x. x x,,, x, f (x) = 1., x, x = x,, x < 0, f (x) = x/x = 1., f (x) = x /x x > 0, f (x) = 1 x < 0, f (x) = 1. y = f (x) 4-2.
62 4,, x = 0, x x 0 x = 1, x x 0 + x = 1.,, : x 4-2 x 0 x DNE., x,. x + 2 x ( 2) x + 2.,, x + 2 0 x + 2 < 0. x 2 x < 2., x + 2 = x + 2,, x + 2 = (x + 2)., x > 2, x + 2 / (x + 2) 1; x < 2, 1., y = x + 2 / (x + 2) y = x /x 2, 4-3. 4-3, 1( 1, ).
5,.. (,,,,,.,,.,,.,,., ; ; ; ;, ; ; ;. 5.1,.,. y = x 2, y = 1/x,., x = 0,., f (x) = 1/x,, x = 0, f.,.,. 5.1.1 f x a. y = f (x), (a, f (a)).
64 5,, (a, f (a))., (x, f (x)) ( ) (a, f (a))., x a, f (x) f (a).,,,. x a f (x) = f (a), f x = a.,,., f x = a, f (a),, (a, f (a)),, (1) x a f (x) ( ). (2) x = a ; f (a) ( ). (3), f(x) = f(a). x a,,. 5-1. 5-1 1, x = a, ;, x = a. 2,,, ;, x = a,, x = a. 3,, x = a, ; x = a., 4, x = a, f (a),,, x = a., x = a. 5.1.2.,, (a, b).,, f x = a x = b., f (x) = 1/x,, f (0, ), f (0). (, 0), ( 2, 3)
5.1 65, 0, f. [a, b]., 5-2 [a, b] ; [a, b]., x = a x = b x = a,, x = b,. ;., f [a, b], 5-2 (1) f (a, b) ; (2) f x = a ;, f (x) ( ), f (a), x a + ; (3) f x = b ;, f (x) ( ), f (b), x b.,,. /,,. 5.1.3.,.,,., f (x) = 1 f, x, a., f(x) = f(a). x a x f (x) = 1, f (a) = 1,, 1 = 1. x a, x a., g (x) = x. g g(x) = g(a). x a g (x) = x g (a) = a, x = a. x a x a, x a, ;,, ( A A.4.1 ).,,
66 5., x = 0, 1/x, x.,. g (x) = x x, g, x 2 x. x, x ( x ).,,,,,,, ( )., 5.2.11.,. f, f (x) = x sin (1/x). 3.6 ( x > 0 )., x < 0, f., sin (x) x, ( ) 1 f( x) = ( x) sin = ( x) x ( sin ( )) 1 = x sin x ( ) 1 = f(x). x, f, y, f ( 5-3 0.3 < x < 0.3 ). 5-3,. x,, x = 0, 1/x.,,,, x = 0, sin (1/x)., x sin (1/x) ( x ) x = 0, f., x = 0, f x = 0, ( ). g, ( ) 1 x sin x 0, g(x) = x 0, x = 0.
5.1 67, x = 0 ( g 0, f ), g (x) = f (x)., g, f x = 0. x = 0. g (0),., 3.6 ( ) 1 g(x) = x sin = 0. x 0 + x 0 + x ( ), 0. 0 ( ) 1 g(x) = x sin = 0. x 0 x 0 x g(x) = g(0) x 0 0., g x = 0,.,., 4. x 1 x 2 3x + 2, x 2 x = 1 2. x = 1, x = 1.., f (x) = ( x 2 3x + 2 ) / (x 2),,, 0, f., x = 2, f., f x = 1,, f, f(x) = f( 1). x 1 x 2 3x + 2 = ( 1)2 3( 1) + 2 = 2. x 1 x 2 ( 1) 2.,, 5.1.4.., IVT. f [a, b]., f (a) < 0 f (b) > 0., y = f (x), (a, f (a)) x, (b, f (b)) x, 5-4.
68 5 5-4, ( ),,, x a b. a b, a b ;., x a b. f [a, b], ; f, 5-5. x x., [a, b]. x x, ; f (a) > 0 f (b) < 0, f [a, b],, [a, b], x,. x f (c) = 0, 5-5 f [a, b],, f (a) < 0 f (b) > 0,, (a, b) c, f (c) = 0. f (a) > 0 f (b) < 0,. A A.4.2.,., p (x) = x 5 + x 4 + 3x + 1 x = 1 x = 2 x.,, p ( [1, 2]);, p (1) = 4 > 0 p (2) = 9 < 0. p (1) p (2), p [1, 2], (1, 2) c p (c) = 0. c p x.. x = cos (x),. y = x y = cos (x)., π/4 x..., x cos (x) = 0, x = cos(x).,, f (x) = x cos (x). c f (c) = 0. f (c) = 0, c cos (c) = 0, c = cos (c), x = cos (x), x = c.,. a b, f (a) f (b). ( ) π/4, a = 0 b = π/2. f (0) f (π/2).
5.1 69, f (0) = 0 cos (0) = 0 1 = 1,,, f (π/2) = π/2 cos (π/2) = π/2 0 = π/2,. f ( ),, (0, π/2) c f (c) = 0, x = cos (x).,, (0, π/2). (, π/4,.)., f (a) < 0 f (b) > 0( ),, (a, b) c f (c) = 0., M 0,., f [a, b] ; f (a) < M f (b) > M( ),, (a, b) c f (c) = M., f (x) = 3 x + x 2,, f (x) = 5 f ; 0 2, f (0) = 1 f (2) = 13. 1 13 5( ),, (0, 2) c f (c) = 5., f (x) = 5., g, g (x) = 3 x + x 2 5. f (x) = 5 c, c g (x) = 0. g (0) < 0 g (2) > 0,,,. 5.1.5 IVT., p,, c p (c) = 0. (., x 2 + 1, x.),, 4.3.,, p (x), a n x n,, x p(x) a n x n = 1 x p(x) a n x n = 1., x, p (x) a n x n ( 1).,,, 1. x., A, p (A) a n A n., B p (B) a n B n., a n A n a n B n. n,., a n > 0, a n B n a n A n. (
70 5 n n,,.), p(a) a n A n a n B n p(b)., p (A) p (B). p, ;, A B c, p (c) = 0., p,,, p,. 5.1.6., f, [a, b]. (,.), (a, f (a)),,, (b, f (b)).,,,,.,, [a, b] f x = c, f (c) f [a, b]., x, f (c) f (x)., [a, b] [a, b].,., f x = c, f (c) f [a, b]., [a, b] x, f (c) f (x)., [a, b]., --, -- f [a, b], f [a, b]., [a, b] (, ), 5-6. 5-6 5-6, x = c x = d., x = c x = a., x = b, x = c x = d. (, ).,,
5.2 71 ;, C, [a, b]., f, (a, b) 5-7 5-7, f [a, b],.,.,,.. (a, b)., x = c, x = b,. x = b,., b., b b., b, b b.,.,,.,,.,, (a, b), x = c x = d.. [a, b],. 5.2...,..,,. 5.2.1,.,,.,,
72 5 =.., ( ) ( ).,.,,,,, 60 /.,,..,,.,, 10, 0.25.,, 10 1.5 90 /., 10,. 90 /,,.,,. 0.000 1,.,,.,. 5.2.2,., 0,, 1. 0., 5-8. 5-8, 2 5., 3. 2 1, 3.,,. = ( ) ( ). 2 5, 5 2 = 3. 2 1, ( 1) 2 = 3.,,.,,.,. 2 11, 5, 9 + 6 = 15, 3. 2 4
5.2 73 2, 0, 12.,,,,.,.,., =.,,,.,,. 0,.,,, 0,,,,. 5.2.3,,,,,,.. t., 2: 00p.m.,, 0., 2: 03p.m.,, t = 180., u t. v t u t u., u t.., t = u t + v t u. t u t,., t = u t v t u.,.., t, f (t)., f (t) = t. v t u u t f (u) f (t) v t u = =. u t u t, u t ( u t )., u t u t, t u, f (t) f (u),,
74 5 f (u) f (t) t =. u t u t,, u = t,, 0/0... h = u t., u t, h., u t, h 0., u = t + h, f (t + h) f (t) t =. h 0 h,.., 7,, t = 0., t 15t 2 + 7( 15 )., f (t) = 15t 2 + 7, t., t = h 0 f(t + h) f(t) h (15(t + h) 2 + 7) (15t 2 + 7) =. h 0 h, (t + h) 2 = t 2 + 2th + h 2,, 15t 2 + 30th + 15h 2 + 7 15t 2 7 30th + 15h 2 = = (30t + 15h). h 0 h h 0 h h 0, h,,., h = 0 t = h 0 (30t + 15h) = 30t., 0, 30 0 = 0 /., t = 1/2, 30 1/2 = 15 /., 30., t 30t,, 30 /., 30 /, 30. 5.2.4., f (t) t. t, t u. y = f (t), (t, f (t)) (u, f (u)), 5-9.
5.2 75 5-9 f(u) f(t) =, u t v t u., t u, (t, f (t)) (u, f (u)).. u t,,, u t, 5-10. 5-10 (t, f (t)). u t,, (t, f (t)). 5.2.5, f x., (x, f (x)) y = f (x).,.,, (x, f (x))., 5-11 (x, f (x)), 5-11
76 5,., y = x, 5-12. (0, 0),.,,,. 5-12 5.2.10. (x, f (x)),,,.,., (x, f (x)).,.,,. x( ) z, (z, f (z))., (x, f (x)) (z, f (z)), 5-13., 5-13 f(z) f(x) z x, z x, x,.,,, f(z) f(x) (x, f(x)) =. z x z x h = z x,,, z x, h 0, f(x + h) f(x) (x, f (x)) =. h 0 h, 5.2.6 5-14,
5.2 77 5-14., x., (x, f (x)) x. f, f., f x f.,, f (x) = h 0 f(x + h) f(x) h, f x. x,, x f,, f x..,, y = x., x f,, (x, f (x)),., 5.2.3 f(t + h) f(t) t =. h 0 h, f (t) t. f (x), x t, v (t) t, v (t) = f (t).,.. f (x) = x 2, f (x) 5.2.3 f f(x + h) f(x) (x + h) 2 x 2 (x) = = h 0 h h 0 h x 2 + 2xh + h 2 x 2 2xh + h 2 = = h 0 h h 0 h = h 0 (2x + h) = 2x., f (x) = x 2 f (x) = 2x., y = x 2 ( x, x 2) 2x., 5-15.
78 5 5-15 x = 1 2, f (x) = 2x. (2 1 2 ), x 2. 5.2.7 f (x), f (x + h)., y = f (x), x x + h,, f (x + h) y. h x,, x.,, x x. ( x x, ), x h, f (x) f f(x + x) f(x) (x) =. x 0 x,. (x, y), y = f (x)., x, x. y y, f ( x ).,,, x = x x y = y y. x = x + x, y = y y = f(x ) f(x) = f(x + x) f(x). f (x) f y (x) = x 0 x., x f (x) y., y = f (x) = x 2,, f (x) = 2x., x = 6.,, f (x) f (6) = 2 6 = 12., 6 2 = 36 6, 36 12., 0.01 6, 0.12 36., (6.01) 2 36.12., 36.120 1,.
5.2 79,, f (x) y x, x 0., 6,. (6.000 4) 2. x 6 0.000 4,, y 12, 0.004 8. (6.000 4) 2 36.004 8., 36.004 800 16,, 6,., 12 x = 6. x = 13, f (13), 2 13 = 26., 13 2 = 139; (13.000 2) 2 13 13.000 2, 0.000 2. 26, 26 0.000 2 169., 0.005 2 169 169.005 2. (13.000 2) 2 169.005 200 04., 13., f y (x) = x 0 x., x, y x., x. x, x, dx, x, y., dx dy ;,, y = f (x),, dy dx f (x)., y = x 2, dy dx = 2x., x2 y, f (x) = dy dx = d(x2 ) dx = d dx (x2 ) = 2x., 5.2.3,, t f (t) = 15t 2 + 7,, 30t. f (t), f (t) = 30t. p, p = 15t 2 + 7, dp = 30t., dt x y,., dy dy y x. y = f (x), dx dx f (x).,, dy y, x 0 dx x.,
80 5 5.2.8,, f., m b, f (x) = mx + b. f (x), y = f (x) (x, f (x))., y = mx + b m, y b.,,, x, f (x) m y = mx + b m. f f(x + h) f(x) (m(x + h) + b) (mx + b) (x) = = h 0 h h 0 h mh = h 0 h = m = m. h 0, x, f (x) = m.,., ( ; 11.3.1 )., f, f (x) = b,, 0., x, f (x) = 0., 0. 5.2.9 f, f,,,.,, f.,, f (x) = x 2,, f (x) = 2x.,. g (x) = 2x, g (x). g, 2, g (x) = 2., f 2,, x, f (x) = 2. y = f (x),, dy dx f (x). y = f (x),, d2 y dx 2 f (x)., y = f (x) = x 2, f (x) = d2 y dx 2 = d2 (x 2 ) dx 2 = d2 dx 2 (x2 ) = 2. f (x) = x 2 ( x) 2. f f., f f, f (x), f (3) (x), d 3 y dx 3 d3 dx 3 (y).
5.2 81 f (3) (x),.,,,, f (4) (x) f (x).,, f (2) (x) f (x). f (1) (x) f (x),,, f (0) (x) f (x) ( )., n, f (n) (x). 5.2.10 5.2.5, f (x) = x., x = 0.,., f f(x + h) f(x) x + h x (x) = =. h 0 h h 0 h, x = 0,, 0 x f f(0 + h) f(0) 0 + h 0 h (0) = = = h 0 h h 0 h h 0 h., 4.6,., f (0), 0 f.,,,., 1, 1., f(x + h) f(x) h 0 + h f(x + h) f(x), h 0 h, (, h 0).,,,.,,,.,, f (x) = x,, x = 0 1, 1. 5-16, 1 (, 1,, x > 0, f (x) = 1).,, 1 (, x < 0, f (x) = 1)., x = 0. 5-16,
82 5,.,,.,,, x,,,,,,.,,.,,. 5.2.11.,.,,., f x,, x., 7, sin (x) x. x., ( ).,. f x, f(u) = f(x), u x, 5.1.1,,, h = u x,., u = x+h,, u x, h 0., f(x + h) = f(x). h 0,.,. f x ;, f (x),, f, f(x + h) f(x) h 0 h.,, f (x),,,,., f (x).., ( ) f(x + h) f(x) h. h 0 h,,
5.2 83 ( ) f(x + h) f(x) f(x + h) f(x) h = h = f (x) 0 = 0. h 0 h h 0 h h 0,. ( f (x),.), h ( ) f(x + h) f(x) h = (f(x + h) f(x)). h 0 h h 0, (f(x + h) f(x)) = 0. h 0, f (x), (, ) f(x + h) f(x) = 0. h 0, f (x), f(x + h) = f(x) h 0,.,.,,
6,.,.,., ; ; ; ; ; ;. 6.1, f (x) = 1/x x. f f(x + h) f(x) (x) =, h 0 h,, 1 f (x) = x + h 1 x. h 0 h, 0 h, 0., 0.,. f (x) = h 0 x (x + h) x(x + h) h h = h 0 hx(x + h). h,, h = 0, f 1 (x) = h 0 x(x + h) = 1 x(x) = 1 x 2.
d dx ( ) 1 = 1 x x 2. 6.1 85, f (x) = x, 4.2. f f(x + h) f(x) x + h x (x) = =, h 0 h h 0 h 0. 0 x + h x x + h + x f (x + h) x (x) = = h 0 h x + h + x h 0 h( x + h + x) ;, x, h,, f (x) = h 0, h h( x + h + x) = h 0 d dx ( x) = 1 2 x. 1 x + h + x = 1 x + x = 1 2 x.,, f (x) = x + x 2,,, f f(x + h) f(x) ( x + h + (x + h) 2 ) ( x + x 2 ) (x) = =. h 0 h h 0 h,, f (x) = h 0 x + h x (x + h) 2 x 2 +. h h 0 h ; 1/2 x, 5.2.6 2x., f (x) = 1 2 x + 2x. x n x, n. f (x) = x n ; f f(x + h) f(x) (x + h) n x n (x) = =. h 0 h h 0 h (x + h) n. ;
86 6, (x + h) n = (x + h)(x + h) (x + h). n.,,,. x, n x,, x n. x,, (x + h) n = (x + h)(x + h) (x + h) = x n + h,. h, x,, h (n 1) x,,, hx n 1. h x ( h, x,, h, x, )., n h x,, n hx n 1. nhx n 1., h,, h 2., (x + h) n = (x + h)(x + h) (x + h) = x n + nhx n 1 + h 2 h 2 ( ) h 2, x h., (x + h) n = (x + h)(x + h) (x + h) = x n + nhx n 1 + h 2 ( )., f (x + h) n x n x n + nhx n 1 + h 2 ( ) x n (x) = =. h 0 h h 0 h x n,, h f nhx n 1 + h 2 ( ) (x) = = (nx n 1 + h ( )). h 0 h h 0 h 0, 0( ). nx n 1.,, n, d dx (xn ) = nx n 1, 9.5.1, a, d dx (xa ) = ax a 1,, 1.
6.2 ( ) 87., a = 0,, x a 1. 0x 1, 0. 5.2.8 ;, C, d (C) = 0. dx, a = 1,, x a x., 1x 0, 1., 5.2.8 ; d (x) = 1. dx a = 2, x 2 x 2x 1, 2x.., a = 1, x 1 1 x 2., 1/x 1/x 2,,.,. a = 1 2, x1/2 x 1 2 x 1/2. ( 9.1.1 ), x 1/2 x,.,,,, 1 2 1., 2 a = 1. 3 d dx (x1/3 ) = 1 3 x1/3 1 = 1 3 x 2/3. ( 9.1.1 ), d dx ( 3 x) = 1 3 3 x 2.,. x a x. 6.2 ( ).,,. f, f(x) = 3x7 + x 4 2x 5 + 15x 4/3 23x + 9 6x 2 4.. 6.2.6,
88 6 ( ) x a f,., ;, f f (x). ( A A.6, 6.2.7.) 6.2.1,., x 2 2x,, 7x 2 7 2x, 14x. x 2 2x, 1., x a.,,,., 7x 2, 2, 7 14,, x 14x 1, 14x., 13x 4, 4 13, 52,, x 52x 3. 6.2.2,,., x 3x 5 2x 2 + 7 x + 2?, 1/ x x 1/2,, 3x 5 2x 2 +7x 1/2 +2., 3x 5 15x 4., 2x 2 4x, 7x 1/2 7 2 x 3/2., 2 0, 2.,, +2., ( d 3x 5 2x 2 + 7 ) + 2 = d dx x dx (3x5 2x 2 + 7x 1/2 + 2) = 15x 4 4x 7 2 x 3/2., x 3/2 x x, 15x 4 4x 7 1 2 x x., x 5/2 x 2 x, x 7/2 x 3 x,. 6.2.3,., ( ),
6.2 ( ) 89 h(x) = (x 5 + 2x 1)(3x 8 2x 7 x 4 3x). f (x) = x 5 + 2x 1 g (x) = 3x 8 2x 7 x 4 3x. h f g. f g, f (x) = 5x 4 + 2 g (x) = 24x 7 14x 6 4x 3 3., h,. h (x) ( 5x 4 + 2 ) ( 24x 7 14x 6 4x 3 3 ). h (x),., f g ( g )., g f.,. ( 1) h(x) = f(x)g(x), h (x) = f (x)g(x) + f(x)g (x)., h (x) = ( x 5 + 2x 1 ) ( 3x 8 2x 7 x 4 3x ), h f g,., f g f(x) = x 5 + 2x 1 g(x) = 3x 8 2x 7 x 4 3x f (x) = 5x 4 + 2 g (x) = 24x 7 14x 6 4x 3 3.,., f (x) g (x), f (x) g (x),. h (x) = f (x)g(x) + f(x)g (x) = (5x 4 + 2)(3x 8 2x 7 x 4 3x) +(x 5 + 2x 1)(24x 7 14x 6 4x 3 3)., h...,, y = x, f (x)., y = ( x 3 + 2x ) (3x + x + 1), dy/dx, u = ( x 3 + 2x ) v = (3x + x + 1).,, u f (x), du/dx f (x); v g (x),,, ( 2) y = uv, dy dx = v du dx + u dv dx.
90 6 u = x 3 + 2x v = 3x + x + 1 du dx = 3x2 + 2 dv dx = 3 + 1 2 x. dy dx = v du dx + u dv dx = (3x + ( x + 1)(3x 2 + 2) + (x 3 + 2x) 3 + 1 ) 2. x, dy/dx. y = (x 2 + 1)(x 2 + 3x)(x 5 + 2x 4 + 7), ( ) y = uvw, dy dx = du dv dw vw + u w + uv dx dx dx., uvw,, d/dx. ( ),, u = x 2 + 1, v = x 2 + 3x w = x 5 + 2x 4 + 7,, y uvw. du/dx = 2x, dv/dx = 2x + 3 dw/dx = 5x 4 + 8x 3., dy dx = du dv dw vw + u w + uv dx dx dx = (2x)(x 2 + 3x)(x 5 + 2x 4 + 7) + (x 2 + 1)(2x + 3)(x 5 + 2x 4 + 7) +(x 2 + 1)(x 2 + 3x)(5x 4 + 8x 3 ). y,,,..,, x sin (x).., x,. 6.2.4,. h(x) = 2x3 3x + 1 x 5 8x 3 + 2 x. f (x) = 2x 3 3x + 1 g (x) = x 5 8x 3 + 2,,
6.2 ( ) 91 h f g, h (x) = f (x) /g (x). ( 1) h(x) = f(x) g(x), h (x) = f (x)g(x) f(x)g (x) (g(x)) 2.,,., f g f(x) = 2x 3 3x + 1 g(x) = x 5 8x 3 + 2 f (x) = 6x 2 3 g (x) = 5x 4 24x 2., h (x) = f (x) /g (x), h (x) = f (x)g(x) f(x)g (x) (g(x)) 2 = (6x2 3)(x 5 8x 3 + 2) (2x 3 3x + 1)(5x 4 24x 2 ) (x 5 8x 3 + 2) 2.,. y = 3x2 + 1 2x 8 7, dy/dx,, u = 3x 2 + 1 v = 2x 8 7, y = u/v. ( 2) y = u v, du dy v dx = dx u dv dx v 2., u = 3x 2 + 1 v = 2x 8 7 du dx = 6x dv dx = 16x7. du dy v dx = dx u dv dx v 2 = (2x8 7)(6x) (3x 2 + 1)(16x 7 ) (2x 8 7) 2., ( ). 6.2.5 h (x) = ( x 2 + 1 ) 99, h (x). (
92 6, x 2 + 1 99, ).,., h f g,, g (x) = x 2 + 1, f (x) = x 99., x, g, x 2 + 1., f, ( x 2 + 1 ) 99, h (x)., h (x) f (g (x)). ( 1.3.), ( 1) h(x) = f(g(x)), h (x) = f (g(x))g (x)..., g., f, g (x), x., f (x) = x 99, f (x) = 99x 98. g (x) = x 2 + 1, g (x) = 2x. 2x., f (x), x, x 2 + 1 ( g (x))., f (g (x)) = f ( x 2 + 1 ) = 99 ( x 2 + 1 ) 98., h (x) = f (g(x))g (x) = 99(x 2 + 1) 98 (2x) = 198x(x 2 + 1) 98.,.. y = ( x 2 + 1 ) 99 (, dy/dx. x 2 + 1 ), u. y = u 99, u = x 2 + 1., ( 2),, y u, u x, dy dx = dy du du dx. y = u 99 u = x 2 + 1 dy du = 99u98, du dx = 2x. dy dx = dy du du dx = 99u98 2x = 198xu 98., x 2 + 1 u, dy/dx = 198 ( x 2 + 1 ) 98,.. y = x 3 7x, dy/dx u = x 3 7x, y = u. y = u dy du = 1 2 u u = x 3 7x du dx = 3x2 7.
6.2 ( ) 93,, dy dx = dy du du dx = 1 2 u (3x2 7) = 3x2 7 2 u., u. u = x 3 7x, dy dx = 3x2 7 2 x 3 7x.,.., x, u;, u y., u x y.,., du., dy/du du/dx, ( 5.2.7 ). ( ).,., y = ((x 3 10x) 9 + 22) 8. dy/dx u = x 3 10x v = u 9 + 22, y = v 8., dy dx = dy dv du dv du dx., y v, v u, u x.,, y = v 8 v = u 9 + 22 u = x 3 10x dy dv = 8v7, dv du = 9u8 du dx = 3x2 10. dy dx = dy dv du dv du dx = (8v7 )(9u 8 )(3x 2 10)., u v., u 99 + 22 v dy dx = (8v7 )(9u 8 )(3x 2 10) = (8(u 9 + 22) 7 )(9u 8 )(3x 2 10)., x 3 10x u, 8 9, dy dx = (8(u9 + 22) 7 )(9u 8 )(3x 2 10) = 72((x 3 10x) 9 + 22) 7 (x 3 10x) 8 (3x 2 10).,., g h, h (x) = g (x),
94 6 g (5) = 4 g (5) = 7,, h (5). f (x) = x, h (x) = f (g (x)), h (x) = f (g (x)) g (x). f (x) = x, f (x) = 1/2 x;, x = 5, h (x) = f (g(x))g (x) = h (5) = 1 2 g(5) g (5). 1 2 g(x) g (x). g (5) = 4 g (5) = 7, h (5) = 1 2 4 (7) = 7 4. j (x) = g ( x), g. j (25), j (x) = g (f (x)), f (x) = x. j (x) = g (f(x))f (x) = g ( 1 x) 2 x., x = 25, g (5) = 7, j (25) = g ( 1 25) 2 25 = g (5) 1 10 = 7 10 6.2.6 f f(x) = 3x7 + x 4 2x 5 + 15x 4/3 23x + 9 6x 2. 4 f (x), f. ( ).,. y = f (x), dy/dx. y u = 3x 7 + x 4 2x 5 + 15x 4/3 23x + 9 v = 6x 2 4., du/dx dv/dx., 12x.. u = 3x 7 + x 4 2x 5 + 12x 4/3 23x + 9 v = 6x 2 4 du dx =??? dv dx = 12x. du/dx,., du/dx., u q = 3x 7 r, r r = x 4 2x 5 + 15x 4/3 23x + 9.. q,