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利用Bolzano定理逼近函数零点

利用Bolzano定理逼近函数零点
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2013-11-28 18:44
函数微积分几何作图
①本范例演示如何利用Bolzano定理动态逼近函数的零点；②涉及核心数学概念包括Bolzano定理、函数零点、区间套；③作图要点：使用GeoGebra命令Function（函数）、If（条件判断）、Segment（线段）；对象包括b, a, f, A1, B1, D1, C1, F1等；④适合高中及以上学生及微积分教学使用。

GeoGebra 命令及参数

  1. Function(cos(x), a, b) → f
  2. If(D1 * F1 < 0, A1, C1) → A2
  3. If(D1 * F1 < 0, C1, B1) → B2
  4. If(D2 * F2 < 0, A2, C2) → A3
  5. If(D2 * F2 < 0, C2, B2) → B3
  6. If(D3 * F3 < 0, A3, C3) → A4
  7. If(D3 * F3 < 0, C3, B3) → B4
  8. If(D4 * F4 < 0, A4, C4) → A5
  9. If(D4 * F4 < 0, C4, B4) → B5
  10. If(D5 * F5 < 0, A5, C5) → A6
  11. If(D5 * F5 < 0, C5, B5) → B6
  12. If(D6 * F6 < 0, A6, C6) → A7
  13. If(D6 * F6 < 0, C6, B6) → B7
  14. If(D7 * F7 < 0, A7, C7) → A8
  15. If(D7 * F7 < 0, C7, B7) → B8
  16. If(D8 * F8 < 0, A8, C8) → A9
  17. If(D8 * F8 < 0, C8, B8) → B9
  18. If(D9 * F9 < 0, A9, C9) → A10
  19. If(D9 * F9 < 0, C9, B9) → B10
  20. If(D10 * F10 < 0, A10, C10) → A11
  21. If(D10 * F10 < 0, C10, B10) → B11
  22. If(D11 * F11 < 0, A11, C11) → A12
  23. If(D11 * F11 < 0, C11, B11) → B12
  24. If(D12 * F12 < 0, A12, C12) → A13
  25. If(D12 * F12 < 0, C12, B12) → B13
  26. If(D13 * F13 < 0, A13, C13) → A14
  27. If(D13 * F13 < 0, C13, B13) → B14
  28. If(D14 * F14 < 0, A14, C14) → A15
  29. If(D14 * F14 < 0, C14, B14) → B15
  30. If(D15 * F15 < 0, A15, C15) → A16
  31. If(D15 * F15 < 0, C15, B15) → B16
  32. If(D16 * F16 < 0, A16, C16) → A17
  33. If(D16 * F16 < 0, C16, B16) → B17
  34. If(D17 * F17 < 0, A17, C17) → A18
  35. If(D17 * F17 < 0, C17, B17) → B18
  36. If(D18 * F18 < 0, A18, C18) → A19
  37. If(D18 * F18 < 0, C18, B18) → B19
  38. If(D19 * F19 < 0, A19, C19) → A20
  39. If(D19 * F19 < 0, C19, B19) → B20
  40. If(D20 * F20 < 0, A20, C20) → A21
  41. If(D20 * F20 < 0, C20, B20) → B21
  42. If(D21 * F21 < 0, A21, C21) → A22
  43. If(D21 * F21 < 0, C21, B21) → B22
  44. If(D22 * F22 < 0, A22, C22) → A23
  45. If(D22 * F22 < 0, C22, B22) → B23
  46. If(D23 * F23 < 0, A23, C23) → A24
  47. If(D23 * F23 < 0, C23, B23) → B24
  48. If(D24 * F24 < 0, A24, C24) → A25
  49. If(D24 * F24 < 0, C24, B24) → B25
  50. If(D25 * F25 < 0, A25, C25) → A26
  51. If(D25 * F25 < 0, C25, B25) → B26
  52. If(D26 * F26 < 0, A26, C26) → A27
  53. If(D26 * F26 < 0, C26, B26) → B27
  54. If(D27 * F27 < 0, A27, C27) → A28
  55. If(D27 * F27 < 0, C27, B27) → B28
  56. If(D28 * F28 < 0, A28, C28) → A29
  57. If(D28 * F28 < 0, C28, B28) → B29
  58. If(D29 * F29 < 0, A29, C29) → A30
  59. If(D29 * F29 < 0, C29, B29) → B30
  60. If(D30 * F30 < 0, A30, C30) → A31
  61. If(D30 * F30 < 0, C30, B30) → B31
  62. If(D31 * F31 < 0, A31, C31) → A32
  63. If(D31 * F31 < 0, C31, B31) → B32
  64. If(D32 * F32 < 0, A32, C32) → A33
  65. If(D32 * F32 < 0, C32, B32) → B33
  66. If(D33 * F33 < 0, A33, C33) → A34
  67. If(D33 * F33 < 0, C33, B33) → B34
  68. If(D34 * F34 < 0, A34, C34) → A35
  69. If(D34 * F34 < 0, C34, B34) → B35
  70. If(D35 * F35 < 0, A35, C35) → A36
  71. If(D35 * F35 < 0, C35, B35) → B36
  72. If(D36 * F36 < 0, A36, C36) → A37
  73. If(D36 * F36 < 0, C36, B36) → B37
  74. If(D37 * F37 < 0, A37, C37) → A38
  75. If(D37 * F37 < 0, C37, B37) → B38
  76. If(D38 * F38 < 0, A38, C38) → A39
  77. If(D38 * F38 < 0, C38, B38) → B39
  78. If(D39 * F39 < 0, A39, C39) → A40
  79. If(D39 * F39 < 0, C39, B39) → B40
  80. If(D40 * F40 < 0, A40, C40) → A41
  81. If(D40 * F40 < 0, C40, B40) → B41
  82. If(D41 * F41 < 0, A41, C41) → A42
  83. If(D41 * F41 < 0, C41, B41) → B42
  84. If(D42 * F42 < 0, A42, C42) → A43
  85. If(D42 * F42 < 0, C42, B42) → B43
  86. If(D43 * F43 < 0, A43, C43) → A44
  87. If(D43 * F43 < 0, C43, B43) → B44
  88. If(D44 * F44 < 0, A44, C44) → A45
  89. If(D44 * F44 < 0, C44, B44) → B45
  90. If(D45 * F45 < 0, A45, C45) → A46
  91. If(D45 * F45 < 0, C45, B45) → B46
  92. If(D46 * F46 < 0, A46, C46) → A47
  93. If(D46 * F46 < 0, C46, B46) → B47
  94. If(D47 * F47 < 0, A47, C47) → A48
  95. If(D47 * F47 < 0, C47, B47) → B48
  96. If(D48 * F48 < 0, A48, C48) → A49
  97. If(D48 * F48 < 0, C48, B48) → B49
  98. If(D49 * F49 < 0, A49, C49) → A50
  99. If(D49 * F49 < 0, C49, B49) → B50
  100. Segment(G1, K1) → M1
  101. Segment(H1, L1) → N1
  102. Segment(I1, (C1, f(C1))) → O1
  103. Segment(G2, K2) → M2
  104. Segment(H2, L2) → N2
  105. Segment(I2, (C2, f(C2))) → O2
  106. Segment(G3, K3) → M3
  107. Segment(H3, L3) → N3
  108. Segment(I3, (C3, f(C3))) → O3
  109. Segment(G4, K4) → M4
  110. Segment(H4, L4) → N4
  111. Segment(I4, (C4, f(C4))) → O4
  112. Segment(G5, K5) → M5
  113. Segment(H5, L5) → N5
  114. Segment(I5, (C5, f(C5))) → O5
  115. Segment(G6, K6) → M6
  116. Segment(H6, L6) → N6
  117. Segment(I6, (C6, f(C6))) → O6
  118. Segment(G7, K7) → M7
  119. Segment(H7, L7) → N7
  120. Segment(I7, (C7, f(C7))) → O7
  121. Segment(G8, K8) → M8
  122. Segment(H8, L8) → N8
  123. Segment(I8, (C8, f(C8))) → O8
  124. Segment(G9, K9) → M9
  125. Segment(H9, L9) → N9
  126. Segment(I9, (C9, f(C9))) → O9
  127. Segment(G10, K10) → M10
  128. Segment(H10, L10) → N10
  129. Segment(I10, (C10, f(C10))) → O10
  130. Segment(G11, K11) → M11
  131. Segment(H11, L11) → N11
  132. Segment(I11, (C11, f(C11))) → O11
  133. Segment(G12, K12) → M12
  134. Segment(H12, L12) → N12
  135. Segment(I12, (C12, f(C12))) → O12
  136. Segment(G13, K13) → M13
  137. Segment(H13, L13) → N13
  138. Segment(I13, (C13, f(C13))) → O13
  139. Segment(G14, K14) → M14
  140. Segment(H14, L14) → N14
  141. Segment(I14, (C14, f(C14))) → O14
  142. Segment(G15, K15) → M15
  143. Segment(H15, L15) → N15
  144. Segment(I15, (C15, f(C15))) → O15
  145. Segment(G16, K16) → M16
  146. Segment(H16, L16) → N16
  147. Segment(I16, (C16, f(C16))) → O16
  148. Segment(G17, K17) → M17
  149. Segment(H17, L17) → N17
  150. Segment(I17, (C17, f(C17))) → O17
  151. Segment(G18, K18) → M18
  152. Segment(H18, L18) → N18
  153. Segment(I18, (C18, f(C18))) → O18
  154. Segment(G19, K19) → M19
  155. Segment(H19, L19) → N19
  156. Segment(I19, (C19, f(C19))) → O19
  157. Segment(G20, K20) → M20
  158. Segment(H20, L20) → N20
  159. Segment(I20, (C20, f(C20))) → O20
  160. Segment(G21, K21) → M21
  161. Segment(H21, L21) → N21
  162. Segment(I21, (C21, f(C21))) → O21
  163. Segment(G22, K22) → M22
  164. Segment(H22, L22) → N22
  165. Segment(I22, (C22, f(C22))) → O22
  166. Segment(G23, K23) → M23
  167. Segment(H23, L23) → N23
  168. Segment(I23, (C23, f(C23))) → O23
  169. Segment(G24, K24) → M24
  170. Segment(H24, L24) → N24
  171. Segment(I24, (C24, f(C24))) → O24
  172. Segment(G25, K25) → M25
  173. Segment(H25, L25) → N25
  174. Segment(I25, (C25, f(C25))) → O25
  175. Segment(G26, K26) → M26
  176. Segment(H26, L26) → N26
  177. Segment(I26, (C26, f(C26))) → O26
  178. Segment(G27, K27) → M27
  179. Segment(H27, L27) → N27
  180. Segment(I27, (C27, f(C27))) → O27
  181. Segment(G28, K28) → M28
  182. Segment(H28, L28) → N28
  183. Segment(I28, (C28, f(C28))) → O28
  184. Segment(G29, K29) → M29
  185. Segment(H29, L29) → N29
  186. Segment(I29, (C29, f(C29))) → O29
  187. Segment(G30, K30) → M30
  188. Segment(H30, L30) → N30
  189. Segment(I30, (C30, f(C30))) → O30
  190. Segment(G31, K31) → M31
  191. Segment(H31, L31) → N31
  192. Segment(I31, (C31, f(C31))) → O31
  193. Segment(G32, K32) → M32
  194. Segment(H32, L32) → N32
  195. Segment(I32, (C32, f(C32))) → O32
  196. Segment(G33, K33) → M33
  197. Segment(H33, L33) → N33
  198. Segment(I33, (C33, f(C33))) → O33
  199. Segment(G34, K34) → M34
  200. Segment(H34, L34) → N34
  201. Segment(I34, (C34, f(C34))) → O34
  202. Segment(G35, K35) → M35
  203. Segment(H35, L35) → N35
  204. Segment(I35, (C35, f(C35))) → O35
  205. Segment(G36, K36) → M36
  206. Segment(H36, L36) → N36
  207. Segment(I36, (C36, f(C36))) → O36
  208. Segment(G37, K37) → M37
  209. Segment(H37, L37) → N37
  210. Segment(I37, (C37, f(C37))) → O37
  211. Segment(G38, K38) → M38
  212. Segment(H38, L38) → N38
  213. Segment(I38, (C38, f(C38))) → O38
  214. Segment(G39, K39) → M39
  215. Segment(H39, L39) → N39
  216. Segment(I39, (C39, f(C39))) → O39
  217. Segment(G40, K40) → M40
  218. Segment(H40, L40) → N40
  219. Segment(I40, (C40, f(C40))) → O40
  220. Segment(G41, K41) → M41
  221. Segment(H41, L41) → N41
  222. Segment(I41, (C41, f(C41))) → O41
  223. Segment(G42, K42) → M42
  224. Segment(H42, L42) → N42
  225. Segment(I42, (C42, f(C42))) → O42
  226. Segment(G43, K43) → M43
  227. Segment(H43, L43) → N43
  228. Segment(I43, (C43, f(C43))) → O43
  229. Segment(G44, K44) → M44
  230. Segment(H44, L44) → N44
  231. Segment(I44, (C44, f(C44))) → O44
  232. Segment(G45, K45) → M45
  233. Segment(H45, L45) → N45
  234. Segment(I45, (C45, f(C45))) → O45
  235. Segment(G46, K46) → M46
  236. Segment(H46, L46) → N46
  237. Segment(I46, (C46, f(C46))) → O46
  238. Segment(G47, K47) → M47
  239. Segment(H47, L47) → N47
  240. Segment(I47, (C47, f(C47))) → O47
  241. Segment(G48, K48) → M48
  242. Segment(H48, L48) → N48
  243. Segment(I48, (C48, f(C48))) → O48
  244. Segment(G49, K49) → M49
  245. Segment(H49, L49) → N49
  246. Segment(I49, (C49, f(C49))) → O49
  247. Segment(G50, K50) → M50
  248. Segment(H50, L50) → N50
  249. Segment(I50, (C50, f(C50))) → O50

AI 绘图提示语参考

画一个函数图像，并演示如何使用二分法动态逼近函数的零点。初始区间为[a, b]，函数为cos(x)。在区间内，根据函数值正负不断二分区间，并在每一步绘制连接区间端点的线段，最终线段会逐渐逼近函数的零点，展示Bolzano定理的应用。
以上提示语为根据原始绘图命令整理而来，仅供参考。