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Approach a root using Bolzano's Theorem

Approach a root using Bolzano's Theorem
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2013-11-28 18:44
FunctionCalculusGeometric Construction
This example demonstrates how to dynamically approach a zero of a function using Bolzano's Theorem. Key math concepts include Bolzano's Theorem, function roots, and interval halving. GeoGebra commands used are Function, If, Segment; objects include b, a, f, A1, B1, etc. Suitable for high school students and calculus instruction.

GeoGebra Commands & Parameters

  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 Drawing Prompt Reference

Draw a function graph of cos(x) and dynamically demonstrate how to approach the zero of the function using Bolzano's Theorem. Start with an interval [a, b]. Continuously bisect the interval based on the sign of the function values, drawing line segments connecting the endpoints at each step. The segments will gradually converge to the zero of the function, illustrating the application of Bolzano's Theorem.
The prompt above was compiled from the original drawing commands and is for reference only.