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Grade 12 Mathematics Unit 2 Introduction To calculus, Rate of Change #EthiopanEducation #zsecrettrai — Transcript

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  1. 0:00Sal
  2. 0:03tic un un 2 introduction
  3. 0:15tot
  4. 0:26uc derivat define
  5. 0:31rate Of Change rate Of
  6. 0:33Change average rate Of Change instant
  7. 0:36rate Of Change
  8. 0:39tical for matic
  9. 0:42definition instant rate Of
  10. 0:46Change amp ex tic
  11. 1:01appliation appliation Unit 1 2 un 3 un
  12. 1:064un 5
  13. 1:11ex end
  14. 1:19EX
  15. 1:24exam appliation
  16. 1:28ut
  17. 1:38Tion
  18. 1:41ent
  19. 1:43exam uc
  20. 2:19ud inflation of curen inflation of curen
  21. 2:44Tim
  22. 2:50ous is one of the components of maema
  23. 2:53that concern with change and
  24. 2:58motion
  25. 3:03ul
  26. 3:07[Muzică]
  27. 3:11ut end
  28. 3:15ovt
  29. 3:19timat
  30. 3:25u derivat diff
  31. 3:30is the rate at which One quantity change
  32. 3:32with respect to another geom graphic
  33. 3:39deriv
  34. 3:41x
  35. 3:48y
  36. 3:50x y y
  37. 3:54fx
  38. 3:58fx
  39. 4:01deriv this Point deriv
  40. 4:07ur at this Point
  41. 4:14ent fx
  42. 4:18Point
  43. 4:23ent
  44. 4:24so
  45. 4:26derat pr diff
  46. 4:34ent
  47. 4:35ent
  48. 4:39teic ING
  49. 4:47appliation rate Of Change rate Of Change
  50. 4:56qu k per
  51. 5:0812 K
  52. 5:15m
  53. 5:17k 24
  54. 5:21km Of Change
  55. 5:35rate Of Change con
  56. 5:39apply rate Of
  57. 5:42Change ne
  58. 5:450 POS rate Of Change Of
  59. 5:49Change
  60. 5:52x
  61. 5:56y x y
  62. 6:01x
  63. 6:03y x
  64. 6:07yate Of Change POS rate Of Change neate
  65. 6:12Of
  66. 6:13Change rate Of
  67. 6:15Change x
  68. 6:18y x
  69. 6:22y x
  70. 6:25y Of Change neate Of Change 0 rate Of
  71. 6:31Change x
  72. 6:38yx
  73. 6:41yate Of Change 0 rate Of
  74. 6:45Change no
  75. 6:49fate Of
  76. 6:53Change Of Change ne
  77. 6:58Tion x
  78. 7:06y x y
  79. 7:12fx
  80. 7:14x
  81. 7:18y
  82. 7:21fx x
  83. 7:26[Muzică]
  84. 7:34of
  85. 7:38ne
  86. 7:41x
  87. 7:43y
  88. 7:48of rate Of Change avage rate Of
  89. 7:52Change instant rate Of
  90. 7:55Change average rate Of Change average
  91. 7:59rate of
  92. 8:02definition y equ fx
  93. 8:07function the average rate Of Change of y
  94. 8:10with respect to x on the interval
  95. 8:14abate Of Change equal to change in y
  96. 8:19over change in
  97. 8:21X change in y change in
  98. 8:25X change in y over change in X
  99. 8:31x
  100. 8:33y
  101. 8:35ab
  102. 8:38ab a X1 b
  103. 8:45X2 x
  104. 8:48fx X1 x
  105. 8:54fx1 a
  106. 8:56X1
  107. 8:58X1 x
  108. 9:01fx1 b X2 x X2 f x 2 average rate Of
  109. 9:09Change change in y over change in
  110. 9:13X this equ to change in y y2 - y1 over
  111. 9:19change in X X2 - X1 X1 X2 y2
  112. 9:26fx2 y1 f
  113. 9:32this equ
  114. 9:33to f
  115. 9:36X2
  116. 9:38F1 over X2
  117. 9:43X1
  118. 9:45of example Find the average rate Of
  119. 9:49Change of a function f of x to 2x min 9
  120. 9:53between X1 equ 3 and X2 6
  121. 9:59rate Of Change average rate Of Change
  122. 10:02equ to change in y over change in X
  123. 10:07change y y2 - y1 over X2 - X1 y2 f
  124. 10:15X2 f X1 over X2
  125. 10:23X1 fx2 fx1 X1
  126. 10:27X2 X1
  127. 10:31X2 fx 2 fx2 X2
  128. 10:386 f
  129. 10:426 var x 6 2 6 -
  130. 10:509 6 12 12
  131. 10:549
  132. 10:56fx1 fx1 X1
  133. 11:00F 3 2 3 -
  134. 11:069 -
  135. 11:133 fx 2 3 f X1 3 6 3
  136. 11:27this 3
  137. 11:30[Muzică]
  138. 11:33function average rate Of Change between
  139. 11:35the interval X1 3 and x
  140. 11:416
  141. 11:44example rate Of Change gra
  142. 11:48discuss examine the motion of a Car that
  143. 11:51travels along a straight road time in se
  144. 11:55distance
  145. 11:580
  146. 11:590
  147. 12:03se se
  148. 12:0710
  149. 12:128 Find the average velocity during the
  150. 12:15time interval 2 L than or equ to t or
  151. 12:18equ to 4 rate Of
  152. 12:21Change equal change in y over change in
  153. 12:26X average average
  154. 12:32average rate Of Change equ to f b f a
  155. 12:38over b -
  156. 12:41a
  157. 12:43fx2 f X1 over X2 -
  158. 12:52x1 2 X1 x1 2 a 4 X2 equ
  159. 13:014age
  160. 13:03velocity avage rate Of Change rate Of
  161. 13:06Change velocity rate Of Change of
  162. 13:09distance rate Of Change of
  163. 13:11distance velocity equ to rate Of Change
  164. 13:15of distance change in s over change in v
  165. 13:21s function
  166. 13:24y f
  167. 13:27x s Tion
  168. 13:31of so X1
  169. 13:36X2 s As A function of
  170. 13:40time As A function of
  171. 13:46time
  172. 13:52for s
  173. 13:55T1 S2
  174. 14:02t111
  175. 14:05t21 2 s
  176. 14:162
  177. 14:1810
  178. 14:19S2
  179. 14:22s22
  180. 14:254 4
  181. 14:304
  182. 14:344age velocity dur This is time interval
  183. 14:39average velocity equal to s of T2 min s
  184. 14:44of T1 over T2 -
  185. 14:52T1 S2 43 43 s 10 over T2
  186. 15:104age vity
  187. 15:12V
  188. 15:18per V for this time
  189. 15:23Inter per
  190. 15:29avage rate Of Change rate Of
  191. 15:32Change
  192. 15:37apply y fx
  193. 15:41Tion
  194. 15:45function Find the average rate Of Change
  195. 15:48of the function f over the interval 0
  196. 15:53x9
  197. 15:55num function
  198. 16:07rate Of Change for equ change in y over
  199. 16:10change in X change in y
  200. 16:14fx2
  201. 16:16fx1 over X2 -
  202. 16:20X1 X2
  203. 16:23X1 0 X1
  204. 16:259 x 2
  205. 16:30X1 X1 0
  206. 16:330 x
  207. 16:40077 f
  208. 16:43X1 f X1 0 0 f 0
  209. 16:487 7 fx2
  210. 16:579 3
  211. 17:053 x 3 3
  212. 17:10fx1 7 over 9-0 this
  213. 17:199 avage rate Of Change Tion
  214. 17:38instant rate Of
  215. 17:41Change rate Of
  216. 17:44Change avage rate Of Change instant rate
  217. 17:48Of Change
  218. 17:52instant
  219. 17:56so X1 X2
  220. 17:59instant f at aular moment Point
  221. 18:04f instant Of Change is the change at the
  222. 18:08particular moment instant rate Of
  223. 18:12Change instant rate Of Change equ
  224. 18:16average rate Of Change form change in y
  225. 18:19over change in
  226. 18:21X change
  227. 18:24y y2 - y1 over X2 - X1 this equ to y2 f
  228. 18:33X2 y1 f
  229. 18:36X1 over X2 -
  230. 18:44X1 average rate Of Change instant rate
  231. 18:47Of Change
  232. 18:51for foc change in
  233. 18:55X 0 so
  234. 18:59change in X Gets Closer and Closer to 0
  235. 19:03change diff
  236. 19:050 X1
  237. 19:10X2 average rate Of Change instant rate
  238. 19:13Of
  239. 19:14Change aage rate Of Change is the rate
  240. 19:17Of Change over the entire per over the
  241. 19:20interval over the interval ab over the
  242. 19:23interval X1 and
  243. 19:25X2 avage rate Of Change instant rate Of
  244. 19:29Change is the change at a certain per of
  245. 19:32time so instant rate Of Change
  246. 19:36specific X1
  247. 19:42X2 X2
  248. 19:44X1
  249. 19:460
  250. 19:480 change in X change in
  251. 19:52X h h
  252. 19:55so 0
  253. 19:59change x h h
  254. 20:020
  255. 20:08SPF the instant rate Of Change of a
  256. 20:11function f f function y equ fx
  257. 20:17function function instant rate Of Change
  258. 20:20of this funtion at x x
  259. 20:29change in y over change x H over h
  260. 20:36h h appr to
  261. 20:390 instant rate Of Change form
  262. 20:44example instant rate Of Change equ
  263. 20:53to instant rate Of Change
  264. 21:00instant rate Of Change change in y over
  265. 21:04H change in y change y equ
  266. 21:11fx2 f
  267. 21:14X1
  268. 21:16form
  269. 21:18fx2
  270. 21:20fx1 over
  271. 21:24H X1 x2m
  272. 21:30X1
  273. 21:32X2
  274. 21:33x
  275. 21:37x change
  276. 21:39x X2 - X1 which h
  277. 21:48h X2
  278. 21:53X1
  279. 21:55X2 X1 + h
  280. 22:02X2 X1 X1 X1 X2
  281. 22:09x
  282. 22:12fx2 X1 + H f X1
  283. 22:18x over h a instan rate Of Change instant
  284. 22:25rate Of Change of Any
  285. 22:27function
  286. 22:32[Muzică]
  287. 22:36amp evaluate the instant rate Of Change
  288. 22:39of a function f of x equ x+ 9 at x min 3
  289. 22:46solution instant rate Of
  290. 22:50Change equ to CH y over CH x CH x
  291. 22:58y over H
  292. 23:02instant change in
  293. 23:05y
  294. 23:08instant at a particular Point a c this
  295. 23:12equ form f of x not + H min f of x not
  296. 23:20over
  297. 23:22h
  298. 23:25function
  299. 23:26function fx x 2x +
  300. 23:339 f x + h
  301. 23:37fx
  302. 23:39x
  303. 23:41x
  304. 23:453
  305. 23:46f x not + h a fx - 3 + hx
  306. 23:59x not + h x + H 3 +
  307. 24:06h 2
  308. 24:07- 3 + h +
  309. 24:139
  310. 24:15fx
  311. 24:18fxx
  312. 24:213 f x which F 3
  313. 24:282 - 3 + 9
  314. 24:3837
  315. 24:4227
  316. 24:4427 the instant rate Of
  317. 24:47Change fx not + h
  318. 24:52- 3+
  319. 24:56h s+
  320. 25:0127 fx 27 over
  321. 25:06H - 3 +
  322. 25:10H
  323. 25:12perfect a +
  324. 25:17b a + 2ab + b
  325. 25:282 times of min 3
  326. 25:32s+ 2 times of min 3 times h + h
  327. 25:39+ + 9 -
  328. 25:4327 over
  329. 25:46h
  330. 25:49h
  331. 25:56= h + 2 h + 9 - 27 over
  332. 26:06h 9 8
  333. 26:0997 -
  334. 26:1227 - 12 2 of H so this 2 H - 12 h h
  335. 26:25h h 2 H
  336. 26:29- over
  337. 26:32h
  338. 26:34h exactly 0 as H appr to
  339. 26:390 exactly
  340. 26:450 2 H -
  341. 26:4912
  342. 26:50h
  343. 26:540 this 2 0
  344. 26:580 0
  345. 27:0112
  346. 27:0312
  347. 27:05Ev the instant rate Of Change of the
  348. 27:08function 2x + 9 at x 3 function instant
  349. 27:13rate Of Change at x
  350. 27:183
  351. 27:20- instant Of
  352. 27:27Change
  353. 27:40ect instant Of Change concept
  354. 27:45apply
  355. 27:46vity V equ velocity rate Of Change of
  356. 27:51displacement at time change of
  357. 27:54displacement ching over ching
  358. 28:05oc instant rate Of Change con
  359. 28:10apply at a part
  360. 28:14time change
  361. 28:17diff 0
  362. 28:230 rate Of Change of displacement at a
  363. 28:27part
  364. 28:32velocity
  365. 28:34b
  366. 28:48back BT
  367. 29:01BT
  368. 29:08b instant rate Of Change
  369. 29:13apply b
  370. 29:15bom polynomial function f of inad of x
  371. 29:24var BT b b
  372. 29:28as function of
  373. 29:30time b Culture b polynomial
  374. 29:35function f of
  375. 29:38repr instant rate Of Change at a
  376. 29:41particular
  377. 29:43time at a part
  378. 29:53time instant rate Of Change concept
  379. 29:58of
  380. 29:59time f oft equal to 71 tq +
  381. 30:0663 What is the instant rate of growth of
  382. 30:10the Culture When t equal To Two
  383. 30:15minutes
  384. 30:19Fun
  385. 30:21kog BTG
  386. 30:32fg rate of
  387. 30:36gr mg
  388. 30:44perg mg per
  389. 30:48Sot p
  390. 30:51example example the biomass of a
  391. 30:54bacterial Culture in milligrams As A
  392. 30:57function of time in minut is given by f
  393. 31:00of t To 71 t cu plus 63 What is the
  394. 31:05instant rate of Grow of the Culture When
  395. 31:08t equ To 2 minutes
  396. 31:11sol What is the instant rate of growth
  397. 31:15of the Culture When t equals To 2
  398. 31:19minut function f of t function of
  399. 31:24time
  400. 31:2671 tq
  401. 31:36x
  402. 31:42not 2 minut instant rate Of
  403. 31:46Change equal to change in y over change
  404. 31:55x x
  405. 31:57by changing t so change in y y
  406. 32:02function change y over change t equ
  407. 32:06change y
  408. 32:10instant f x not +
  409. 32:13h x t not + H f
  410. 32:21xx over H as H approach to0 instant
  411. 32:28H
  412. 32:310 instant rate Of Change so f t not + H
  413. 32:38f t
  414. 32:41not
  415. 32:43f t not +
  416. 32:46H
  417. 32:48not h h Tion function
  418. 32:56aq +
  419. 33:00+
  420. 33:03H 71
  421. 33:06+ H Power of 3+
  422. 33:1163 f of
  423. 33:19t
  424. 33:22times t not of 3+
  425. 33:31so instant rate Of
  426. 33:33Change 71
  427. 33:36+ H Power of 3+
  428. 33:4063
  429. 33:42min f t not f
  430. 33:46t 71 times t not Cube + 63 over
  431. 33:54h t not + H Power of 3
  432. 33:58t not + H Power
  433. 34:023 pas
  434. 34:09trle 3 + 3t H + 3 h +
  435. 34:25h 71 Times t +
  436. 34:303 H + 3 h+
  437. 34:40H
  438. 34:4371
  439. 34:49plus over
  440. 34:52H this equ
  441. 34:59t
  442. 35:02q t
  443. 35:04not
  444. 35:13t t
  445. 35:17q t not
  446. 35:24q
  447. 35:263 h
  448. 35:30t times H
  449. 35:34+ 3 times not
  450. 35:40H not h +
  451. 35:46+
  452. 35:486 +
  453. 35:5063
  454. 35:536 H 3 H 3
  455. 36:00H 3 over
  456. 36:03h h
  457. 36:07h h
  458. 36:10times t
  459. 36:13not
  460. 36:16+ not H + 71 h
  461. 36:23h
  462. 36:25h h
  463. 36:29this
  464. 36:31equ not
  465. 36:34+ not H + 71
  466. 36:42h h
  467. 36:510
  468. 36:532+ 2 Tim
  469. 36:5871
  470. 36:590
  471. 37:020 0 0
  472. 37:110
  473. 37:198
  474. 37:21b As A function of
  475. 37:39per ID
  476. 38:00the rate of Grow of the biomass of a
  477. 38:03bacterial Culture When t equ
  478. 38:082g
  479. 38:10per
  480. 38:15b 2
  481. 38:22Tion function f of t as a function of
  482. 38:26time 71 tq +
  483. 38:3863
  484. 38:478g Grant of curve and rate Of
  485. 38:56Change y

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