Ex 6.1, 11 - A particle moves along the curve 6y = x3 + 2 - Finding rate of change

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  1. Chapter 6 Class 12 Application of Derivatives
  2. Serial order wise
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Ex 6.1,11 A particle moves along the curve 6𝑦 = 𝑥3 +2. Find the points on the curve at which the y-coordinate is changing 8 times as fast as the 𝑥−coordinate. Given that A particular Moves along the curve 6𝑦 = 𝑥3 + 2 We need to find points on the curve at which 𝑦 coordinate is changing 8 times as fast as the 𝑥 –coordinate i.e. we need to find 𝑥,𝑦﷯ for which 𝑑𝑦﷮𝑑𝑡﷯=8 𝑑𝑥﷮𝑑𝑡﷯ 6𝑦 = 𝑥3 +2 Diff Both Side w.r.t 𝑡 𝑑 6𝑦﷯﷮𝑑𝑡﷯= 𝑑 𝑥﷮3﷯+2﷯﷮𝑑𝑡﷯ 6 𝑑𝑦﷮𝑑𝑡﷯= 𝑑 𝑥﷮3﷯﷯﷮𝑑𝑡﷯+ 𝑑 2﷯﷮𝑑𝑡﷯ 6 𝑑𝑦﷮𝑑𝑡﷯= 𝑑 𝑥﷮3﷯﷯﷮𝑑𝑡﷯ × 𝑑𝑥﷮𝑑𝑥﷯+0 6 𝑑𝑦﷮𝑑𝑡﷯= 𝑑 𝑥﷮3﷯﷯﷮𝑑𝑥﷯ × 𝑑𝑥﷮𝑑𝑡﷯ 6 𝑑𝑦﷮𝑑𝑡﷯=3 𝑥﷮2﷯ . 𝑑𝑥﷮𝑑𝑡﷯ 𝑑𝑦﷮𝑑𝑡﷯= 3 𝑥﷮2﷯﷮6﷯ 𝑑𝑥﷮𝑑𝑡﷯ 𝑑𝑦﷮𝑑𝑡﷯= 𝑥﷮2﷯﷮2﷯ 𝑑𝑥﷮𝑑𝑡﷯ We need to find point for which 𝑑𝑦﷮𝑑𝑡﷯=8 𝑑𝑥﷮𝑑𝑡﷯ Putting 𝑑𝑦﷮𝑑𝑡﷯= 𝑥﷮2﷯﷮2﷯ . 𝑑𝑥﷮𝑑𝑡﷯ 𝑥﷮2﷯﷮2﷯ . 𝑑𝑥﷮𝑑𝑡﷯=8 𝑑𝑥﷮𝑑𝑡﷯ 𝑥﷮2﷯﷮2﷯=8 𝑥﷮2﷯=8 ×2 𝑥﷮2﷯=16 𝑥=± ﷮16﷯ 𝑥=± 4 𝑥=4 , −4 Putting values of 𝑥 in (1) 6𝑦 = 𝑥3 +2 When 𝑥=4 6𝑦= 4﷯﷮3﷯+2 6𝑦=64+2 6𝑦=66 𝑦= 66﷮6﷯ 𝑦=11 Points is 4 , 11﷯ When 𝑥=− 4 6𝑦= − 4﷯﷮3﷯+2 6𝑦=− 64+2 6𝑦=− 62 𝑦= − 62﷮6﷯ 𝑦= − 31﷮3﷯ Points is − 4, − 31﷮3﷯﷯ Hence Required points on the curve are 𝟒 , 𝟏𝟏﷯ & − 𝟒, − 𝟑𝟏﷮𝟑﷯﷯

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