Example 22 - Use differential to approximate (25)1/3 - Examples

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  1. Chapter 6 Class 12 Application of Derivatives
  2. Serial order wise
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Example 22 Use differential to approximate ﷐(25)﷮﷐1﷮3﷯﷯ Let 𝑦 =﷐𝑥﷮﷐1﷮3﷯﷯ Let 𝑥=27 and △𝑥=−2 Since 𝑦 =﷐𝑥﷮﷐1﷮3﷯﷯ ﷐𝑑𝑥﷮𝑑𝑦﷯=﷐𝑑﷐﷐𝑥﷮﷐1﷮3﷯﷯﷯﷮𝑑𝑥﷯ ﷐𝑑𝑥﷮𝑑𝑦﷯=﷐1﷮3﷯ ﷐𝑥﷮﷐1﷮3﷯ − 1﷯ ﷐𝑑𝑥﷮𝑑𝑦﷯=﷐1﷮3﷯ ﷐𝑥﷮﷐−2﷮3﷯﷯ ﷐𝑑𝑥﷮𝑑𝑦﷯=﷐1﷮3﷐𝑥﷮﷐2﷮3﷯﷯﷯ Now, ∆𝑦=﷐𝑑𝑦﷮𝑑𝑥﷯∆𝑥 ∆𝑦= ﷐1﷮3﷐𝑥﷮﷐2﷮3﷯﷯﷯∆𝑥 Putting values ∆𝑦= ﷐1﷮3﷐﷐27﷯﷮﷐2﷮3﷯﷯﷯﷐−2﷯ ∆𝑦= ﷐−2﷮3 ×﷐﷐﷐3﷮3﷯﷯﷮﷐2﷮3﷯﷯﷯ ∆𝑦= ﷐−2﷮3 × ﷐3﷮2﷯﷯ ∆𝑦= ﷐−2﷮27﷯ ∆𝑦=−0.074 Also, ∆𝑦=𝑓﷐𝑥+∆𝑥﷯ −𝑓﷐𝑥﷯ ∆𝑦=﷐﷐𝑥+∆𝑥﷯﷮﷐1﷮3﷯﷯−﷐𝑥﷮﷐1﷮3﷯﷯ Putting values −0.074=﷐﷐27−2﷯﷮﷐1﷮3﷯﷯−﷐27﷮﷐1﷮3﷯﷯ −0.074=﷐﷐25﷯﷮﷐1﷮3﷯﷯−﷐3﷮3 × ﷐1﷮3﷯﷯ −0.074=﷐﷐25﷯﷮﷐1﷮3﷯﷯−3 ﷐﷐25﷯﷮﷐1﷮3﷯﷯=−0.074+3 ﷐﷐25﷯﷮﷐1﷮3﷯﷯=2.926 Hence, approximate value of ﷐﷐25﷯﷮﷐1﷮3﷯﷯ is 𝟐.𝟗𝟐𝟔

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