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Ex 6.3, 17 - Find points on y = x3 at which slope of tangent - Ex 6.3

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  1. Chapter 6 Class 12 Application of Derivatives
  2. Serial order wise
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Ex 6.3,17 Find the points on the curve 𝑦=𝑥3 at which the slope of the tangent is equal to the y-coordinate of the point Let the Point be ﷐ℎ , 𝑘﷯ on the Curve 𝑦=𝑥3 Where Slope of tangent at ﷐ℎ , 𝑘﷯=𝑦−𝑐𝑜𝑜𝑟𝑑𝑖𝑛𝑎𝑡𝑒 𝑜𝑓 ﷐ℎ, 𝑘﷯ i.e. ﷐﷐𝑑𝑦﷮𝑑𝑥﷯│﷮﷐ℎ, 𝑘﷯﷯=𝑘 Given 𝑦=﷐𝑥﷮3﷯ Differentiating w.r.t.𝑥 ﷐𝑑𝑦﷮𝑑𝑥﷯=3﷐𝑥﷮2﷯ ∴ Slope of tangent at ﷐ℎ , 𝑘﷯ is ﷐﷐𝑑𝑦﷮𝑑𝑥﷯│﷮﷐ℎ, 𝑘﷯﷯=3﷐ℎ﷮2﷯ From (1) ﷐﷐𝑑𝑦﷮𝑑𝑥﷯│﷮﷐ℎ, 𝑘﷯﷯=𝑘 3﷐ℎ﷮2﷯=𝑘 Also Point ﷐ℎ , 𝑘﷯ is on the Curve 𝑦=﷐𝑥﷮3﷯ ⇒ Point ﷐ℎ , 𝑘﷯ must Satisfy the Equation of Curve i.e. 𝑘=﷐ℎ﷮3﷯ Now our equations are 3﷐ℎ﷮2﷯=𝑘 …(1) & 𝑘=﷐ℎ﷮3﷯ …(2) Putting Value of 𝑘=3﷐ℎ﷮2﷯ in (3) 3﷐ℎ﷮2﷯=﷐ℎ﷮3﷯ ﷐ℎ﷮3﷯−3﷐ℎ﷮2﷯=0 ﷐ℎ﷮2﷯﷐ℎ−3﷯=0 Hence the Required Point on the Curve are ﷐𝟎 , 𝟎﷯ & ﷐𝟑 , 𝟐𝟕﷯

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