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  1. Chapter 6 Class 12 Application of Derivatives
  2. Serial order wise

Transcript

Example 41 An Apache helicopter of enemy is flying along the curve given by ๐‘ฆ= ๐‘ฅ^2 + 7. A soldier, placed at (3, 7), wants to shoot down the helicopter when it is nearest to him. Find the nearest distance. The curve is given as y = x2 + 7 Let helicopter be at point (x, y) = (x, x2 + 7) Let d be the distance between helicopter soldier at (3, 7) d = โˆš((๐‘ฅ2โˆ’๐‘ฅ1)2+(๐‘ฆ2โˆ’๐‘ฆ1)2) d = โˆš((๐‘ฅโˆ’3)2+(ใ€–(๐‘ฅใ€—^2+7)โˆ’7)2) d = โˆš((๐‘ฅโˆ’3)2+(๐‘ฅ)4) We need to find nearest distance i.e. minimum value of d Let f(x) = d2 f(x) = (x โˆ’ 3)2 + x4 When f(x) is minimum, d is minimum Finding fโ€™(x) fโ€™(x) = 2(x โˆ’ 3) + 4x3 = 2x โˆ’ 6 + 4x3 = 4x3 โˆ’ 2x โˆ’ 6 (To make calculation easy) Factorizing fโ€™(x) fโ€™(1) = 4(1)3 โˆ’ 2(1) โˆ’ 6 = 4 + 2 โˆ’ 6 = 0 Hence, (x โ€“ 1) is a factor of 4x3 โˆ’ 2x โˆ’ 6 Thus, fโ€™(x) = (x โˆ’ 1) (4x2 + 4x + 6) Hence fโ€™ (x) = 0 gives x โˆ’ 1 = 0 x = 1 2x2 + 2x + 3 = 0 x = (โˆ’2 ยฑ โˆš(4 โˆ’ 4(2)(3)))/4 x = (โˆ’2 ยฑ โˆš(4 โˆ’ 24))/4 x = (โˆ’2 ยฑ โˆš(โˆ’20))/4 This is not possible as there are no real roots. Hence there is only one point x = 1 This is either the maxima or minima. Hence we find fโ€(x) fโ€™โ€™ (x) = (4x3 + 2x โˆ’ 6)โ€™ fโ€™โ€™(x) = 12x2 + 2 Finding value at x = 1,. fโ€™โ€™(1) = 12(1)2 + 2 = 12 + 2 = 14 Since fโ€™โ€™(x) > 0 โˆด x = 1 is the minima. The value of f(1) is f(1) = (1 โˆ’ 3)2 + 14 = (โˆ’2)2 + 1 = 4 + 1 = 5 Hence, minimum distance between soldier & Helicopter d = โˆš(๐‘“(1)) d = โˆš๐Ÿ“

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Davneet Singh
Davneet Singh is a graduate from Indian Institute of Technology, Kanpur. He has been teaching from the past 9 years. He provides courses for Maths and Science at Teachoo.