Check Full Chapter Explained - Continuity and Differentiability - Application of Derivatives (AOD) Class 12






Last updated at Jan. 7, 2020 by Teachoo
Check Full Chapter Explained - Continuity and Differentiability - Application of Derivatives (AOD) Class 12
Transcript
Ex 6.3, 15 Find the equation of the tangent line to the curve ๐ฆ=๐ฅ2 โ2๐ฅ+7 which is : (a) parallel to the line 2๐ฅโ๐ฆ+9=0 We know that Slope of tangent is ๐๐ฆ/๐๐ฅ ๐ฆ=๐ฅ2 โ2๐ฅ+7 Differentiating w.r.t.๐ฅ ๐๐ฆ/๐๐ฅ=2๐ฅโ2 Finding Slope of line 2๐ฅโ๐ฆ+9=0 2๐ฅโ๐ฆ+9=0 ๐ฆ=2๐ฅ+9 ๐ฆ=2๐ฅ+9 The Above Equation is of form ๐ฆ=๐๐ฅ+๐ where m is Slope of line Hence, Slope of line 2๐ฅโ๐ฆ+9 is 2 Now, Given tangent is parallel to 2๐ฅโ๐ฆ+9=0 Slope of tangent = Slope of line 2๐ฅโ๐ฆ+9 = 0 ๐๐ฆ/๐๐ฅ=2 2๐ฅโ2=2 2(๐ฅโ1)=2 ๐ฅ=2 Finding y when ๐ฅ=2 , ๐ฆ=๐ฅ^2โ2๐ฅ+7= (2)^2โ2(2)+7=4โ4+7=7 We need to find Equation of tangent passes through (2, 7) & Slope is 2 Equation of tangent is (๐ฆโ7)=2(๐ฅโ2) ๐ฆโ7=2๐ฅโ4 ๐ฆโ2๐ฅโ7+4=0 ๐ฆโ2๐ฅโ3=0 Hence Required Equation of tangent is ๐โ๐๐โ๐=๐ We know that Equation of line at (๐ฅ1 , ๐ฆ1) & having Slope m is ๐ฆโ๐ฆ1=๐(๐ฅโ๐ฅ1) Ex 6.3,15 Find the equation of the tangent line to the curve ๐ฆ=๐ฅ2โ2๐ฅ+7 which is (b) perpendicular to the line 5๐ฆโ15๐ฅ=13 We know that Slope of tangent is ๐๐ฆ/๐๐ฅ ๐ฆ=๐ฅ2 โ2๐ฅ+7 Differentiating w.r.t.๐ฅ ๐๐ฆ/๐๐ฅ=2๐ฅโ2 Finding Slope of line 5๐ฆโ15๐ฅ=13 5๐ฆโ15๐ฅ=13 5๐ฆ=15๐ฅ+13 ๐ฆ=1/5 (15๐ฅ+13) ๐ฆ=15/5 ๐ฅ+13/5 ๐ฆ=3๐ฅ+13/5 Above Equation is of form ๐ฆ=๐๐ฅ+๐ , where m is Slope of a line โด Slope = 3 Now, Given tangent is perpendicular to 5๐ฆโ15๐ฅ=13 Slope of tangent ร Slope of line = โ1 ๐๐ฆ/๐๐ฅ ร 3=โ1 ๐๐ฆ/๐๐ฅ=(โ1)/( 3) 2๐ฅโ2=(โ1)/( 3) 2๐ฅ=(โ1)/( 3)+2 2๐ฅ=(โ1 + 6)/3 2๐ฅ=5/3 ๐ฅ=5/6 Finding y when ๐ฅ=5/6 ๐ฆ=๐ฅ^2โ2๐ฅ+7=(5/6)^2โ2(5/6)+7=25/36โ10/6+7=217/36 โด Point is (5/6 ,217/36) Equation of tangent passing through (5/6 ,217/36) & having Slope (โ1)/( 3) (๐ฆโ217/36)=(โ1)/( 3) (๐ฅโ5/6) (36๐ฆ โ217)/36=(โ1)/( 3) (๐ฅโ5/6) 36๐ฆ โ217=(โ36)/( 3) (๐ฅโ5/( 6)) 36๐ฆ โ217=โ12(๐ฅโ5/( 6)) 36๐ฆ โ217=โ12๐ฅ+(12 ร 5)/6 36๐ฆ โ217=โ12๐ฅ+10 36๐ฆ โ217=โ12๐ฅ+10 36๐ฆ+12๐ฅโ217โ10=0 ๐๐๐+๐๐๐โ๐๐๐=๐ is Required Equation of tangent We know that Equation of line at (๐ฅ1 , ๐ฆ1) & having Slope m is ๐ฆโ๐ฆ1=๐(๐ฅโ๐ฅ1)
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