HW 7 Solutions - Mathematics 408K Homework 7 Gagan Tara...

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408 K: Homework 7 Gagan Tara Nanda 05 th October, 2004 HW 7 : # s 3 , 11 , 17 3. We are given the expression 3 x 2 +3 y 2 =4 , so we shall f rst implicitly di f erentiate with respect to x .W ege t 6 x +6 yy 0 =0 .( 1 ) S incewewantto f nd y 00 ,wew i l limp l ic it lyd i f erentiate expression (1) , rather than solve for y 0 and then di f erentiate y 0 . The reason is that it is easier and quicker to di f erentiate (1) andthensolvefor y 00 ,rather than solve (1) for y 0 andthend i f erentiate to get y 00 . So implicitly di f erentiating (1) with respect to x , we get, by the Product Rule, 6+ ¡ 6 y 0 ¢¡ y 0 ¢ +(6 y ) ¡ y 00 ¢ =0 . In other words, 6 yy 00 = 6 6 ¡ y 0 ¢ 2 yy 00 = 1 ¡ y 0 ¢ 2 .( 2 ) Going back to (1) , we realize that we actually need to express y 0 in terms of x and y ,soastosubst itute that expression in (2) .Sof rom (1) ,weget y 0 = x y .( 3 ) Putting (3) into (2) gives yy 00 = 1 μ x y 2
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This note was uploaded on 02/28/2010 for the course M 56200 taught by Professor Radin during the Fall '09 term at University of Texas.

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HW 7 Solutions - Mathematics 408K Homework 7 Gagan Tara...

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