Chapter_11.5 - y x dx dy y x dx dy xy y x dx dy xy dx dy-=...

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Chapter 11.5 Page 621. Given 0 2 3 2 = - + y x , find dx dy . x dx dy x y y x y x y x 6 6 ' 0 ' 6 0 0 ' 6 )' 0 ( )' 2 ( )' ( )' 3 ( 2 - = - = = + = - + = - + The above process is called implicit differentiation. Page 623. Example 1: Given 0 25 2 2 = - + y x , find y ’ and the slope of the graph at x = 3. The given equation is a circle with radius 5; and there are two points on the graph that have x coordinate equal to 3. Find y ’. )' 0 ( )' 25 ( )' ( )' ( 2 2 = - + y x dx dy y y y y 2 ' 2 )' ( 2 = × = y x y x dx dy x dx dy y dx dy y x - = - = - = = - + 2 2 2 2 0 0 2 2 At 3 = x , 4 16 0 25 9 0 25 2 2 2 2 ± = = = - + = - + y y y y x (3,0) (5,0) 1
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In this case y ’ has a unique value at each point: y x y - = ' at ) 4 , 3 ( gives the slope , 4 3 ' - = y ; The point-slope form of the tangent line: ) 3 ( 4 3 4 - - = - x y . y x y - = ' at ) 4 , 3 ( - gives the slope , 4 3 4 3 ' = - - = y ; The point-slope form of the tangent line: ) 3 ( 4 3 4 - = + x y . Page 624. Example 2: Find the equation(s) of the tangent line(s) to the graph 0 1 2 2 = + + - x xy y at the points where x = 1. 1 st . Find y ’: 2 xy is a product of two functions, dx dy xy y y x y x xy 2 )' ( )' ( )' ( 2 2 2 2 + = + = x dx dy xy y dx dy x dx dy xy y dx dy x xy y 2 2 0 2 ) 2 ( )' 0 ( )' 1 ( )' ( )' ( )' ( 2 2 2 2 - = - - = + + - = + + - ) 2 1 ( 2 2 ) 2 1 ( 2 2 2 2 2 xy
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Unformatted text preview: y x dx dy y x dx dy xy y x dx dy xy dx dy-+-= ⇒ +-=-⇒ +-=-(3,4) (3,- 4) 2 2 nd . Find the y-values that corresponded to x = 1: ) 1 )( 2 ( 2 1 1 1 2 2 2 2 = +-⇒--= ⇒ = + +-⇒ = + +-y y y y y y x xy y . So 1 , 2-= = y y ; ) 2 , 1 ( and ) 1 , 1 (-are the required points. 3 rd . The slope at ) 2 , 1 ( : 3 2 3 2 ) 2 )( 1 ( 2 1 ) 2 ( ) 1 ( 2 ) 2 1 ( 2 2 2-=-=-+-=-+-= xy y x dx dy ; The point-slope form of the tangent line: ) 1 ( 3 2 2--=-x y . 4 th . The slope at ) 1 , 1 (-: 3 1 3 1 ) 1 )( 1 ( 2 1 ) 1 ( ) 1 ( 2 ) 2 1 ( 2 2 2-=-=---+-=-+-= xy y x dx dy ; The point-slope form of the tangent line: ) 1 ( 3 1 1--= + x y . 3...
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Chapter_11.5 - y x dx dy y x dx dy xy y x dx dy xy dx dy-=...

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