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Unformatted text preview: 108 Section 3.7
x 2 xy ( 2y)(y) 4y 2 2y 2 y2 y2 y2 3y 2 y
7 , 3 41. Find the two points: The curve crosses the xaxis when y 0, so the equation becomes x 2 0x 0 7, or x 2 7. The solutions are x 7, so the points are ( 7, 0). Show tangents are parallel: x2 xy y2 7
d (7) dx ( 2y) 2 7 7 7 7 7 3 7 . 3 The points are 2 7 7 and 2 , 3 3 d 2 d d 2 (x ) (xy) (y ) dx dx dx dy dy 2x x (y)(1) 2y dx dx dy (x 2y) dx dy dx 2 7 7 2( 7 0 2(0) 7) 0 (2x
2x y x 2y Note that these are the same points that would be obtained by interchanging x and y in the solution to part (a). y) 43. First curve: 2x 2
d (2x 2) dx 3y 2 5
d (5) dx Slope at ( 7, 0): Slope at ( 7, 0): 2
0 2(0) 2 d (3y 2) dx dy 4x 6y dx dy dx 0
4x 6y 2x 3y The tangents at these points are parallel because they have the same slope. The common slope is 2. 42.
d 2 (x ) dx Second curve: y2 x3
d 3 x dx x2
d (xy) dx xy y2 7
d (7) dx d 2 (y ) dx 2x x dy dx (y)(1) (x 2y dy dx 0 (2x
2x y x 2y d 2 y dx dy 2y dx dy dx 3x 2
3x 2 2y 2 3 and respectively. At (1, 3 2 dy 2y) dx dy dx At (1, 1), the slopes are y) the slopes are
2 and 3 1), 3 respectively. In both cases, the 2 tangents are perpendicular. To graph the curves and normal lines, we may use the following parametric equations for (a) The tangent is parallel to the xaxis when
dy dx 2x y x 2y t 0, or y 2x. 2x for y in the original equation, we have :
5 cos t, y 2
3 2 First curve: x Second curve: x
2 2 Substituting x2 5 sin t 3 t ,y t 1 1 2t 2t x xy y (x)( 2x) ( 2x)2 x 2 2x 2 4x 2 3x 2 x 7 7 7 7 7 3 7 , 3 Tangents at (1, 1): x 1 3t, y x 1 2t, y 1 3t Tangents at (1, 1): x 1 3t, y x 1 2t, y 1 3t 2
7 . 3 The points are 7 7 ,2 and 3 3 (b) Since x and y are interchangeable in the original
dx can be obtained by interchanging x and y dy dy dx 2y x in the expression for . That is, . The dx dy x 2y dx tangent is parallel to the yaxis when 0, or dy equation, [ 2.4, 2.4] by [ 1.6, 1.6] x 2y. Substituting 2y for x in the original equation, we have: ...
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This note was uploaded on 10/05/2011 for the course MAC 1147 taught by Professor German during the Fall '08 term at University of Florida.
 Fall '08
 GERMAN

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