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Pre-Calc Homework Solutions 79

# Pre-Calc Homework Solutions 79 - Section 3.3 d(uv dx 79...

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24. (a) At x 5 2, } d d x } ( uv ) 5 u (2) v 9 (2) 1 v (2) u 9 (2) 5 (3)(2) 1 (1)( 2 4) 5 2 (b) At x 5 2, } d d x } 1 } u v } 2 5 5 } (1)( 2 4 ( ) 1 2 ) 2 (3)(2) } 5 2 10 (c) At x 5 2, } d d x } 1 } u v } 2 5 5 } (3)(2) 2 (3) ( 2 1)( 2 4) } 5 } 1 9 0 } (d) Use the result from part (a) for } d d x } ( uv ). At x 5 2, } d d x } (3 u 2 2 v 1 2 uv ) 5 3 u 9 (2) 2 2 v 9 (2) 1 2 } d d x } ( uv ) 5 3( 2 4) 2 2(2) 1 2(2) 5 2 12 25. y 9 ( x ) 5 2 x 1 5 y 9 (3) 5 2(3) 1 5 5 11 The slope is 11. (iii) 26. The given equation is equivalent to y 5 } 3 2 } x 1 6, so the slope is } 3 2 } . (iii) 27. y 9 ( x ) 5 3 x 2 2 3 y 9 (2) 5 3(2) 2 2 3 5 9 The tangent line has slope 9, so the perpendicular line has slope 2 } 1 9 } and passes through (2, 3). y 5 2 } 1 9 } ( x 2 2) 1 3 y 5 2 } 1 9 } x 1 } 2 9 9 } Graphical support: [ 2 4.7, 4.7] by [ 2 2.1, 4.1] 28. y 9 ( x ) 5 3 x 2 1 1 The slope is 4 when 3 x 2 1 1 5 4, at x 5 6 1. The tangent at x 5 2 1 has slope 4 and passes through ( 2 1, 2 2), so its equation is y 5 4( x 1 1) 2 2, or y 5 4 x 1 2. The tangent at x 5 1 has slope 4 and passes through (1, 2), so its equation is y 5 4( x 2 1) 1 2, or y 5 4 x 2 2. The smallest slope occurs when 3 x 2 1 1 is minimized, so the smallest slope is 1 and occurs at x 5 0. Graphical support: [ 2 4.7, 4.7] by [ 2 3.1, 3.1] 29. y 9 ( x ) 5 6 x 2 2 6 x 2 12 5 6( x 2 2 x 2 2) 5 6( x 1 1)( x 2 2) The tangent is parallel to the x -axis when y 9 5 0, at x 5 2 1 and at x 5 2. Since y ( 2 1) 5 27 and y (2) 5 0, the two points where this occurs are ( 2 1, 27) and (2, 0). Graphical support: [ 2 3, 3] by [ 2 10, 30] 30. y 9 ( x ) 5 3 x 2 y 9
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