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Pre-Calc Homework Solutions 248

Pre-Calc Homework Solutions 248 - 248 Section 6.2 d[F(x dx...

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46. (a) } d d x } [ F ( x ) 1 C ] should equal f ( x ). (b) The slope field should help you visualize the solution curve y 5 F ( x ). (c) The graphs of y 1 5 F ( x ) and y 2 5 E x 0 f ( t ) dt should differ only by a vertical shift C . (d) A table of values for y 1 2 y 2 should show that y 1 2 y 2 5 C for any value of x in the appropriate domain. (e) The graph of f should be the same as the graph of NDER of F ( x ). (f) First, we need to find F ( x ). Let u 5 x 2 1 1, du 5 2 x dx . E } ˇ x 2 w x 1 w 1 w } dx 5 E } 1 2 } u 2 1/2 du 5 u 1/2 5 ˇ x 2 w 1 w 1 w 1 C Therefore, we may let F ( x ) 5 ˇ x 2 w 1 w 1 w . a) } d d x } ( ˇ x 2 w 1 w 1 w 1 C ) 5 } 2 ˇ x 1 2 w 1 w 1 w } (2 x ) 5 } ˇ x 2 w x 1 w 1 w } 5 f ( x ) b) [ 2 4, 4] by [ 2 3, 3] c) [ 2 4, 4] by [ 2 3, 3] d) e) [ 2 4, 4] by [ 2 3, 3] 47. Let u 5 x 4 1 9, du 5 4 x 3 dx . (a) E 1 0 } ˇ x x 3 4 w d 1 w x 9 w } 5 E 10 9 } 1 4 } u 2 1/2 du 5 } 1 2 } u 1/2 4 5 } 1 2 } ˇ 1 w 0 w 2 } 1 2 } ˇ 9 w 5 } 1 2 } ˇ 1 w 0 w 2 } 3 2 } < 0.081 (b) E } x 4 x 1 3 9 } dx 5 E } 1 4 } u 2 1/2 du 5 } 1 2 } u 1/2 1 C 5 } 1 2 } ˇ x 4 w 1 w 9 w 1 C E 1 0 } x 4 x 1 3 9 } dx 5 } 1 2 } ˇ x 4 w 1 w 9 w 4 5 } 1 2 } ˇ 1 w 0 w 2 } 1 2 } ˇ 9 w 5 } 1 2 } ˇ 1 w 0 w 2 } 3 2 } < 0.081 48. Let u 5 1 2 cos 3 x , du 5 3 sin 3 x dx . (a) E p /3 p /6 (1 2 cos 3 x ) sin 3 x dx 5 E 2 1 } 1 3 } u du 5 } 1 6 } u 2 4 5 } 1 6 } (2) 2 2 } 1 6 } (1) 2 5 } 1 2 } (b) E (1 2 cos 3 x ) sin 3 x dx 5 E } 1 3 } u du 5 } 1 6 } u 2 1 C 5 } 1 6 } (1 2 cos 3 x ) 2 1 C E p /3 p /6 (1 2 cos 3 x ) sin 3 x dx 5 } 1 6 } (1 2
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