Find the expected value variance and standard

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6. Find the expected value, variance, and standard deviation for each of the following probability density functions: (a) f ( x ) = 12 x 2 - 12 x 3 on [0 , 1]. (b) g ( x ) = 4 - 2 x on [1 , 2]. (c) h ( x ) = 3 /x 4 on [1 , ). (d) i ( x ) = 5 /x 6 on [1 , ). (e) j ( x ) = 3 x 4 on [1 , ). (f) k ( x ) = 5 2 x 3 / 2 on [0 , 1]. (g) l ( x ) = 1 4 e x/ 4 on [0 , ). 7. Find each indicated Taylor polynomial at the given value. (a) f ( x ) = ln(1 + 2 x ); p 4 ( x ) at 0. (b) g ( x ) = e 2 x ; p 3 ( x ) at 1. (c) h ( x ) = ln(2 - x ); p 3 ( x ) at 1. (d) i ( x ) = (1 - x ) 4 ; p 3 ( x ) at 0. (e) j ( x ) = x + 4; p 2 ( x ) at 0. Use this to approximate 5. (f) k ( x ) = x 3 ; p 3 ( x ) at 4. Use this to approximate (3 . 9) 3 . (g) l ( x ) = x 2 e x ; p 2 ( x ) at 0. (h) m ( x ) = 4 e x - 2 sin x ; p 3 ( x ) at 0. Use this to approximate m (0 . 1). 8. (a) Use a third degree Taylor polynomial to approximate integraldisplay 0 . 1 0 1 1 + x dx .
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(b) Use the fourth degree Taylor polynomial for ln(1 + x ) to approximate ln(1 . 1). (c) Use the fourth degree Taylor polynomial for e x to approximate e 0 . 1 . (d) Use the third degree Taylor polynomial of x to approximate 15 . 9. (e) Use the fourth degree Taylor polynomial for x 1 / 3 to approximate 3 8 . 1. (f) Use the third degree Taylor polynomial for ln( x + 1) at x = 0 to estimate the value of integraldisplay 1 / 2 0 ln( x + 1) dx . Compare your result with the exact value found using integration by parts.
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