3.6 Prelim Answers - Math 106 Prelim 3 Answers R Durrett 1a...

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Math 106 Prelim 3 Answers April 24, 2007 R. Durrett 1a. ∂f ∂x = y 2 - 3 2 · 2 x ( x 2 + y 3 ) 1 / 2 - 9 ( x + y ) - x ( x + y ) 2 = 4 - 3 2 · 2 9 1 / 2 - 9 · 2 9 = 1 ∂f ∂y = 2 xy - 3 2 · 3 y 2 ( x 2 + y 3 ) 1 / 2 + 9 x ( x + y ) 2 = 4 - 3 2 · 12 9 1 / 2 + 9 · 1 9 = - 1 1b. ∂g ∂t = - 1 2 t - 3 / 2 e - x 2 / 4 t + t - 1 / 2 x 2 4 t 2 e - x 2 / 4 t ∂g ∂x = t - 1 / 2 - 2 x 4 t e - x 2 / 4 t 2 g ∂x 2 = t - 1 / 2 - 2 4 t e - x 2 / 4 t + t - 1 / 2 - 2 x 4 t 2 e - x 2 / 4 t ∂g ∂t - 2 g ∂x 2 = 0 2. a. Let u = ln x , du = dx/x . e 1 x (ln x ) 3 / 2 dx = 1 u - 3 / 2 du = - 2 u - 1 / 2 1 = 2 b. f ( x ) = ln x , g ( x ) = x - 2 , f ( x ) = 1 /x , g ( x ) = - x - 1 e 1 ln x x 2 dx = - x - 1 ln x e 1 + e 1 x - 2 dx = - 1 /e + ( - x - 1 ) e 1 = 1 - 2 /e 3. EX = 1 0 (5 / 2) x 5 / 2 dx = (5 / 7) x 7 / 2 1 0 = 5 / 7 E ( X 2 ) = 1 0 (5 / 2)
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Unformatted text preview: 1 = 5 / 9 Variance = (5 / 9)-(5 / 7) 2 = (245-225) / 441 = 20 / 441. 4. The volume is π µ 4 1 x 2 + 2 x 1 / 2 + 1 x dx . The integral is x 3 3 + 4 3 x 3 / 2 + ln x ´ ´ ´ ´ 4 1 = 64 3 + 32 3 + ln 4-1 3-4 3 = 91 3 + ln 4...
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