hw1_sol - t ∈ (0 , 1 / 3). Solution Note that F ( x ) ∈...

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ISyE 2027 Homework 1: Solutions due on Wednesday, May 26 Include all intermediate steps of the computations in your answers Derivatives 1. Find d dx ( x 9 - x 8 ), d dx x 8 ( x 2 - 1) and d dx e - x 2 Solution d dx ( x 9 - x 8 ) = 9 x 8 - 8 x 7 , d dx x 8 ( x 2 - 1) = 10 x 9 - 8 x 7 , d dx e - x 2 = e - x 2 ( - 2 x ) 2. Let g be a differentiable function of R with derivative g 0 . Express the derivative of the function f ( x ) = x 2 g ( x ) in terms of g 0 . Solution f 0 ( x ) = ( x 2 ) 0 g ( x ) + g 0 ( x )( x 2 ) = 2 xg ( x ) + g 0 ( x ) x 2 Integration 3. Compute R 2 0 3 x 4 dx , R 0 (1 / 3) e - (1 / 3) x dx and R 0 (1 / 3) xe - (1 / 3) x dx Solution Z 2 0 3 x 4 dx = 3 Z 2 0 x 4 dx = 3 5 x 5 | 0 = 96 5 Z 0 (1 / 3) e - (1 / 3) x dx = Z 0 e - y dy = - e - y | 0 = 1 Z 0 (1 / 3) xe - (1 / 3) x dx = 3 Z 0 ye - y dy = 3[ - ye - y | 0 + Z 0 e - y dy ] = 3[0 + 1] = 3 4. Compute R -∞ F ( x ) dx , where F ( a ) = 0 , a < 0 a 2 , 0 a < 2 1 , 2 a < 3 0 , a 3 Solution Z -∞ F ( x ) dx = Z 2 0 F ( x ) dx + Z 3 2 F ( x ) dx = Z 2 0 x 2 + Z 3 2 1 dx = 2 5. Compute R 3 x =0 R x y =0 3 xy dydx and R x =0 R y =0 e 2 x + y dydx . 1
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Solution Z 3 x =0 Z x y =0 3 xy dydx = Z 3 x =0 3 x Z x y =0 y dydx = Z 3 x =0 3 x 3 / 2 dx = 3 x 4 / 8 | 3 0 = 243 / 8 Z x =0 Z y =0 e 2 x + y dydx = Z x =0 e 2 x dx Z y =0 e y dy = Solving For Unknowns 6. Let the function F be given by F ( x ) = 0 , x < 0 x 4 3 , 0 x < 1 1 , x 3 Find a function φ such that F ( φ ( t )) = t for
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Unformatted text preview: t ∈ (0 , 1 / 3). Solution Note that F ( x ) ∈ (0 , 1 / 3), for 0 ≤ x < 1, we use the expression F ( x ) = x 4 / 3. Hence, t = F ( φ ( t )) = φ ( t ) 4 / 3 Then φ ( t ) = (3 t ) 1 / 4 Series 7. Compute ∑ ∞ n =1 (1 / 5) n and ∑ ∞ n =1 (1 / 5) n /n !. Solution ∞ X n =1 (1 / 5) = 1 5-1 = 1 / 4 Note that 1 + ∞ X n =1 (1 / 5) n /n ! = ∞ X n =0 (1 / 5) n /n ! = e 1 / 5 Then ∑ ∞ n =1 (1 / 5) n /n ! = e 1 / 5-1 8. Determine whether ∑ ∞ n =1 1 /n < ∞ and whether ∑ ∞ n =1 1 /n 2 < ∞ Solution ∞ X n =1 1 /n = ∞ , ∞ X n =1 1 /n 2 < ∞ 2...
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This note was uploaded on 01/23/2011 for the course ISYE 2027 taught by Professor Zahrn during the Spring '08 term at Georgia Institute of Technology.

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hw1_sol - t ∈ (0 , 1 / 3). Solution Note that F ( x ) ∈...

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