solCal3samplemid2

solCal3samplemid2 - SOLUTIONS TO CALCULUS III SAMPLE...

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Unformatted text preview: SOLUTIONS TO CALCULUS III SAMPLE MIDTERM 2 Problem 1. (20 points) (a) Find the indicated power using De Moivres Theorem: (2 3 + 2 i ) 5 ; (1- i ) 8 (b) Find the indicated roots and sketch the roots in the complex plane: The fifth roots of 32; The cube roots of 1 + i Solution: (a) (2 3 + 2 i ) 5 = 4 e i/ 6 5 = 4 5 e i 5 6 = 1024 - 3 2 + 1 2 i =- 512 3 + 512 i (1- i ) 8 = 2 e- i/ 4 8 = ( 2) 8 e- i 2 = 16 1 = 16 (b) Fifth roots of 32: write 32 = 2 5 e i , then the five fifth roots of 32 are 2, 2 e i 2 / 5 , 2 e i 4 / 5 , 2 e i 6 / 5 , and 2 e i 8 / 5 . Cube roots of 1+i: write 1 + i = 2 e i/ 4 , then the three cube roots are 6 2 e i/ 12 , 6 2 e i ( / 12+2 / 3) = 6 2 e i 3 / 4 , and 6 2 e i ( / 12+4 / 3) = 6 2 e i (17 / 12) Problem 2. (20 points) (a) Solve the initial-value problem 2 y 00 + 5 y + 3 y = 0 , y (0) = 3 , y (0) =- 4 (b) Solve the boundary-value problem y 00 + 4 y + 13 y = 0...
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solCal3samplemid2 - SOLUTIONS TO CALCULUS III SAMPLE...

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