solCal3samplemid2

solCal3samplemid2 - SOLUTIONS TO CALCULUS III SAMPLE...

Info iconThis preview shows pages 1–2. Sign up to view the full content.

View Full Document Right Arrow Icon

Info iconThis preview has intentionally blurred sections. Sign up to view the full version.

View Full DocumentRight Arrow Icon
This is the end of the preview. Sign up to access the rest of the document.

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...
View Full Document

Page1 / 3

solCal3samplemid2 - SOLUTIONS TO CALCULUS III SAMPLE...

This preview shows document pages 1 - 2. Sign up to view the full document.

View Full Document Right Arrow Icon
Ask a homework question - tutors are online