Review Packet - Math 222: Midterm 2 Review Packet Complex...

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Unformatted text preview: Math 222: Midterm 2 Review Packet Complex Numbers 1. Show that sin(3 ) = 3cos 2 ( )sin( )- sin 3 ( ) , and derive a similar formula for cos(3 ) . 2. Find all real and complex solutions to the fol- lowing polynomials: a) z 3- 6 z 2 + 11 z- 6 = 0 b) z 4- 9 z 2 + 20 = 0 c) z 4- 16 = 0 d) 3 z 6 = 10 z 3- 3 e) z 4 + 2 z 3- z- 2 3. Compute the derivatives of the following func- tions: a) f ( x ) = tan( x ) + i tan( x ) b) g ( x ) = sin( ix ) , where sin( z ) := e iz- e- iz 2 i for all complex numbers z. c) h ( x ) = e ik ( x ) , where k ( x ) is a differen- tiable function. 4. Compute the following integrals (using complex methods): a) Z cos 2 ( x ) dx b) Z 4 sin(3 x )cos( x ) dx c) Z - sin 2 (3 x )cos 3 (2 x ) dx Differential Equations Instructions. Find the solutions for the following differential equations and IVPs. Solve the word problems as well. 1. y = x 2 y- y y + 1 , y (3) =- 1 ....
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Review Packet - Math 222: Midterm 2 Review Packet Complex...

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