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

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Math 222: Midterm 2 Review Packet Complex Numbers 1. Show that sin(3 θ ) = 3 cos 2 ( θ ) sin( θ ) - sin 3 ( θ ) , and derive a similar formula for cos(3 θ ) . 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 π 4 0 sin(3 x ) cos( x ) dx c) Z π - π sin 2 (3 x ) cos 3 (2 x ) dx Differential Equations Instructions. Find the solutions for the following differential equations and IVP’s. Solve the word problems as well. 1. y 0 = x 2 y - y y + 1 , y (3) = - 1 . 2. y 0 + y 4 = 1 . 3. sin( x ) + yy 0 = 0 , y (0) = - 2 . 4. x sec( x ) y + cos( x ) y 0 = 0 . 5. y 0 - 7 y = sin(2 x ) . 6. p 0 - 1 t p = t 2 + 3 t - 2 . 7. y 0 + 2 x y = x, y (1) = 0 .
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