436FinalF04 - b Show that the sum of ALL these 6th roots is...

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MATH 436 - Final Exam       NAME: _______________________   SHOW ALL WORK 1)  a) State the Generalized  Cauchy  Integral formula  for finding  the kth derivative of the  real-valued  function f(x) at x = a.  b) Use the Cauchy  Integral formula  to find f ) 6 / ( ) 3 ( π for f(x) = xcosx.
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2) Use the Midpoint Riemann sum, the Trapezoidal Rule and Simpson’s Rule to approximate the  integral  - - 2 2 4 5 . 1 ) 735 cos(. dx x e x accurate to 6 decimal places.  Which is the best of these 3  methods and why do you believe it is the best?
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3) a) Find ALL of the 6th roots of z = - 9 + 4 I.
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Unformatted text preview: b) Show that the sum of ALL these 6th roots is 0. 4) Let y = f(x) = x on the interval [0, 2]. Give the even extension of this function to the entire interval [-2, 2] and give this extension’s infinite order Fourier series. Now use Parseval’s identity on this even extension to show that 90 1 4 1 4 π = ∑ ∞ = N N . 5) Use the Method of Steepest Descent to solve the system for its first quadrant solution (do one iteration) (x - 1)+ (y +1)= 25 and (x - 3)+ (y +2)= 36....
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