# Prelim 2 Short Answers - Prelim 2 Math 2930 Spring 2011...

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Prelim 2 Math 2930 Spring 2011 show work, 5 problems, no calculators 1) (a) (4 points) Write your name and section number on your exam booklet. In parts (b) and (c) we consider the function of period 2 π with f ( x ) = x on the interval - π 2 , π 2 , and f ( x ) = 0 for other points in [ - π, π ]. (b) (8 points) Find the value f (6). (c) (8 points) Find the coefficient b 3 , of sin(3 x ) in the Fourier series of f . 2) The weight of a 1000 kg mass compresses a spring 9.8 millimeter. (a) (5 points) Tell why the spring constant k = 10 6 N/m. (b) (5 points) Write the Newton F = ma equation for frictionless un- forced oscillations of this system, in terms of the vertical displacement from equilibrium y ( t ), and verify that it can be rewritten as y + 1000 y = 0. (c) (5 points) Write the general solution to the equation of part (b). (d) (5 points) Will the mass oscillate slower, or faster, than 29.30 cycles per second? 3) (20 points) Solve the diff eq x + 4 x + 3 x = 5 sin( t ) + 7 sin(2 t ) for x ( t ). 4) Consider the partial differential equation u t = 5 u xx - u x , where u is a function of x and t and subscripts denote partial derivatives.

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