09SpringHwk01_Soln - Math 3301 Homework Set 1 Solutions 10...

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Math 3301 Homework Set 1 – Solutions 10 Points 1. (2 pts) From the differential equation we can see that the derivative will be zero at : 2,2,4 y = − . A sketch of the direction field and a few solutions is shown to the right. From this we can see that the long term behaviors are, ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) 0 4 0 4 4 2 0 4 2 0 2 2 y y t y y t y y t y y t > → ∞ = = < < ≤ − → − 4. (3 pts) ( ) ( ) ( ) ( ) ( ) ( ) 2 3 3 6 3 2 2 3 2 2 3 3 2 3 3 1 3 2 6 2 3 1 3 ln 3 3 3 t dt t t t t t t t t t t t y y t t t t y dt t dt t y t c y t t t ct µ − − ′− + = = = = = + = + e e e e e e e e e e ( ) ( ) ( ) ( ) 6 3 3 2 6 2 3 3 4 4 1 4 3 3 3 3 0 1 t t y c c y t t t t = = + = = + e e e e e 5. (2 pts) We know from Calc I that relative extrema occur at critical points and critical points are those points where the derivative is zero or doesn’t exist. However, we’ve been told that the derivative exists and is continuous everywhere so that means that at the critical point that gives the relative maximum,
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