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Unformatted text preview: Practice Problems 2 for MTH 131: Fall 2006 1). Evaluate the following limits: (i) lim x 1 1- x 2 1- x (ii) lim x x- sin( x ) x- tan( x ) 2). Solve the following initial value problem: dy dx = 2 x ( y + 1) , y (0) = 0 3). Compute the indefinite integrals of these functions: (i) x- 5 / 2 (ii) x ( x- 1)(2 x +3) (iii) x x 2 + 2 x + 5 4). Solve the following initial value problem: dy dx = 2 xy 2 1 + x 2 , y ( x ) = y Find the interval of existence for the cases where y > 0 and y < 0. 5). By changing variables to y = u 1- n , show that the ODE du dt = a ( t ) u + g ( t ) u n is transformed into a linear ODE for y . 6). Here is a modified harvesting equation: N ( t ) = (1- N ) N- h (1 + N 3 ) , N (0) = N The harvesting rate depends on N , and increases as N increases ( h 0 is constant.) a) To find the equilibrium solutions, you must find the zeroes of the function f ( u ) = (1- u ) u- h (1 + u 3 ). Rather than looking for exact values, argue as follows. Sketch the two graphsas follows....
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