Exam1_S2010 Solutions

# We are given a function describing the rate of change

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We are given a function describing the rate of change of the population of pumas, so to find the function describing the actual population r ( t ), we must take the integral of R ( t ): r ( t ) = Z R ( t ) dt = 500 Z te 0 . 08 t dt Use integration by parts. Choose u = t and dv = e 0 . 08 t dt . Then du = dt and v = e 0 . 08 t 0 . 08 . Then integration by parts gives: 500 Z te 0 . 08 t dt = 500 te 0 . 08 t 0 . 08 - Z e 0 . 08 t 0 . 08 dt A simple u -substitution in the second integral gives r ( t ) = 500 te 0 . 08 t 0 . 08 - e 0 . 08 t (0 . 08) 2 + C We are told that the initial population of pumas is 1000. That is, r (0) = 500 (0) e 0 . 08(0) 0 . 08 - e 0 . 08(0) (0 . 08) 2 + C = 1000 This reduces to - 500 (0 . 08) 2 + C = 1000 so C = 1000 + 500 (0 . 08) 2 Thus the population function is given by r ( t ) = 500 te 0 . 08 t 0 . 08 - e 0 . 08 t (0 . 08) 2 + 1000 + 500 (0 . 08) 2 and the population in year 6 is r (6) = 500 (6) e 0 . 08(6) 0 . 08 - e 0 . 08(6) (0 . 08) 2 + 1000 + 500 (0 . 08) 2 5

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9. Suppose a function f ( x ) has y -intercept (0 , - 8), and the tangent line to the graph of f ( x ) at x = 3 has the equation 6 x - 3 y = 12. Compute Z 3 0 xf 00 ( x ) dx . First do the indefinite integral, using integration by parts. Choose u = x and dv = f 00 ( x ) dx . Then du = dx and v = f 0 ( x ). Then integration by parts gives Z xf 00 ( x ) dx = xf 0 ( x ) - Z f 0 ( x ) dx. Evaluating the second integral, we get Z xf 00 ( x ) dx = xf 0 ( x ) - f ( x ) + C. Taking C = 0 to evaluate the definite integral, we get Z 3 0 xf 00 ( x ) dx = ( xf 0 ( x ) - f ( x )) 3 0 = 3 f 0 (3) - f (3) + f (0) . We must the rest of the information we know to find the values of f 0 (3) , f (3) and f (0). We know that the y -intercept of f is (0 , - 8), so f (0) = - 8. The tangent line to f at x = 3 has slope f 0 (3), so we put the equation given above in standard form, giving us y = 2 x - 4. The slope of this line is 2, which means f 0 (3) = 2. Finally, we know that the tangent line to f at x = 3 must touch the graph of f at x = 3, so plugging in
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