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tut09 - The University of Sydney Math1003 Integral Calculus...

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The University of Sydney Math1003 Integral Calculus and Modelling Semester 2 Exercises for Week 9 2011 Assumed Knowledge Integration techniques. Objectives (8a) To be able to write a rational expression as a sum of partial fractions. (8b ) To be able to solve a separable equation using partial fractions. Preparatory Questions 1. Use partial fractions to express the following as a sum of terms: ( i ) 1 ( x + 1)( x - 2) ( ii ) 7 2 x 2 + 5 x - 3 ( iii ) 1 ( x + a )( x + b ) ( iv ) x x - 5 Practice Questions 2. Evaluate the integrals: ( i ) integraldisplay 1 ( x + 1)( x - 2) dx ( ii ) integraldisplay 7 2 x 2 + 5 x - 3 dx ( iii ) integraldisplay 1 ( x + a )( x + b ) dx ( iv ) integraldisplay x x - 5 dx 3. (Suitable for group work and discussion.) A trout farmer believes that, if she doesn’t remove any fish from a certain tank, the population of fish is modelled by the differential equation dP dt = 2 P - 0 . 01 P 2 , where P is the number of fish after t years. ( i ) Suppose the farmer initially stocks the tank with 50 fish.
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