# tut07 - 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 7 2011 Assumed Knowledge Proportionality and inverse proportionality. Integration techniques. Taylor series expansion. Objectives (6a) Given a verbal description of a simple model, to be able to express it as a mathematical equation. (6b ) To be able to recognise an ordinary diFerential equation. (6c) To be able to sketch the solution curves for a ±rst-order diFerential equation from its direction ±eld. Preparatory Questions 1. ( i ) The diFerential equation dy dx = f ( x ) has a direction ±eld given by the diagram below. On the direction ±eld draw the graphs of two solutions of dy/dx = f ( x ), where one solution y = g ( x ) passes through the point (0 , 1) and the other solution y = h ( x ) satis±es the equation h (1) = 0. ( ii ) Do the graphs of y = g ( x ) and y = h ( x ) intersect? If not, why not? 1 2 3 1 1 2 3 1

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Practice Questions 2. Heat tends to fow From hot bodies to cold bodies. Newton observed that the rate at which temperature rises or Falls within a body is proportional to the temperature di±erence between the body and its surroundings.
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## tut07 - The University of Sydney Math1003 Integral Calculus...

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