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Growth Authors: Bill Davis, Horacio Porta and Jerry Uhl ©1996-2007 Publisher: Math Everywhere, Inc. Version 6.0 1.07 The Race Track Principle LITERACY · L.1) Here are plots of x and Sin @ x D for 0 § x § 2: 0.5 1.0 1.5 2.0 x 0.5 1.0 1.5 2.0 Use the Race Track Principle to explain why x ¥ Sin @ x D for all x ¥ 0. Remember that - 1 § Cos @ x D § 1 no matter what x is. · L.2) Here are plots of e x and the line function 1 + x for 0 § x § 2: 0.5 1.0 1.5 2.0 x 2 3 4 5 6 7 Use the Race Track Principle to explain why e x ¥ 1 + x for all x ¥ 0. Remember that e x ¥ 1 for x ¥ 0. · L.3) Put f @ x D = Sin @ x D 2 and g @ x D = I 1 - Cos @ x D 2 M and look at f @ 0 D and g @ 0 D : Clear @ f, g, x D ; f @ x _ D = Sin @ x D 2 ; g @ x _ D = 1 - Cos @ x D 2 ; 8 f @ 0 D , g @ 0 D< 8 0, 0 < Now look at f £ @ x D and g £ @ x D : 8 f' @ x D , g' @ x D< 8 2Cos @ x D Sin @ x D , 2Cos @ x D Sin @ x D< What version of the Race Track Principle confirms that Sin @ x D 2 = f @ x D = g @ x D = 1 - Cos @ x D 2 for x ¥ 0? · L.4) Here are plots of the solutions of the differential equations y £ @ x D = 0.6 y @ x D with y @ 0 D = 2 and y £ @ x D = 1.2 y @ x D I 1 - y @ x D 100 M with y @ 0 D = 2: 1
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