By Tracy Kompelien
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Title: 2-D Shapes Are at the back of the Drapes!
Author: Kompelien, Tracy
Publisher: Abdo Group
Publication Date: 2006/09/01
Number of Pages: 24
Binding style: LIBRARY
Library of Congress: 2006012570
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Extra info for 2-D Shapes Are Behind the Drapes!
Thus 3 À jbj > 3 À 1 ¼ 2; Again, we use the Transitive Rule. and we can then deduce from the previous chain of inequalities that & j3 À bj > 2, as desired. Remarks 1. The results of Example 1 can also be stated in the form: (a) j3 þ a3j 4, for jaj 1; (b) j3 À bj > 2, for jbj < 1. 2. The reverse implications 3 þ a3 4 ) jaj 1 and j 3 À bj > 2 ) j bj < 1 are FALSE. For example, try putting a ¼ À32 and b ¼ À2! Problem 1 (a) jaj 1 2 Use the Triangle Inequality to prove that: 3 ; (b) jbj < 1 ) b3 À 1 > 7.
This assumption is P(k). This multiplication is valid since ð1 þ xÞ ! 0. ð1 þ xÞkþ1 ! ð1 þ xÞð1 þ kxÞ ¼ 1 þ ðk þ 1Þx þ kx2 ! 1 þ ðk þ 1Þx: Thus, we have ð1 þ xÞkþ1 ! 1 þ ðk þ 1Þx; in other words the statement P(k þ 1) holds. So, P(k) true for some k ! 1 ) P(k þ 1) true. We decrease the expression if we omit the final nonnegative term. It follows, by the Principle of Mathematical Induction, that ð1 þ xÞn ! & 1 þ nx, for x ! À1, n ! 1: 1 Problem 9 By applying Bernoulli’s Inequality with x ¼ Àð2nÞ , prove 1 1 that 2n !
3 we have taken for granted the existence of rational powers and their properties, without giving formal definitions. pﬃﬃﬃ How can we supply these definitions? For example, how can we define 2 as a decimal? Consider the set È É E ¼ x 2 Q : x > 0; x2 < 2 : 29 The Least Upper Bound Property of R is an example of an existence theorem, one which asserts that a real number exists having a certain property. Analysis contains many such results which depend on the Least Upper Bound Property of R. While these results are often very general, and their proofs elegant, they do not always provide the most efficient methods of calculating good approximate values for the numbers in question.
2-D Shapes Are Behind the Drapes! by Tracy Kompelien