Calculus: An Historical Approach by William McGowen Priestley (auth.)

By William McGowen Priestley (auth.)

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Example text

B) is irrational. Hint. (a) First note that J8 = ~ = ,j4 . j2. j2 is known to be irrational. JI8 8. (In this question, you are not asked to give rigorous justification Jor your answers, only to test your intuition about the notion oJ area. , blown up by a factor of 3). For instance, a square of size 2 by 2 becomes a square of size 6 by 6. By what factor is the area of each figure blown up? ) areas, are blown up by a factor of 9. (b) Suppose, as in part (a), that each length is blown up by a factor of r.

4. What is Limg at -2? Answer: Limg does not exist at -2, beeause gIs) does not exist when s is elose to - 2, giving 9 no elue as to a purpose at - 2. 5. What is Lim 9 at 1202? 6. What is g(1202)? Answer: Sinee the funetion 9 is not allowed to aet at the point 1202, g(1202) does not exist. 7. What is g(O)? 8. What is Lim 9 at O? 9. 1 ? Answer : It does not exist. 10. 1? §8. " "That is doubtful," replied Lim, "for you are stililikely to confuse purpose with action. " said Lim, "A right answer, but tor a wrong reason.

The importance of this interplay between algebra and geometry was first seen by two seventeenth-century Frenchmen, Pierre de Fermat and Rene Descartes. It was they who developed analytie geometry, the name given to the study of this interplay, whose goal is the attainment of a synthesis of algebra and geometry. D. 1637, when Descartes wrote La Geometrie, it has been common knowledge that curves can have equations and that equations determine curves. Why did the Greeks fail to utilize this means of approaching problems in geometry?

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