Augustin-Louis Cauchy and Derivatives

 Warm-Up

 Business – gain or loss in profit  Cars – figure out speed at a certain point  Health – growth in height over a period of time Motivation

 Born in Paris in 1789  Received early education from his father  He initially prepared himself to be a civil engineer  He entered the Ēcole Polytechnique in 1805, where he met Lagrange and Laplace who later persuaded him to study pure science. History

 He was a professor at Ēcole Polytechnique  His main job was teaching calculus and he was not satisfied with the way the foundations of the subject then stood.  His goal was to “do calculus right”  The result was one of the most famous textbooks in the history of mathematics, The Ēcole Polytechnique Course in Analysis. Teachings

 He can be ranked next to Euler in volume of output  He wrote extensively in both pure and applied math  several books  789 papers, some of which are very extensive works  Cauchy wrote such lengthy articles the Academy of Sciences quickly passed a rule limiting all papers to a maximum of four pages because the printing bill was high. Works

 Cauchy gave numerous contributions to advanced mathematics  researches in convergence and divergence of infinite series  real and complex function theory  differential equations  determinants  probability  mathematical physics Contributions

 Cauchy root test and Cauchy ratio test for convergence and divergence  Cauchy product of two given series  Basic concepts of limits and continuity  Definition of derivative  He used the notion of a limit as the basis for a rigorous foundation for calculus Contributions to Calculus

 The Derivative

 Definition of Derivative

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1.Replace x with x+h 2.Simplify numerator 3.Factor h out of numerator 4.Divide by h 5.Plug in 0 for h 6.Evaluate slope Steps

 Example 1

Pascal’s Triangle 

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 Practice

 Answers

 Recap

 Homework:  Pg. 324 #’s 1, 3, 6, 7, 11, 13, 17 Independent Practice