Augustin-Louis Cauchy and Derivatives

Warm-Up  

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

History 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.

Teachings 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.

Works 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.

Contributions 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 to Calculus 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

The Derivative  

Definition of Derivative  

http://www.clas.ucsb.edu/staff/lee/secant,%20tangent,%20and%20derivatives.htm

Steps Replace x with x+h Simplify numerator Factor h out of numerator Divide by h Plug in 0 for h Evaluate slope

Example 1    

Pascal’s Triangle  

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Practice  

Answers  

Recap  

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