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

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

1. Replace x with x+h2. Simplify numerator3. Factor h out of numerator4. Divide by h5. Plug in 0 for h6. 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