Fundamental Theorem of Calculus - Introduction and Part 1 of the Theorem

in MES Science20 days ago (edited)

In this video I introduce one of the most important mathematical discoveries and in fact one of the most important discoveries in general that mankind has discovered, and that is the Fundamental Theorem of Calculus! I introduce the theorem by giving a brief history lesson on it and later illustrate how integral and differential calculus are linked with this Fundamental Theorem by showing that the two branches of calculus are inversely related; i.e. the derivative of the integral is the function it is integrating!


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Fundamental Theorem of Calculus - Introduction and Part 1 of the Theorem

Fundamental theorem of Calculus Part 1.png

Introduction

  • The Fundamental Theorem of Calculus is appropriately named because it forms a connection between the 2 branches of calculus: Differential and Integral Calculus.
  • Differential calculus arose from the tangent problem.
  • Integral calculus arose from the area problem.
  • Newton’s teacher at Cambridge University (UK), Isaac Barrow (1630-1677), discovered that integral and differential calculus are closely related, in fact he realized they were inverse processes.
  • The Fundamental Theorem of Calculus gives the precise inverse relationship between the derivative and the integral.
  • Newton and Leibniz exploited this relationship and used it to develop calculus into a systematic mathematical method.
    • In particular, they saw that the Fundamental Theorem enabled them to compute areas and integrals very easily without having to compute them as limits of sums as in my earlier videos.

Part 1 of the Theorem

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