April 25

Today in class:

  • Two versions of the Fundamental Theorem of Calculus were stated:
    • 1. If you can guess an antiderivative you can calculate the integral (integral here is the sum or area where antiderivative is the function)
      • If $F'(x) = f(x)$
      • Then: $\int\limits_a^b {f(x) dx} = F'(a) - F'(b)$
    • 2. You can find an antiderivative by computing integrals:
      • Let $F(x) = \int \limits_a^x {f(t) dt}$
      • Then: $F'(x) = f(x)$
  • We discussed Leibniz's FTC using the diagram in our text on pg. 133
  • Leibniz's work is best represented by the first version of the FTC stated above but also implies the second.
  • Leibniz's diagram demonstrates the relationship between areas and tangents
  • Conclusion from the diagram is that we can find the area of an unknown curve if we know the function whose tangents give the height at any point of the unknown curve
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