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The First Fundamental Theorem of Calculus

The First Fundamental Theorem of Calculus. First FTOC. ** Notice!! You do not have to include a “C” when you integrate f(x). Example Evaluating an Integral. Example. The Derivative of an Integral. Now let’s see some more examples / practice of definite integral problems.

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The First Fundamental Theorem of Calculus

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  1. The First Fundamental Theorem of Calculus

  2. First FTOC ** Notice!! You do not have to include a “C” when you integrate f(x)

  3. Example Evaluating an Integral

  4. Example

  5. The Derivative of an Integral

  6. Now let’s see some more examples / practice of definite integral problems

  7. The Mean Value Theorem for Definite Integrals

  8. The Mean Value Theorem for Definite Integrals

  9. Mean Value Theorem (for definite integrals) If f is continuous on then at some point c in , The mean value theorem for definite integrals says that for a continuous function, at some point on the interval the actual value will equal the average value. p

  10. Average(Mean) Value

  11. Example Applying the Definition

  12. The average value of a function is the value that would give the same area if the function was a constant:

  13. Now let’s see some examples using the Mean Value Theorem for Integrals and to find the average value of a function

  14. The Second Fundamental Theorem of Calculus

  15. The Second Fundamental Theorem of Calculus

  16. The Derivative of an Integral

  17. 1. Derivative of an integral. Second Fundamental Theorem:

  18. Second Fundamental Theorem: 1. Derivative of an integral. 2. Derivative matches upper limit of integration.

  19. Scond Fundamental Theorem: 1. Derivative of an integral. 2. Derivative matches upper limit of integration. 3. Lower limit of integration is a constant.

  20. Second Fundamental Theorem: New variable. 1. Derivative of an integral. 2. Derivative matches upper limit of integration. 3. Lower limit of integration is a constant.

  21. Example

  22. Example Applying the Fundamental Theorem

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