Section 6'4 Second Fundamental Theorem of Calculus

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Section 6.4Second Fundamental Theorem of Calculus
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The 1st Fundamental Theorem of Calculus

• If f is continuous on the interval a,b, and
  f(t) F(t), then
• Using this theorem lets calculate

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• Now calculate
• Notice any pattern? What can you say about
• assuming F(x) f(x)

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The 2nd Fundamental Theorem of Calculus

• If f is continuous on an interval, and if a is
  any number in that interval, then the function F
  defined as followsis an antiderivative of f.
• What we have done in this case is created a
  function for F for functions, f that have
  difficult/impossible antiderivatives
  (analytically speaking)

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• An example is the function sin(x)/x
• It is not possible to analytically find its
  antiderivative
• Therefore we define the function known as the
  sine-integral to be
• This way we actually have a function that accepts
  inputs, x and returns outputs, Si(x)
• Scientists and engineers use Si all the time
• Optics, harmonic motion and oscillations, etc.

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Example

• Write an expression for g(x) with the given
  properties

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Example

• Consider the following

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Example

• Find the following derivatives