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Section 6.4 Second Fundamental Theorem of Calculus

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If f is continuous on the interval [a,b], and f(t) = F’(t), then Using this theorem let’s calculate The 1st Fundamental Theorem of Calculus

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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 follows is 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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Find the following derivatives Example

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