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lecture29 - Math 121 Lecture 29 Derivatives Definition...

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Math 121 Lecture 29 Derivatives: Definition. Techniques. Applications.
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! Antidifferentiation ! Areas and Riemann Sums ! Definite Integrals and the Fundamental Theorem ! Areas in the xy -Plane ! Applications of the Definite Integral § 6.1 Antidifferentiation
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! Antidifferentiation ! Finding Antiderivatives ! Theorems of Antidifferentiation ! The Indefinite Integral ! Rules of Integration ! Antiderivatives in Application Definition Example Antidifferentiation : The process of determining f ( x ) given f ! ( x ) If , then
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Find all antiderivatives of the given function. The derivative of x 9 is exactly 9 x 8 . Therefore, x 9 is an antiderivative of 9 x 8 . So is x 9 + 5 and x 9 -17.2. It turns out that all antiderivatives of f ( x ) are of the form x 9 + C (where C is any constant) as we will see next.
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Determine the following. Using the rules of indefinite integrals, we have Find the function f ( x ) for which and f (1) = 3. The unknown function f ( x ) is an antiderivative of . One antiderivative is . Therefore, by Theorem I, Now, we want the function f ( x ) for which f (1) = 3. So, we must use that information in our antiderivative to determine C . This is done below.
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