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Unformatted text preview: Lecture 13 - Antiderivatives and the Definite Integral 13.1 Antiderivatives and Indefinite Integrals Definition 13.1 If f and g are two functions defined on an interval I , then we say that f is an antiderivative of g if f ( x ) = g ( x ) for all x in I . Example: Let f ( x ) = 6 e 3 x , and let g 1 ( x ) = 2 e 3 x , g 2 ( x ) = 2 e 3 x + 37, and g 3 ( x ) = 2 e 3 x- 12. Then g 1 ,g 2 , and g 3 are all antiderivatives of f : g 1 ( x ) = 2 e 3 x · 3 = 6 e 3 x g 2 ( x ) = 2 e 3 x · 3 = 6 e 3 x g 3 ( x ) = 2 e 3 x · 3 = 6 e 3 x Note that we can always change the constant and we still have an antiderivative. The notation we use for this is the following: Z f ( x ) d x = g ( x ) + C if and only if g ( x ) = f ( x ) The expression R f ( x ) d x is called an indefinite integral , and C is an arbitrary constant called the constant of integration . So the preceding example could be written as Z 6 e 3 x d x = 2 e 3 x + C Example: A firm knows that the marginal cost function at a production level of x units is M ( x ) = x 2- 2 x + 10. Suppose that the fixed initial cost is $35,000. Find the cost function. solution: Remember, marginal cost is the derivative of the cost function. So M ( x ) = C ( x ). Hence to find C ( x ) we should find the antiderivative of M ( x ). So we calculate C ( x ) = Z M ( x ) d x = Z ( x 2- 2 x + 10) d x For simple functions such as this one can “hunt around” for the antiderivative. In this case,For simple functions such as this one can “hunt around” for the antiderivative....
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