Stewart6e4_9 - MATH 191 Sections 1 and 3 Calculus I Fall 2007 Section 4.9 Antiderivatives For numerous reasons that will become evident next semester

# Stewart6e4_9 - MATH 191 Sections 1 and 3 Calculus I Fall...

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MATH 191, Sections 1 and 3 Calculus I Fall 2007 Section 4.9: Antiderivatives! For numerous reasons that will become evident next semester when you take Calc II, the ability to work backwards , obtaining a function by starting with its derivative, is incredibly important. Definition. A function F is called an of the function f on the interval I if F ( x ) = for all x in I . Examples. 1. Find an antiderivative for x on ( -∞ , ). 2. Find an antiderivative for x 2 on ( -∞ , ). 3. Find an antiderivative for 1 x on (0 , ). 4. Find an antiderivative for 1 x that works both on (0 , ) and on ( -∞ , 0). 5. Find an antiderivative for x n on (0 , ), as long as n = 1. 6. Find an antiderivative for sin( x ) on ( -∞ , ). Remember that if you’ve got two functions f and g satisfying f ( x ) = g ( x ) for every x in an interval, the MVT says that f and g differ by a constant on that interval. This establishes the
Theorem. If F is an antiderivative for f on the interval I , then any other antiderivative G can be obtained from F by adding a constant: G ( x ) = F ( x )+ C .