MATH 191, Sections 1 and 3Calculus IFall 2007Section 4.9: Antiderivatives!For numerous reasons that will become evident next semester when you takeCalc II, the ability to workbackwards, obtaining a function by starting with itsderivative, isincrediblyimportant.Definition.A functionFis called anof the functionfonthe intervalIifF(x) =for allxinI.Examples.1. Find an antiderivative forxon (-∞,∞).2. Find an antiderivative forx2on (-∞,∞).3. Find an antiderivative for1xon (0,∞).4. Find an antiderivative for1xthat works both on (0,∞)andon (-∞,0).5. Find an antiderivative forxnon (0,∞), as long asn= 1.6. Find an antiderivative for sin(x) on (-∞,∞).Remember that if you’ve got two functionsfandgsatisfyingf(x) =g(x) foreveryxin an interval, the MVT says thatfandgdiffer by a constant on thatinterval. This establishes the
Theorem.IfFis an antiderivative forfon the intervalI, then anyotherantiderivativeGcan be obtained fromFby adding a constant:G(x) =F(x)+C.