1Review•Limit laws•Derivative rules2AntiderivativesRecall when we were looking at the motion of a particle, we were given a function for its position at a giventime. We could figure out how fast it was moving by looking at the derivative. What if we had the velocity,but wanted the position? Well, velocity is the derivative of the position, so if we could find a function thathas a derivative of the velocity we’re looking for that’d be the position.Definition 1.A functionFis an antiderivative offin an interval [a, b] ifF(x) =f(x) for allxin [a, b].Example 2.1.f(x) =x2.It’s not hard to see that an antiderivative isF(x) =13x3.But what aboutG(x) =13x3+ 1?A consequence of the mean value theorem is that if two functions have the same derivative on an interval,then they differ by a constant. So ifGandFare both antiderivatives off, thenF(x) =G(x) +C.Theorem 1.IfFis an antiderivative offon an interval [a, b], then the most general antiderivative offonIis:F(x) +CWhereCis an arbitrary constant.