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Unformatted text preview: 1 Review Limit laws Derivative rules 2 Antiderivatives Recall when we were looking at the motion of a particle, we were given a function for its position at a given time. 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 that has a derivative of the velocity were looking for thatd be the position. Definition 1. A function F is an antiderivative of f in an interval [ a, b ] if F ( x ) = f ( x ) for all x in [ a, b ]. Example 2.1. f ( x ) = x 2 . Its not hard to see that an antiderivative is F ( x ) = 1 3 x 3 . But what about G ( x ) = 1 3 x 3 + 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 if G and F are both antiderivatives of f , then F ( x ) = G ( x ) + C ....
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