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**Unformatted text preview: **Lecture 30 1 Lecture 30: (Sec. 6.1) Antiderivatives and the Rules of Integration Up to now we have looked at Differential Calcu- lus : We now consider Integral Calculus : ex. The velocity of a car (in ft/sec) after starting from rest is given by v ( t ) = 2 √ t , 0 ≤ t ≤ 30. Find the position function of the car. How far has the car traveled after 16 seconds? Lecture 30 2 ex. Given the derivative, find the function: 1) dy dx = − 5 2) dy dx = 4 x 3 Lecture 30 3 Def. A function F is an antiderivative of func- tion f on an interval I if for each x in that interval, NOTE: The operation of finding the antiderivative is called ex. Show that F ( x ) = ln( x 2 +3) is an antiderivative of f ( x ) = 2 x x 2 + 3 . Then find another antiderivative of f . Lecture 30 4 Theorem Let F be an antiderivative of a function f . Then if G is another antiderivative of f (that is, G ′ ( x ) = F ′ ( x ) = ), then Lecture 30 5 ex. Find all antiderivatives of f ( x ) = − 2 x and graph....

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