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**Unformatted text preview: **Math 1314 Day 8 The FUNDAMENTAL THEOREM OF CALCULUS (Think this might be important??) Suppose ( ) f x is a continuous function on the interval [a, b]. Then ( ) ( ) ( ) b a f x dx F b F a =- where F is any antiderivative of f . Theres a lot to this theorem. An informal proof can be found on page 986 in Section 14.4 of the Text. But I wont be provin it here. Well use this notation: Heres a very basic example: Example 1 : ( 29 3 1 2 3 x dx- + What happens to the C??? ( ) b a f x dx = Example 2 : Evaluate: - + 3 1 2 ) 7 4 3 ( dx x x Example 3 : Evaluate: 4 2 1 1 6 dx x x - Example 4 : Evaluate: +- 5 2 2 6 4 2 dx x x x Example 5 : Evaluate: dx x e x +- 3 ) 1 ( Example 6 : Evaluate: ( 29 ( 29 dx x x 1 3 2 2-- Example 7 : Find the area of the region under the graph of 2 3 ) ( x x x f- = over the interval [0, 3]. We will need to use substitution to evaluate some problems: Example 8 : Evaluate ( 29 - 3 5 2 3 4 dx x x Example 9 : Evaluate 1 2 3 dx e x x Example 10 : Evaluate + 2 1 3 2 6 3 dx x x Applications Example 11 : A company purchases a new machine for which the rate of depreciation is...

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