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Chap5_Sec3

# Chap5_Sec3 - FTC If f is the function whose graph is shown...

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If f is the function whose graph is shown and , find the values of: g (0), g (1), g (2), g (3), g (4), and g (5). square4 Then, sketch a rough graph of g . We use these values to sketch the graph of g. square4 Notice that, because f ( t ) is positive for t < 3, we keep adding area for t < 3. square4 So, g is increasing up to x = 3, where it attains a maximum value. square4 For x > 3, g decreases because f ( t ) is negative . Example 1 (5.3) 0 ( ) ( ) x g x f t dt = FTC

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Find the derivative of the function square4 As is continuous, the FTC1 gives: Example 2 (5.3) 2 0 ( ) 1 x g x t dt = + 2 ( ) 1 f t t = + 2 '( ) 1 g x x = + FTC1
Find Here, we have to be careful to use the Chain Rule in conjunction with the FTC1. Let u = x 4 . Then, 4 1 sec x d t dt dx Example 4 (5.3) FTC1 ( ) 4 1 1 1 4 3 sec sec (Chain Rule) sec (FTC1) sec( ) 4 = = = = x u u d d t dt t dt dx dx d du sect dt du dx du u dx x x

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FTC2 Evaluate the integral square4 The function f ( x ) = e x is continuous everywhere and we know that an antiderivative is F ( x ) = e x .
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