13112an5.9-11 - c Kendra Kilmer March 1 2012 Section 5.4...

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c circlecopyrt Kendra Kilmer March 1, 2012 Section 5.4 - The Fundamental Theorem of Calculus Example 1: Let g ( x )= integraldisplay x 0 f ( t ) dt where f is the function whose graph is shown. 1 2 3 4 5 1 2 -1 -2 a) Evaluate g ( 0 ) , g ( 2 ) , and g ( 4 ) . b) On what interval(s) is g increasing? decreasing? c) Where does g have an absolute maximum value? Absolute minimum value? Example 2: If g ( x )= integraldisplay x 1 t 3 dt , a) Find a formula for g ( x ) . b) What does your answer represent? c) Find g ( x ) . 9
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c circlecopyrt Kendra Kilmer March 1, 2012 Fundamental Theorem of Calculus Suppose f is continuous on [ a , b ] . 1. If g ( x )= integraldisplay x a f ( t ) dt , then g is an antiderivative of f , that is g ( x )= f ( x ) for a < x < b . Alternate Notation: d dx integraldisplay x a f ( t ) dt = f ( x ) 2. integraldisplay b a f ( x ) dx = F ( b ) F ( a ) , where F is any antiderivative of f , that is F = f Example 3: Use Part I of the Fundamental Theorem of Calculus to find the derivative of the following functions.
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