82 another proof of the evaluation theorem a let a x

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82. Another proof of the Evaluation Theorem a. Let a = x 0 6 x 1 6 x 2 g 6 x n = b be any partition of 3 a , b 4 and let F be any antiderivative of ƒ. Show that F ( b ) - F ( a ) = a n i = 1 3 F ( x i ) - F ( x i - 1 ) 4 . b. Apply the Mean Value Theorem to each term to show that F ( x i ) - F ( x i - 1 ) = ƒ( c i )( x i - x i - 1 ) for some c i in the interval ( x i - 1 , x i ). Then show that F ( b ) - F ( a ) is a Riemann sum for ƒ on 3 a , b 4 . F ( x ) = 1 a ƒ( t ) dt for the specified function ƒ and interval 3 a , b 4 . Use a CAS to perform the following steps and answer the questions posed. a. Plot the functions ƒ and F together over 3 a , b 4 b. Solve the equation F ( x ) = 0. What can you see to be true about the graphs of ƒ and F at points where F ( x ) = 0? Is your obser- vation borne out by Part 1 of the Fundamental Theorem coupled with information provided by the first derivative? Explain your answer. c. Over what intervals (approximately) is the function F increasing and decreasing? What is true about ƒ over those intervals? , c. From part (b) and the definition of the definite integral, show that 83. Suppose that ƒ is the differentiable function shown in the accom- panying graph and that the position at time t (sec) of a particle moving along a coordinate axis is s = L t 0 ƒ( x ) dx meters. Use the graph to answer the following questions. Give reasons for your answers. y x 0 1 2 3 4 5 6 7 8 9 1 2 3 4 1 2 (1, 1) (2, 2) (5, 2) (3, 3) y = f ( x ) a. What is the particle’s velocity at time t = 5? b. Is the acceleration of the particle at time t = 5 positive, or negative? c. What is the particle’s position at time t = 3? d. At what time during the first 9 sec does s have its largest value? e. Approximately when is the acceleration zero? f. When is the particle moving toward the origin? Away from the origin? g. On which side of the origin does the particle lie at time t = 9? 84. Find lim x Sq 1 2 x L x 1 dt 2 t . COMPUTER EXPLORATIONS In Exercises 85–88, let F ( x ) = 1 a ƒ( t ) dt for the specified function ƒ and interval 3 a , b 4 . Use a CAS to perform the following steps and answer the questions posed. a. Plot the functions ƒ and F together over 3 a , b 4 b. Solve the equation F ( x ) = 0. What can you see to be true about the graphs of ƒ and F at points where F ( x ) = 0? Is your obser- vation borne out by Part 1 of the Fundamental Theorem coupled with information provided by the first derivative? Explain your answer. c. Over what intervals (approximately) is the function F increasing and decreasing? What is true about ƒ over those intervals? x .

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