Explanation Given information: The derivative f ′ ( x ) is graphed in below figure. Concept used: Fundamental theorem of calculus states that if f ' ( t ) is continuous for a ≤ t ≤ b then ∫ a b f ' ( t ) d t = f ( b ) − f ( a ) . Calculation: To find the values of the function f at x = 1 , 2 , 3 , 4 , 5 , 6 . Divide the interval [ 0 , 6 ] as [ 0 , 1 ] , [ 1 , 2 ] , [ 2 , 3 ] , [ 3 , 4 ] , [ 4 , 5 ] and [ 5 , 6 ] . Use the Fundamental Theorem of Calculus in each sub interval to the graph of the function f ' and find the area under the curve in these each interval. Separate the graph of f ' into the intervals [ 0 , 1 ] , [ 1 , 2 ] , [ 2 , 3 ] , [ 3 , 4 ] , [ 4 , 5 ] and [ 5 , 6 ] as shown below. For the interval [ 0 , 1 ] , using fundamental theorem of Calculus, ∫ 0 1 f ' ( t ) d t = f ( 1 ) − f ( 0 ) f ( 1 ) = ∫ 0 1 f ' ( t ) d t + f ( 0 ) Here ∫ 0 1 f ' ( t ) d t represents the area under the curve f ' on [ 0 , 1 ] . In the interval [ 0 , 1 ] , the graph of f ' lies below the x-axis. So, the area under the curve is, ∫ 0 1 f ' ( t ) d t = − 1 2 ( 1 ) ( 1 ) ( For triangular region, area = 1 2 ( height ) × ( base ) ) = − 1 2 ( The negative sign indicates that the area lies below the x − axis ) Therefore, f ( 1 ) = ∫ 0 1 f ' ( t ) d t + f ( 0 ) = − 1 2 + 2 Since f ( 0 ) = 2 = 3 2 Similarly, for the interval [ 1 , 2 ] , ∫ 1 2 f ' ( t ) d t = f ( 2 ) − f ( 1 ) f ( 2 ) = ∫ 1 2 f ' ( t ) d t + f ( 1 ) = ∫ 1 2 f ' ( t ) d t + 3 2 In the interval [ 1 , 2 ] , f ' is rectangular area. So, the area of the rectangle is base × height . f ( 2 ) = − ( 1 ) ( 1 ) + 3 2 negative sign indicates that the rectangular lies below the x-axis = 1 2 Similarly, for the interval [ 2 , 3 ] , ∫ 2 3 f ' ( t ) d t = f ( 3 ) − f ( 2 ) f ( 3 ) = ∫ 2 3 f ' ( t ) d t + f ( 2 ) = ∫ 2 3 f ' ( t ) d t + 1 2 In the interval [ 2 , 3 ] , f ' is rectangular area

6th Edition

ISBN: 9781119267874

Author: Hughes-Hallett

Publisher: WILEY

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Chapter 5.5, Problem 15P

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Chapter 5.5, Problem 14P

Chapter 5.5, Problem 16P

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Which of the functions shown below is differentiable at = 0? Select the correct answer below: -7-6-5-4- -6-5-4-3-21, -7-6-5-4-3-2 -7-6-5-4-3-2-1 2 4 5 6 -1

correct answer is Acould you please show me how to compute using the residue theorem

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Chapter 5 Solutions

APPLIED CALCULUS-WILEYPLUS

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Chapter 5 Solutions

Fill the table:

x 0 1 2 3 4 5 6

f(x) 2