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: Let f ' ( x ) be the derivative function of f ( x ) . Given that f ( 0 ) = − 10 . Use the fundamental theorem of calculus as shown below. ∫ 0 1 f ' ( t ) d t = f ( 1 ) − f ( 0 ) f ( 1 ) = ∫ 0 1 f ' ( t ) d t + f ( 0 ) Now find the value of ∫ 0 1 f ' ( t ) d t . Use right hand sums to approximate ∫ 0 1 f ' ( t ) d t . From the diagram given in the question f ' ( 1 ) = 8 . The length of interval Δ x = 1 . Number of intervals is. n = 1 − 0 1 = 1 Use right hand sum, ∫ 0 1 f ' ( t ) d t = 1 ⋅ f ' ( 1 ) = 1.8 = 8 Therefore, f ( 1 ) = ∫ 0 1 f ' ( t ) d t + f ( 0 ) = 8 + ( − 10 ) = − 2 Thus, the required functional value is f ( 1 ) = − 2 . To find the value of f ( 2 ) Consider, f ( 2 ) = ∫ 0 2 f ' ( t ) d t + f ( 0 ) Now use right hand sums to approximate ∫ 0 2 f ' ( t ) d t From the diagram given in the problem f ' ( 1 ) = 8 and f ' ( 2 ) = 8 Consider, ∫ 0 2 f ' ( t ) d t = 1 ⋅ f ' ( 1 ) + 1 ⋅ f ' ( 2 ) = 1 ⋅ 8 + 1 ⋅ 8 = 16 Therefore, f ( 2 ) = ∫ 0 2 f ' ( t ) d t + f ( 0 ) = 16 − 10 = 6 The required functional value is f ( 2 ) = 6 . To find the value of f ( 3 ) Consider f ( 3 ) = ∫ 0 3 f ' ( t ) d t + f ( 0 ) From the diagram given in the program, f ' ( 1 ) = 8 , f ' ( 2 ) = 8 and f ' ( 3 ) = 4 Use right hand sums to approximate ∫ 0 3 f ' ( t ) d t . ∫ 0 3 f ' ( t ) d t = 1 ⋅ f ' ( 1 ) + 1 ⋅ f ' ( 2 ) + 1 ⋅ f ' ( 3 ) = 1 ⋅ 8 + 1 ⋅ 8 + 1 ⋅ 4 = 20 So, f ( 3 ) = ∫ 0 3 f ' ( t ) d t + f ( 0 ) = 20 − 10 = 10 The required functional vales is f ( 3 ) = 10 . To find the value of f ( 4 ) . Consider f ( 4 ) = ∫ 0 4 f ' ( t ) d t + f ( 0 ) Use right hand sums to approximate ∫ 0 4 f ' ( t ) d t

Applied Calculus, 6e WileyPLUS + Loose-leaf

6th Edition

ISBN: 9781119408901

Author: Deborah Hughes-Hallett, Andrew M. Gleason, Patti Frazer Lock, Daniel E. Flath

Publisher: Wiley (WileyPLUS Products)

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Question

Chapter 5.5, Problem 16P

To determine

Fill the table:

x 0 1 2 3 4 5 6

f(x) −10 2

Expert Solution & Answer

Chapter 5 Solutions

Applied Calculus, 6e WileyPLUS + Loose-leaf

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