Explanation Given: • N = 5 • Rate of marine animal families during the Phanerozoic Eon: r ( t ) = 3130 ( t + 262 ) f o r 0 ≤ t ≤ 544 • t is the time elapsed (in millions of years) since the beginning of the eon 544 millions of years ago • Periods: 100 ≤ t ≤ 150 a n d 350 ≤ t ≤ 400 • t = 544 is 4 million years ago which refers to the present time Key Concepts Used: We can make use of three rectangular approximations to area namely, the Left-endpoint approximation, Right-endpoint approximation and Mid-point approximation. Endpoint Approximation: 1 : Right-endpoint approximation ( R N ): It is equal to Δ x times the sum of the function values at the right endpoints of the subintervals. R N = Δ x ∑ j = 1 N f ( a + j Δ x ) 2 : Left-endpoint approximation ( L N ): It is equal to Δ x times the sum of the function values at the left endpoints of the subintervals. L N = Δ x ∑ j = 1 N − 1 f ( a + j Δ x ) 3 : Mid-point approximation ( M N ): It is equal to Δ x times the sum of the function values at the midpoints of the subintervals. M N = Δ x ∑ j = 1 N f ( a + ( j − 1 2 ) Δ x ) Here in this problem, we have to find average of R N and L N with given value of N and the total number of families that became extinct in the given periods 100 ≤ t ≤ 150 a n d 350 ≤ t ≤ 400 For this we have to write the table to corresponding periods and find the values of R N and L N by means of which we can find their averages. Formulas Used: N t h Right-endpoint approximation: R N = Δ x ∑ j = 1 N f ( a + j Δ x ) N t h Left-endpoint approximation: L N = Δ x ∑ j = 1 N − 1 f ( a + j Δ x ) N t h Mid-point approximation: M N = Δ x ∑ j = 1 N f ( a + ( j − 1 2 ) Δ x ) Tables: Table 1 : For the period 100 ≤ t ≤ 150 , N = 5 , Δ t = 150 − 100 5 = 50 10 = 10 t 100 110 120 130 140 150 r ( t ) 8.6464 8.4139 8.1937 7.9847 7.7860 7.5970 Table 2 : For the period 350 ≤ t ≤ 400 , N = 5 , Δ t = 400 − 350 5 = 50 5 = 10 t 350 360 370 380 390 400 r ( t ) 5.1143 5.0321 4.9525 4.8753 4.8006 4.7281 Calculation: Description: R N = Δ x ∑ j = 1 N f ( a + j Δ x ) ; L N = Δ x ∑ j = 1 N − 1 f ( a + j Δ x ) Step 1 : From values of table 1 , we can find the values and average of R N and L N for the period 100 ≤ t ≤ 150 ...

3rd Edition

ISBN: 9781464125263

Author: Jon Rogawski, Colin Adams

Publisher: W.H. Freeman & Co

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Chapter 5.6, Problem 27E

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Average of $R_{N}$ and $L_{N}$ with given value of $N$ and the total number of families that became extinct in the given periods

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Chapter 5.6, Problem 26E

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Average of R N and L N with given value of N and the total number of families that became extinct in the given periods

Solution Summary: The author explains how to determine the average of R N and L N with given value of N, and the total number of families that became extinct in the given periods.

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Average of $R_{N}$ and $L_{N}$ with given value of $N$ and the total number of families that became extinct in the given periods

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Average of R N and L N with given value of N and the total number of families that became extinct in the given periods

Solution Summary: The author explains how to determine the average of R N and L N with given value of N, and the total number of families that became extinct in the given periods.

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Average of RNand LNwith given value of Nand the total number of families that became extinct in the given periods

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Chapter 5 Solutions

Average of $R_{N}$ and $L_{N}$ with given value of $N$ and the total number of families that became extinct in the given periods