Explanation Given: f ( x ) = 3 x 3 − x 2 over [1, 5], that R N = 4 N ∑ j = 1 N ( 192 j 3 N 3 + 128 j 2 N 2 + 28 j N + 2 ) Formula used: Sum of the first n natural number is n 2 2 + n 2 . Sum of the square of the first n natural number is n 3 3 + n 2 2 + n 6 . Sum of the cube of the first n natural number is n 4 4 + n 3 2 + n 2 2 . Calculation: Let f ( x ) = 3 x 3 − x 2 on [1, 5]. Let N be a positive integer and set a=1, b=5, and Δ x = ( b − a ) N = 4 N . Also, let x j = a + j Δ x = 1 + 4 j N , j = 1 , 2 , ... , N be the right endpoints of the N subintervals of [1, 5]. Then f ( x j ) = 3 ( 1 + 4 j N ) 3 − ( 1 + 4 j N ) 2 = 3 ( 1 + 12 j N + 48 j 2 N 2 + 64 j 3 N 3 ) − ( 1 + 8 j N + 16 j 2 N 2 ) = 192 j 3 N 3 + 128 j 2 N 2 + 28 j N + 2...

3rd Edition

ISBN: 9781464125263

Author: Jon Rogawski, Colin Adams

Publisher: W.H. Freeman & Co

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Chapter 5.1, Problem 54E

To determine

The function $f (x) = 3 x^{3} - x^{2}$ over [1, 5], that $R_{N} = \frac{4}{N} \sum_{j = 1}^{N} (\frac{192 j^{3}}{N^{3}} + \frac{128 j^{2}}{N^{2}} + \frac{28 j}{N} + 2)$ and evaluate $\lim_{N \to \infty} R_{N}$ .

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Chapter 5.1, Problem 53E

Chapter 5.1, Problem 55E

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The function f ( x ) = 3 x 3 − x 2 over [1, 5], that R N = 4 N ∑ j = 1 N ( 192 j 3 N 3 + 128 j 2 N 2 + 28 j N + 2 ) and evaluate lim N → ∞ R N .

Solution Summary: The author explains how to determine the function f (x) = 3 x 3 + 2 over [1, 5], that R N = 4 N j = 1 N.

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The function $f (x) = 3 x^{3} - x^{2}$ over [1, 5], that $R_{N} = \frac{4}{N} \sum_{j = 1}^{N} (\frac{192 j^{3}}{N^{3}} + \frac{128 j^{2}}{N^{2}} + \frac{28 j}{N} + 2)$ and evaluate $\lim_{N \to \infty} R_{N}$ .

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The function f ( x ) = 3 x 3 − x 2 over [1, 5], that R N = 4 N ∑ j = 1 N ( 192 j 3 N 3 + 128 j 2 N 2 + 28 j N + 2 ) and evaluate lim N → ∞ R N .

Solution Summary: The author explains how to determine the function f (x) = 3 x 3 + 2 over [1, 5], that R N = 4 N j = 1 N.

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The function f(x)=3x3−x2 over [1, 5], that RN=4N∑j=1N(192j3N3+128j2N2+28jN+2) and evaluate limN→∞RN.

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The function $f (x) = 3 x^{3} - x^{2}$ over [1, 5], that $R_{N} = \frac{4}{N} \sum_{j = 1}^{N} (\frac{192 j^{3}}{N^{3}} + \frac{128 j^{2}}{N^{2}} + \frac{28 j}{N} + 2)$ and evaluate $\lim_{N \to \infty} R_{N}$ .