Explanation Given: f ( x ) = 3 x 2 + 4 x over [0, 2], that R N = 2 N ∑ j = 1 N ( 24 j 2 N 2 + 16 j N ) . 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 . Calculation: Let f ( x ) = 3 x 2 + 4 x on [0, 2]. Let N be a positive integer and set a=0, b=2, and Δ x = ( b − a ) N = 2 N . Also, let x j = a + j Δ x = 2 j N , j = 1 , 2 , ... , N be the right endpoints of the N subintervals of [0, 3]. Then R N = Δ x ∑ j = 1 N f ( x j ) = 2 N ∑ j = 1 N ( 3 ( 2 j N ) 2

3rd Edition

ISBN: 9781464125263

Author: Jon Rogawski, Colin Adams

Publisher: W.H. Freeman & Co

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Limits

When it comes to calculus, limits are considered to be a very important topic of discussion. The main importance of limit lies in expressing the value of a function as it gets closer to a certain value. Also, limits can help you to understand other topic…

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Chapter 5.1, Problem 53E

To determine

The function $f (x) = 3 x^{2} + 4 x$ over [0, 2], that $R_{N} = \frac{2}{N} \sum_{j = 1}^{N} (\frac{24 j^{2}}{N^{2}} + \frac{16 j}{N})$ and evaluate $\lim_{N \to \infty} R_{N}$

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Chapter 5.1, Problem 52E

Chapter 5.1, Problem 54E

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The function f ( x ) = 3 x 2 + 4 x over [0, 2], that R N = 2 N ∑ j = 1 N ( 24 j 2 N 2 + 16 j N ) and evaluate lim N → ∞ R N

Solution Summary: The author analyzes the function f (x) = 3 x 2 + 4 over [0, 2], that R N = 2 N j = 1 N.

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The function $f (x) = 3 x^{2} + 4 x$ over [0, 2], that $R_{N} = \frac{2}{N} \sum_{j = 1}^{N} (\frac{24 j^{2}}{N^{2}} + \frac{16 j}{N})$ and evaluate $\lim_{N \to \infty} R_{N}$

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The function f ( x ) = 3 x 2 + 4 x over [0, 2], that R N = 2 N ∑ j = 1 N ( 24 j 2 N 2 + 16 j N ) and evaluate lim N → ∞ R N

Solution Summary: The author analyzes the function f (x) = 3 x 2 + 4 over [0, 2], that R N = 2 N j = 1 N.

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The function f(x)=3x2+4x over [0, 2], that RN=2N∑j=1N(24j2N2+16jN) and evaluate limN→∞RN

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The function $f (x) = 3 x^{2} + 4 x$ over [0, 2], that $R_{N} = \frac{2}{N} \sum_{j = 1}^{N} (\frac{24 j^{2}}{N^{2}} + \frac{16 j}{N})$ and evaluate $\lim_{N \to \infty} R_{N}$