Explanation Given: f ( x ) = x 2 + 7 x over [6, 11] 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 ) = x 2 + 7 x on [6, 11]. Then Δ x = ( 11 − 6 ) N = 5 N and a = 6. Hence, R N = Δ x ∑ j = 1 N f ( 6 + j Δ x ) = 5 N ∑ j = 1 N ( ( 6 + 5 j N ) 2 + 7 ( 6 + 5 j N ) ) = 5 N ∑ j = 1 N ( 78 + 95 j N + 25 j 2 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 58E

To determine

The area under the graph as a limit and compute a formula for R_N

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Chapter 5.1, Problem 57E

Chapter 5.1, Problem 59E

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Q1. Let f : N × N +R be defined by (a + 1)(a + 2b) f(a, b) = 2 (a) Show that the image of f is contained in N. (b) Determine exactly which natural numbers are in the image of f.

(b) Let 8 > 0. Show that there is a limit point æ of E such that x - a| < .

Define the function f(x) to be 1/x for x for all x that are (2n,2n+1) for even n and -1/x for x for all x that are you (2n, 2n+1) for odd n. Graph the function and describe for which points limit do not exist. Does the limit exist as x goes to infinity? (The n here represents natural numbers e.g. 0,1,2,...27,... and hence the first interval is (0,1], the second is (1,2], etc.)

Chapter 5 Solutions

Calculus - Standalone book

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