3.3 How many subintervals are required to ensure an approximation error of less thanx 10-5 when approximating the integral of cosa on [0, 1] and employing3.3.1 the composite trapezoidal rule?3.3.2 the composite Simpson's rule?

While approximating the integral: ∫abf(x)dx, the error bound for: Composite trapezoidal rule is…

Answered: How many subintervals are required to…

Functions and Change: A Modeling Approach to College Algebra (MindTap Course List)

6th Edition

ISBN:9781337111348

Author:Bruce Crauder, Benny Evans, Alan Noell

Publisher:Bruce Crauder, Benny Evans, Alan Noell

Chapter2: Graphical And Tabular Analysis

Section2.1: Tables And Trends

Problem 1TU: If a coffee filter is dropped, its velocity after t seconds is given by v(t)=4(10.0003t) feet per...

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Question

3.3 How many subintervals are required to ensure an approximation error of less than
x 10-5 when approximating the integral of cosa on [0, 1] and employing
3.3.1 the composite trapezoidal rule?
3.3.2 the composite Simpson's rule?

Expert Solution

Step 1

While approximating the integral: $\int_{a}^{b} f (x) d x$ , the error bound for:

Composite trapezoidal rule is $\frac{M}{12} \frac{{(b - a)}^{3}}{n^{2}}$ . Here, $|f'' (x)| \leq M$ for $a \leq x \leq b$ .
Composite Simpson's rule is $\frac{L}{180} \frac{{(b - a)}^{5}}{n^{4}}$ . Here, $|f^{4} (x)| \leq L$ for $a \leq x \leq b$ .

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If a coffee filter is dropped, its velocity after t seconds is given by v(t)=4(10.0003t) feet per second. What is the terminal velocity, and how long does it take the filter to reach 99 of terminal velocity? Use a table increment of 0.1 and given your answer to the nearest tenth of a second.

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How many subintervals are required to ensure an approximation error of less than x 10-5 when approximating the integral of cos on [0, 1] and employing 3.3.1 the composite trapezoidal rule? 3.3.2 the composite Simpson's rule?