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Suppose you need to estimate the value of| 5(2) dz using the midpoint method. How many subintervals are required to guarantee that your estimate is within .0001 of the true value given that the graph of f"(r) is as shown below? f"(1)

Suppose you need to estimate the value of| 5(2) dz using the midpoint method. How many subintervals are required to guarantee that your estimate is within .0001 of the true value given that the graph of f"(r) is as shown below? f"(1)

Advanced Engineering Mathematics
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
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**Problem Statement:** Suppose you need to estimate the value of \(\int_{0}^{8} f(x) \, dx\) using the midpoint method. How many subintervals are required to guarantee that your estimate is within 0.0001 of the true value given that the graph of \(f''(x)\) is as shown below? **Graph Explanation:** The graph depicts \(f''(x)\) over the interval from 0 to 8 on the x-axis. The y-axis ranges from -4 to 2. Key observations from the graph include: - The graph starts above the x-axis, then crosses it, dips, and eventually goes back above it. - Between x = 0 and x = 2, \(f''(x)\) is positive. - Between x = 2 and x = 5, \(f''(x)\) is negative. - After x = 5, \(f''(x)\) becomes positive again. The curve initially peaks at around \(x = 1\), dips to its lowest at approximately \(x = 4\), and rises again. This information can be used to assess the concavity behavior of \(f(x)\), which impacts the midpoint approximation accuracy and the number of required subintervals for the desired precision.
Transcribed Image Text:**Problem Statement:** Suppose you need to estimate the value of \(\int_{0}^{8} f(x) \, dx\) using the midpoint method. How many subintervals are required to guarantee that your estimate is within 0.0001 of the true value given that the graph of \(f''(x)\) is as shown below? **Graph Explanation:** The graph depicts \(f''(x)\) over the interval from 0 to 8 on the x-axis. The y-axis ranges from -4 to 2. Key observations from the graph include: - The graph starts above the x-axis, then crosses it, dips, and eventually goes back above it. - Between x = 0 and x = 2, \(f''(x)\) is positive. - Between x = 2 and x = 5, \(f''(x)\) is negative. - After x = 5, \(f''(x)\) becomes positive again. The curve initially peaks at around \(x = 1\), dips to its lowest at approximately \(x = 4\), and rises again. This information can be used to assess the concavity behavior of \(f(x)\), which impacts the midpoint approximation accuracy and the number of required subintervals for the desired precision.
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