Define the function p: [0, ∞) R by: sin(x2), x≥0, p(x) = 10, x < 0. Tasks: 1. Boundedness Analysis: • a. Prove that p(x) is bounded on R. ⚫ b. Provide a graph of p(x) over [0, 10] to illustrate its bounded oscillatory behavior. 2. Integrability Investigation: ⚫ a. Determine whether p(x) is Riemann integrable on [0, ∞). ⚫ b. Determine whether p(x) is Lebesgue integrable on [0, ∞). 3. Improper Integral Evaluation: a. Evaluate p(x) da if it exists. ⚫ b. Discuss the convergence or divergence of the improper integral based on your analysis. 4. Histogram Representation: ⚫ a. Construct a histogram of p(x) values over [0, 10] with appropriate binning to capture the oscillatory nature. ⚫ b. Analyze the distribution of p(x) values and relate it to the integrability findings. 5. Analysis of Function Behavior: a. Examine the decay of oscillations in p(x) as a increases.

Define the function p: [0, ∞) R by: sin(x2), x≥0, p(x) = 10, x < 0. Tasks: 1. Boundedness Analysis: • a. Prove that p(x) is bounded on R. ⚫ b. Provide a graph of p(x) over [0, 10] to illustrate its bounded oscillatory behavior. 2. Integrability Investigation: ⚫ a. Determine whether p(x) is Riemann integrable on [0, ∞). ⚫ b. Determine whether p(x) is Lebesgue integrable on [0, ∞). 3. Improper Integral Evaluation: a. Evaluate p(x) da if it exists. ⚫ b. Discuss the convergence or divergence of the improper integral based on your analysis. 4. Histogram Representation: ⚫ a. Construct a histogram of p(x) values over [0, 10] with appropriate binning to capture the oscillatory nature. ⚫ b. Analyze the distribution of p(x) values and relate it to the integrability findings. 5. Analysis of Function Behavior: a. Examine the decay of oscillations in p(x) as a increases.

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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