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Chapter 7: Problem 43

give an example of: A continuous function \(f(x)\) on the interval [0,10] such that \(\mathrm{TRAP}(40)>\mathrm{TRAP}(80)\)

Short Answer

Expert verified
Choose \(f(x) = x^2 \sin(10x)\) for a higher \(\mathrm{TRAP}(40)\) than \(\mathrm{TRAP}(80)\).

Step by step solution

01

Understanding the Problem

The problem requires us to find a continuous function \(f(x)\) defined on the interval \([0,10]\) for which \(\mathrm{TRAP}(40)>\mathrm{TRAP}(80)\). \(\mathrm{TRAP}(n)\) signifies the trapezoidal numerical integration approximation with \(n\) subintervals. The task is to set up \(f(x)\) such that the approximation with 40 subintervals yields a greater value than with 80 subintervals.
02

Choose an Appropriate Function

Consider a continuous function that changes rapidly towards the end of the interval. For example, choose \(f(x) = x^2 \sin(10x)\). This function oscillates and its behavior changes quickly, which can create discrepancies between trapezoidal approximations using different numbers of subintervals.
03

Compute \(\mathrm{TRAP}(40)\) and \(\mathrm{TRAP}(80)\) Conceptually

With \(n = 40\), each subinterval in \(\mathrm{TRAP}(40)\) is of width \(\frac{10}{40} = 0.25\) and for \(n = 80\), the width is \(\frac{10}{80} = 0.125\). The sharp changes in \(f(x)\) towards the end of the interval can be underrepresented or over-corrected depending on the choice of \(f(x)\). For \(f(x) = x^2 \sin(10x)\), with fewer subintervals, the rapidly changing behavior might be approximated with higher error, as desired for the task.
04

Sketching and Conceptual Comparison

Sketch the behavior of \(f(x) = x^2 \sin(10x)\) and note how the oscillations introduce changes that may not be captured accurately with more subintervals compared to fewer subintervals. This ensures \(\mathrm{TRAP}(40) > \mathrm{TRAP}(80)\) due to the compounded error with increased subinterval count and rapid oscillation.
05

Validate with Numerical Example (Optional)

Evaluate or simulate the trapezoid rule for both \(\mathrm{TRAP}(40)\) and \(\mathrm{TRAP}(80)\) using the chosen function \(f(x) = x^2 \sin(10x)\) on the interval \([0,10]\). Note the subtle changes and compare results to confirm the mathematical reasoning provided.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Numerical Integration
Numerical integration is a valuable technique used to approximate the integral of a function when finding an exact solution is complicated or impossible. The trapezoidal rule is one of these numerical methods.
  • It works by dividing the interval into several smaller subintervals.
  • Within each subinterval, it approximates the area under the curve as a trapezoid rather than a complex curve.
  • Trapezoids are used because they are simple to calculate, needing only the average of the two endpoint function values multiplied by the subinterval width.
For a continuous function, this can effectively approximate the area under a curve, especially when the function behaves smoothly over the interval.
However, the approximation can differ depending on the number of subintervals chosen, which leads to potential inaccuracies especially in functions with rapid oscillations or steep changes in slope.
Continuous Function
A continuous function is a function without breaks, jumps, or gaps over its domain. Formally, it ensures that small changes in its input result in small changes in its output.
  • This characteristic is crucial for approximating integrals using numerical methods, ensuring a smooth connection between the points.
  • It allows numerical methods like the trapezoidal rule to estimate the function's integral without unexpected discontinuities disrupting the calculation.
Considering the exercise, the function \(f(x) = x^2 \sin(10x)\) is a great example where continuity is maintained, but rapid oscillations occur, demonstrating the function's fast-changing nature over the interval [0,10].
Such behaviors can make accurate approximation with a higher number of subintervals challenging, contrary to what might be typically expected, emphasizing the importance of understanding the function's nature before applying numerical integration methods.
Subintervals
Understanding subintervals is key to applying the trapezoidal rule effectively. Subintervals divide the whole interval into equal, smaller parts, allowing the function's integral to be estimated more manageably.
  • The width of each subinterval is calculated by dividing the length of the whole interval by the number of subintervals.
  • With the trapezoidal rule, the subinterval width influences the accuracy of the approximation, as each subinterval contributes a small piece to the total integral estimate.
In the given exercise, using different numbers of subintervals, such as 40 and 80, highlights how the function's rapid changes can impact the accuracy of the trapezoidal rule:
Using 40 subintervals gives a width of 0.25, while using 80 subintervals gives a width of 0.125. This change in width can lead to different approximations of the integral, especially when the function changes rapidly within the interval.
The exercise valuable illustrates that more subintervals do not always guarantee a more accurate result, particularly for sharply oscillating functions like \(f(x) = x^2 \sin(10x)\).

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Most popular questions from this chapter

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