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Chapter 5: Problem 17

Use a finite sum to estimate the average value of \(f\) on the given interval by partitioning the interval into four sub intervals of equal length and evaluating \(f\) at the sub interval midpoints. $$f(t)=(1 / 2)+\sin ^{2} \pi t \quad \text { on } \quad[0,2]$$ (Graph cant copy)

Short Answer

Expert verified
The estimated average value of the function on the interval is 1.

Step by step solution

01

Divide the Interval

The interval given is \([0, 2]\). We will divide it into four equal sub-intervals. The length of each sub-interval is \( \frac{2-0}{4} = 0.5\). So the sub-intervals are:\( [0, 0.5], [0.5, 1], [1, 1.5], [1.5, 2] \).
02

Find the Midpoints

Next, we find the midpoints of the sub-intervals:1. Midpoint of \([0, 0.5]\) is \(0.25\).2. Midpoint of \([0.5, 1]\) is \(0.75\).3. Midpoint of \([1, 1.5]\) is \(1.25\).4. Midpoint of \([1.5, 2]\) is \(1.75\).
03

Evaluate the Function at Midpoints

We evaluate the function \(f(t) = \frac{1}{2} + \sin^2(\pi t)\) at each midpoint:1. \(f(0.25) = \frac{1}{2} + \sin^2(\pi \times 0.25) = \frac{1}{2} + \sin^2(\frac{\pi}{4}) = \frac{1}{2} + \left(\frac{\sqrt{2}}{2}\right)^2 = \frac{1}{2} + \frac{1}{2} = 1\).2. \(f(0.75) = \frac{1}{2} + \sin^2(\pi \times 0.75) = \frac{1}{2} + \sin^2(\frac{3\pi}{4}) = \frac{1}{2} + \left(\frac{\sqrt{2}}{2}\right)^2 = \frac{1}{2} + \frac{1}{2} = 1\).3. \(f(1.25) = \frac{1}{2} + \sin^2(\pi \times 1.25) = \frac{1}{2} + \sin^2(\frac{5\pi}{4}) = \frac{1}{2} + \left(-\frac{\sqrt{2}}{2}\right)^2 = \frac{1}{2} + \frac{1}{2} = 1\).4. \(f(1.75) = \frac{1}{2} + \sin^2(\pi \times 1.75) = \frac{1}{2} + \sin^2(\frac{7\pi}{4}) = \frac{1}{2} + \left(-\frac{\sqrt{2}}{2}\right)^2 = \frac{1}{2} + \frac{1}{2} = 1\).
04

Calculate the Average Value

The average value of \(f(t)\) over the interval can be estimated as the average of the function values at the midpoints:\[ \text{Average value} = \frac{1}{4} (f(0.25) + f(0.75) + f(1.25) + f(1.75)) \]\[ = \frac{1}{4} (1 + 1 + 1 + 1) = \frac{4}{4} = 1 \].

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Key Concepts

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

Finite Sum Approximation
In mathematics, the finite sum approximation is a technique used to estimate the average value of a function over a specific interval. This process involves breaking down the interval into smaller sub-intervals, evaluating the function at certain key points within these sub-intervals, usually midpoints, and then averaging these values.
The goal of the finite sum approximation is to provide a simpler way to compute an integral's value in situations where performing an exact integration might be too complex or infeasible for the functions in question.
To apply finite sum approximation, follow these general steps:
  • Divide the given interval into a specified number of sub-intervals.
  • Find a point within each sub-interval, often the midpoint.
  • Evaluate the function at these points.
  • Compute the average of these values to estimate the average value of the entire interval.
Midpoint Evaluation
Midpoint evaluation is a specific technique within the finite sum approximation process. It involves assessing the function at the midpoints of each sub-interval, rather than using endpoints or other points. This can help make the approximation more accurate than other methods, especially when the function behaves smoothly over the interval.
The step-by-step procedure for midpoint evaluation is as follows:
  • Identify the length of each sub-interval by taking the total length of the interval and dividing it by the number of sub-intervals.
  • Determine the midpoint of each sub-interval. This is generally done by averaging the starting and ending points of the sub-interval.
  • Plug these midpoint values into the function to evaluate it at these points.
This method tends to balance any variations in the function's values, smoothing out higher and lower function values you might encounter.
Partitioning Intervals
In order to carry out either finite sum approximation or midpoint evaluation, it is crucial first to partition the interval appropriately. Partitioning intervals means dividing a given interval into a number of smaller, equal-length sub-intervals. This division provides structure to the approximation or integration process and is foundational to accurately estimating values.
The process of partitioning intervals involves:
  • Identifying the total length of the interval you need to analyze.
  • Deciding on the number of sub-intervals you wish to create, which contributes to the accuracy of your approximation.
  • Calculating the length of each sub-interval by dividing the total interval length by the number of sub-intervals.
  • Specifying the boundaries of each sub-interval. These are critical to locating midpoints for evaluating the function.
Partitioning intervals properly allows for equally spaced evaluations, which leads to a more consistent and reliable approximation outcome.

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Most popular questions from this chapter

Express the solutions of the initial value problems in terms of integrals. $$\frac{d y}{d x}=\sqrt{1+x^{2}}, \quad y(1)=-2$$

Find the areas of the regions enclosed by the lines and curves in Exercises \(73-80\). $$x=y^{2} \quad \text { and } \quad x=y+2$$

Let \(F(x)=\int_{a}^{x} f(t) d t\) for the specified function \(f\) and interval \([a, b] .\) Use a CAS to perform the following steps and answer the questions posed. a. Plot the functions \(f\) and \(F\) together over \([a, b]\) b. Solve the equation \(F^{\prime}(x)=0 .\) What can you see to be true about the graphs of \(f\) and \(F\) at points where \(F^{\prime}(x)=0\) ? Is your observation borne out by Part 1 of the Fundamental Theorem coupled with information provided by the first derivative? Explain your answer. c. Over what intervals (approximately) is the function \(F\) increasing and decreasing? What is true about \(f\) over those intervals? d. Calculate the derivative \(f^{\prime}\) and plot it together with \(F .\) What can you see to be true about the graph of \(F\) at points where \(f^{\prime}(x)=0 ?\) Is your observation borne out by Part 1 of the Fundamental Theorem? Explain your answer. $$f(x)=\sin 2 x \cos \frac{x}{3}, \quad[0,2 \pi]$$

Find \(\lim _{x \rightarrow \infty} \frac{1}{\sqrt{x}} \int_{1}^{x} \frac{d t}{\sqrt{t}}\).

Use a CAS to perform the following steps: a. Plot the functions over the given interval. b. Partition the interval into \(n=100.200\), and 1000 subintervals of equal length, and evaluate the function at the midpoint of each subinterval. c. Compute the average value of the function values generated in part (b). d. Solve the equation \(f(x)=\) (average value) for \(x\) using the average value calculated in part (c) for the \(n=1000\) partitioning. $$f(x)=\sin x \quad \text { on } \quad[0, \pi]$$
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