Average Value of a Function
The average value of a function over a given interval is the total accumulated value of the function over that interval divided by the length of the interval. It represents the constant value that the function would need to have over the interval to give the same total value. This concept is commonly used in calculus and real analysis.
3 Key excerpts on "Average Value of a Function"
eBook - ePub- Gregory Hill, Mel Friedman(Authors)
- 2013(Publication Date)
- Research & Education Association(Publisher)
...The integral on the right side of the equation then finds the total area under the graph of f (x) and above the x -axis. On the left side of the equation, (b – a) is the width of the interval, and f avg is a height above the x -axis. So the equation is really height times width equals area. f avg is the height of the single rectangle with width (b – a) which contains the same amount of area as the area under the curve. Figure 7.2 illustrates this fascinating idea. Figure 7.2 The function was chosen to be positive to simplify the introduction of the idea, but there is absolutely no reason f avg cannot be computed for any continuous function on a given interval. In fact, it is certainly possible for an average value to be negative or even zero. The geometric interpretation of average value lends itself to a logical conclusion, which is the integral calculus version of the Mean Value Theorem. The height of the single rectangle will fall somewhere between the maximum value of the function, and the minimum value of the function on the interval [ a, b ]. As a result, the upper edge of the rectangle must intersect the function somewhere between the endpoints of the interval [ a, b ]. This guarantees there will be at least one value in (a, b) that generates f avg. Mean Value Theorem for Definite Integrals If a function f (x) is continuous on an interval [ a, b ], then there exists at least one value x = c in (a, b) such that EXAMPLE 7.12 Find the average value of on the interval [1, 5]. SOLUTION In physics, force times distance equals work. When force is constant, calculating work is simple. To raise a 3-pound book 2 feet takes 6 foot-pounds of work. But when force is constantly varying, calculus is useful to find the work done. Any given force can be applied only over an infinitely small distance and then must change. As a result, an infinite number of products of force times distance must be summed up...
eBook - ePubStatistical Literacy at School
Growth and Goals
- Jane M. Watson(Author)
- 2013(Publication Date)
- Routledge(Publisher)
...4 Average—What Does It Tell Us? It is difficult to understand why statisticians commonly limit their inquiries to Averages, and do not revel in more comprehensive views. Their souls seem as dull to the charm of variety as that of the native of one of our flat English counties, whose retrospect of Switzerland was that, if its mountains could be thrown into its lakes, two nuisances would be got rid of at once. 1 4.1 Background Francis Galton made this observation at the turn of the 20th century and his criticism continued to be appropriate in many mathematics classrooms throughout the century. What he could not foresee, however, was the appropriation of the term average by the general populace in a wide range of social contexts to describe a variety of conditions related to typicality. In the English language the word average has many connotations, from the colloquial “mediocre” to the mean algorithms taught in mathematics classrooms, both arithmetic and geometric. The association of average with the arithmetic mean by most high school mathematics teachers probably reflects their backgrounds and the history of the mean in the curriculum, but it is unlikely to reflect the everyday connections made by their students. The arithmetic mean has had a checkered history that has left it by default as the major summary statistic employed at the school level. For that reason, average is the focus of attention in the chapter on the data reduction phase of the statistical investigation process. This does not mean that other ideas for reducing the complexity of data are not relevant and some of these are considered. Historically the arithmetic mean has probably been associated with the mathematics curriculum longer than any other idea or tool used by statisticians...
eBook - ePub- Jim Fowler, Lou Cohen, Philip Jarvis(Authors)
- 2013(Publication Date)
- Wiley(Publisher)
...5 MEASURING THE AVERAGE 5.1 What is an average? One meaning of statistics is to do with describing and summarizing quantitative data. Any description of a sample of observations must include an aspect which relates to central tendency. That is to say, we need to identify a single number close to the centre of the distribution of observations which represents them all. We call this number the average ; it is often referred to as a measure of location because it indicates, on what might be a scale of infinite magnitude, just where a cluster of observations is located. The average is described by one of three commonly used statistics: the mean, the median and the mode. Each has its own application. 5.2 The mean The mean or, more precisely, the arithmetic mean is the most familiar and useful measure of the average. It is calculated by dividing the sum of a set of observations by the number of observations. If it is possible to obtain an observation from every single item or sampling unit in a population, for example, the mass of every remaining living Californian condor, then the mean is symbolized by µ (mu) and is called the population mean. More usually we have to be content with observations from a sample, in which case the sample mean is symbolized by (‘x-bar’). The sample mean is a direct estimate of the population mean (thus estimates µ). In Chapter 11 we explain how good an estimate it is likely to be. The formulae for calculating the mean are: where x is each observation, N is the number of items (observations) in a population, n is the number of observations in a sample and Σ is the ‘sum of’. Example 5.1 The maximum diameters of the pileus (cap) of five specimens of an edible fungus are recorded below. 8.5cm 9.2cm 7.3cm 6.8cm 10.1cm Calculate the mean diameter. 1. Obtain the sum of the observations (∑ x): 2...
