Related Rates

4 Key excerpts on "Related Rates"

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  eBook - ePub

  Mathematics for the Nonmathematician

  • Morris Kline(Author)
  • 2013(Publication Date)
  • Dover Publications

    (Publisher)

  All these questions and many others that we shall encounter in the present and later chapters bedeviled the mathematicians of the seventeenth century, and hundreds of capable men worked on them. When Newton and Leibniz made their contributions to the calculus, it became clear that all of the above problems and others too could be solved by means of one basic concept, the instantaneous rate of change of one variable with respect to another. Hence we shall begin with this concept.

  16–3  THE CONCEPT OF INSTANTANEOUS RATE OF CHANGE

  There are three closely related ideas: change, average rate of change, and instantaneous rate of change. These three ideas should be carefully distinguished. The concept of change itself is by now a familiar one. When a ball is thrown up into the air, its height above the ground changes. The pursuit of physical problems involving functions soon obliges one to consider not just the mere fact of change but the rate of change of one variable with respect to another. In the case of a ball thrown up into the air, one might wish to know what initial speed will enable the ball to reach a height, say of 100 feet, or what speed the ball has on returning to the ground; that is, information about the speed, which is the rate of change of height with respect to time, is desirable. The statement that the earth travels around the sun in one year is a fact about rate of change rather than about mere change. Our great concern in this age for faster transportation and communication is a concern with rate of change. Circulation of the blood in one’s body means quantity of blood per unit time passing through a specific artery or a collection of arteries, and here, too, it is rate of change which counts. The rate of physiological activity, that is the metabolic rate, measured in terms of the rate of consumption of oxygen per second, is a rate of change. To sum up: the rate of change of one variable with respect to another is a physically useful quantity in many situations.

  The rates of change which are of interest to laymen and even to many specialists are average rates. Thus, if a motorist travels 500 miles in 10 hours, the average speed, i.e., the distance traveled divided by the time of travel, is 50 miles per hour. This average speed is what usually matters, and in most instances it is quite irrelevant that the driver may occasionally have stopped for food and thus had no speed at all during those periods of the trip. Most people like to increase their wealth and are satisfied if the rate of growth, that is the growth in wealth per month or per year, is appreciable. The increase of a country’s population is usually measured per year because this average rate tells the story which is of importance for most purposes.

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  eBook - ePub

  Teaching Secondary School Mathematics

  Research and practice for the 21st century

  • Merrilyn Goos, Gloria Stillman, Sandra Herbert, Vince Geiger(Authors)
  • 2020(Publication Date)
  • Routledge

    (Publisher)

  Figure 12.15 ). Exercises following this introduction usually relate to contexts in which distance and time are the rate- related variables. ‘It appears to be assumed that since speed is experientially real for students, it is also well understood as an abstract concept’ (Herbert, 2010, p. 18). Therefore, speed appears to be taken to be a familiar concept on which to build an understanding of rate and hence, derivative. However, understanding of speed may not transfer to an understanding of rate in non-motion contexts (Herbert & Pierce, 2011).

  Calculus students’ difficulties with rate manifest in many forms. One of the most significant of these is the confusion between the rate and the extensive quantities that constitute it (Thompson, 1994; Rowland & Jovanoski, 2004), for example, understanding of speed as a distance. Other difficulties with understanding of rate include confusion relating to symbols and their use as variables (White & Mitchelmore, 1996); lacking awareness of the relationship between slope, rate and the first derivative (Porzio, 1997); misunderstandings related to average and instantaneous rate (Hassan & Mitchelmore, 2006; Orton, 1984) and related-rates problems in speed (Billings & Kladerman, 2000); and confusion between average and instantaneous rate (Hassan & Mitchelmore, 2006) and geometric contexts (Martin, 2000). Hassan and Mitchelmore (2006) report on their study of 14 Australian senior secondary students’ understanding of average and instantaneous rate. The students were interviewed twice, before and after a teaching intervention designed to address the difficulties with rate expressed in the first interview in which they found most students seemed to have a fair qualitative understanding of rate, but few demonstrated an understanding of either average or instantaneous rate at the start of the first interview. The interview tasks were embedded in the contexts of motion, population growth and cooling rate. None of these rate contexts addresses a rate where time is not a variable.

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  eBook - ePub

  Introduction to Differential Calculus

  Systematic Studies with Engineering Applications for Beginners

  • Ulrich L. Rohde, G. C. Jain, Ajay K. Poddar, A. K. Ghosh(Authors)
  • 2012(Publication Date)
  • Wiley

    (Publisher)

  it will not be difficult to grasp the concept (of derivatives) with our systematic approach .

  The relationship between f (x ) and f ′(x ) is the main theme. We will study what it means for f ′(x ) to be the rate function (or derivative function) derived from f (x ) and what each function says about the other. The important requirement is to understand clearly the meaning of the instantaneous rate (or the actual rate) of change of f(x) with respect to x .

  For this purpose, it is necessary to distinguish between the average rate of change and the actual (or instantaneous) rate of change of a varying (dependent) quantity f (x ) with respect to another varying quantity “x ”, considered to be varying independently.2

  We know that every rate is the ratio of two changes that may occur in two related quantities.

  For example, consider the volume of a sphere, defined by

  Note that, V (r ) will change if “r ” is changed. Now consider the situation when “r ” is increased by 2 units from 1 unit to 3 units. We get

  Average rate of change in V (r ) (for increase in “r ” by 2 units)

  (1)

  Again, consider the situation when r is increased by 2 units from 2 units to 4 units. We get

  Average rate of change in V (r ) (for increase in “r ” by 2 units)

  (2)

  Also, it can be checked that average rate of change in V (r ), for one unit increase in “r ” varies as follows:

  Change in r	Average rate of change (for one unit increase in r )
  From r = 0 to r = 1	k
  From r = 1 to r = 2	7k
  From r = 2 to r = 3	19k
  From r = 3 to r = 4	37k

  From the above data, we observe that for two units increase in “r ”, the average rate of change in V (r ) is not the same as can be seen from (1) and (2) above. Similarly, the average rate of change in V (r ) for a unit change in “r ”, is different for two different values of “r”. This observation indicates that the rate at which V (r ) increases must be different, for different values of

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  eBook - ePub

  Essential Calculus with Applications

  • Richard A. Silverman(Author)
  • 2013(Publication Date)
  • Dover Publications

    (Publisher)

  Chapter 2 DIFFERENTIAL CALCULUS

  2.1 FUNCTIONS

  2.11. Constants and variables. The quantities encountered in mathematics fall into two broad categories, namely “fixed quantities,” called constants , and “changing quantities,” called variables . A constant “takes only one value” in the course of a given problem, while a variable “takes two or more values” in the course of one and the same problem.

  For example, let L be the straight line with slope m and y -intercept b . Then, according to Sec. 1.9 , L has the equation

  Here m and b are constants characterizing the given line L , while x and y are variables, namely the abscissa and ordinate of a point which is free to change its position along L .

  As another example, suppose a stone is dropped from a high tower. Let s be the distance fallen by the stone and t the elapsed time after dropping the stone. Then, according to elementary physics, s and t are variables which are related, at least for a while, by the formula

  where g is a constant known as the “acceleration due to gravity.” To a good approximation, this formula becomes

  if s is measured in feet and t in seconds.

  2.12. Related variables and the function concept

  a. A great many problems arising in mathematics and its applications involve related variables . This means that there are at least two relevant variables, and the value of one of them depends on the value of the other, or on the values of the others if there are more than two. For example, the position of a spy satellite depends on the elapsed time since launching, the cost of producing a commodity depends on the quantity produced, the area of a rectangle depends on both its length and its width, and so on.

  Actually, the situation we have in mind is where knowledge of the values of all but one of the variables uniquely determines the value of the remaining variable, in a crucial sense to be spelled out in a moment (Sec. 2.12b ). The variables whose values are chosen in advance are called independent variables , and the remaining variable, whose value is determined by the values of the other variables, is called the dependent variable . The “rule” or “procedure” leading from the values of the independent variables to the value of the dependent variable, regardless of how this is accomplished, is called a function . We then say that the dependent variable “is a function of” the independent variables. The independent variables are often called the arguments

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