Mathematics
Differentiation
Differentiation is a mathematical process used to find the rate at which a quantity changes. It involves calculating the derivative of a function, which represents the slope of the function at a given point. This process is fundamental in calculus and is used to solve problems related to rates of change, optimization, and finding the behavior of functions.
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eBook - PDFCalculus
Concepts and Contexts, Enhanced Edition
- James Stewart(Author)
- 2018(Publication Date)
- Cengage Learning EMEA(Publisher)
255 Applications of Differentiation We have already investigated some of the applications of derivatives, but now that we know the Differentiation rules we are in a better position to pursue the applications of Differentiation in greater depth. We show how to analyze the behavior of families of functions, how to solve related rates problems (how to calculate rates that we can’t measure from those that we can), and how to find the maximum or minimum value of a quantity. In particular, we will be able to investigate the optimal shape of a can and to explain the location of rainbows in the sky. 4 chirajuti/Shutterstock.com Copyright 2019 Cengage Learning. All Rights Reserved. May not be copied, scanned, or duplicated, in whole or in part. Due to electronic rights, some third party content may be suppressed from the eBook and/or eChapter(s). Editorial review has deemed that any suppressed content does not materially affect the overall learning experience. Cengage Learning reserves the right to remove additional content at any time if subsequent rights restrictions require it. 256 CHAPTER 4 APPLICATIONS OF Differentiation If we are pumping air into a balloon, both the volume and the radius of the balloon are increasing and their rates of increase are related to each other. But it is much easier to measure directly the rate of increase of the volume than the rate of increase of the radius. In a related rates problem the idea is to compute the rate of change of one quantity in terms of the rate of change of another quantity (which may be more easily measured). The procedure is to find an equation that relates the two quantities and then use the Chain Rule to differentiate both sides with respect to time. Inflating a balloon Air is being pumped into a spherical balloon so that its volume increases at a rate of .
eBook - PDF- G. M. Fikhtengol'ts, I. N. Sneddon(Authors)
- 2014(Publication Date)
- Pergamon(Publisher)
CHAPTER 5 Differentiation OF FUNCTIONS OF ONE VARIABLE § 1. Derivative of a function and its computation 76. Problem of calculating the velocity of a moving point. Before proceeding to treat the foundations of the differential and integral calculus we draw the reader's attention to the fact that the ideas of calculus were originated as early as the seventeenth century, i.e. much earlier than the theories investigated in the preceding chapters. In the last chapter of this volume we shall survey the more important facts of the history of mathematical analysis and describe the merits of the two great mathematicians Newton and Leibniz, who completed the works of their predecessors by creating a really new calculus. In our discussion here we shall follow the modern demands of rigour, and not the history of the problem. As an introduction to the differential calculus we shall examine in this subsection the problem of velocity, and in the next subsection the problem of finding a tangent to a curve; both problems are historically connected with the formation of the basic concept of the differential calculus, which was later called the derivative. We begin by a simple example, namely we consider the free fall (in vacuum, when we can disregard the resistance of the air) of a heavy particle. If the time t (seconds) is measured from the beginning of the fall, the distance covered s (metres) is given by the well-known formula where g = 9.81 m/sec 2 . From these facts it is required to determine the velocity v of motion of the point at a given instant of time t, when the point is located at M (Fig. 31). [140] § 1. DERIVATIVE OF A FUNCTION 141 Introduce an increment At of the variable t and consider the instant t + At when the point is located at M ± . The increment MM X of the distance covered in the interval of time At we denote by As. Substituting into (1) t + At instead of t we obtain for the new value of distance the expression s+As = ^(t + Ai) whence As -^-(2t-At + At 2 ).
eBook - PDF- Sebastian J. Schreiber, Karl J. Smith, Wayne M. Getz(Authors)
- 2014(Publication Date)
- Wiley(Publisher)
C H A P T E R 4 Applications of Differentiation Figure 4.1 A great tit is a species of bird whose foraging behavior was studied by biologist Richard Cowie and whose behavior can be predicted by optimal foraging models. Santiago Ba ˜ n´ on / Flickr / Getty Images 4.1 Graphing Using Calculus 4.2 Getting Extreme 4.3 Optimization in Biology 4.4 Decisions and Optimization 4.5 Linearization and Difference Equations Review Questions Group Projects Preview “‘If one way be better than another, that you may be sure is nature’s way.’ Aristotle clearly stated the basic premise of optimization in biology, yet it was almost 2,000 years before the power of this idea was appreciated. The essence of optimization is to calculate the most efficient solution to a given problem, and then to test the prediction. The con- cept has already revolutionized some aspects of biology, but it has the potential for much wider application.” William Sutherland, on “The best solution” in Nature (2005) 435:569 One of the central ideas in physics, chemistry, and biology is that processes act to optimize some physically or biologically meaningful quantity. For example, from physics we know that light in a vacuum travels along a path that is the shortest distance between two points (taking into account that gravity “bends” space), and from biochemistry we know that proteins fold in a way that minimizes the energy of their constituent amino acid configuration. Differential calculus is an important tool for analyzing optimization (maximization or minimization). In this chapter we show how optimization applies to various biological problems and processes. Before we do this, however, we study how calculus can be used to construct the graphs of a variety of functions; in particular, we identify where the graph has turning points corresponding to local minimum or maximum values.
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