Graphs and Differentiation

6 Key excerpts on "Graphs and Differentiation"

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  Foundation Mathematics for the Physical Sciences

  • K. F. Riley, M. P. Hobson(Authors)
  • 2011(Publication Date)
  • Cambridge University Press

    (Publisher)

  3 Differential calculus This and the next chapter are concerned with the formalism of probably the most widely used mathematical technique in the physical sciences, namely the calculus. The current chapter deals with the process of differentiation whilst Chapter 4 is concerned with its inverse process, integration. The topics covered are essential for the remainder of the book; once studied, the contents of the two chapters serve as reference material, should that be needed. Readers who have had previous experience of differentiation and integration should ensure full familiarity by looking at the worked examples in the main text and by attempting the problems at the ends of the two chapters. Also included in this chapter is a section on curve sketching. Most of the mathematics needed as background to this important skill for applied physical scientists was covered in the first two chapters, but delaying our main discussion of it until the end of this chapter allows the location and characterisation of turning points to be included amongst the techniques available. 3.1 Differentiation • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • Differentiation is the process of determining how quickly or slowly a function varies, as the quantity on which it depends, its argument , is changed. More specifically, it is the procedure for obtaining an expression (numerical or algebraic) for the rate of change of the function with respect to its argument. Familiar examples of rates of change include acceleration (the rate of change of velocity) and chemical reaction rate (the rate of change of chemical composition). Both acceleration and reaction rate give a measure of the change of a quantity with respect to time. However, differentiation may also be applied to changes with respect to other quantities, for example the change in pressure with respect to a change in temperature.

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  Teaching and Learning Algebra

  • Doug French(Author)
  • 2004(Publication Date)
  • Continuum

    (Publisher)

  Innovative software packages produced by Tall (1986) linked numerical and graphical approaches and opened the way to the more powerful possibilities available using today's calculators and computers. However, technology does not solve all the problems and needs to be used judiciously to help students come to understand the underlying ideas and the use of algebra in representing and applying them. GRADIENT FUNCTIONS Students who have followed an elementary course in calculus are always very familiar with the fact that when y = jc 2 , the derivative j^ — 2x. They should know that it tells them the gradient of the curve, and that in general this enables them to find the turning points on a curve where it is zero and the rate of change in appropriate applications. They will be able to determine the derivatives of various functions with some degree of fluency, but they are invariably less confident, or even totally ignorant, about where the results come from. As I have suggested in the previous section, the underlying ideas which relate to a limiting process are important and should be emphasized in the introductory stages and subsequently reinforced. Looking at the behaviour of the gradient as you move along a graph gives a feel for the meaning of a derivative as a gradient function. Using a straight edge as the tangent to a graph, students can be asked to observe and describe the way the gradient changes as the straight edge is moved along the curve. From this a graph of the gradient function can be sketched. This is 176 Teaching and Learning Algebra Figure 12.1 Graphing the gradient function illustrated with the graph shown in bold in Figure 12.1, by noting the two points where the gradient of the bold curve is zero, and then considering how the gradient changes and where it is positive and negative.

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  Graphical Methods

  • Carl Runge(Author)
  • 2019(Publication Date)
  • Columbia University Press

    (Publisher)

  CHAPTER III. THE GRAPHICAL METHODS OF THE DIFFERENTIAL AND INTEGRAL CALCULUS. § 13. Graphical Integration. —We have shown how the ele-mentary mathematical operations of adding, subtracting, multi-plying and dividing and the inverse operation of finding the root of an equation can be carried out by graphical methods and how functions of one or more variables may be represented and handled. But the graphical methods would lack generality and would be of very limited use, if they were not applicable to the infinitesimal operations of differentiation and integration. In-deed it is here that they are found of the greatest value. In many cases, where the calculus is applied to problems of natural science or of engineering, the functions concerned are given in a graphical form. Their true analytical structure is not known and as a rule an approximation by analytical expressions is not easily calculated nor easily handled. In these cases it is of vital importance that the operations of the calculus can be performed, although the functions are only given graphically. Let us begin with integration, because it is easier than differ-entiation and of more general application. Suppose a function y = f(x) given by a curve whose ordinate is y and whose abscissa is x. The problem is to find a curve, whose ordinate Y is an integral of the function/(a:), Let us assume the unit of length for the abscissas independent of the unit of length for the ordinates. The value of Y measures the area between the ordinates corresponding to a and x, the curve y = f(x) and the axis of χ in units equal to the rectangle formed by the units of χ and y. 101 102 GRAPHICAL METHODS. In the simple case where f(x) is a constant the equation y = f(x) = c is represented by a line parallel to the axis of χ and Y = cdx = c(x — a). i V i V ^ t. - _ 1 0 * Fio. 72. Y is the ordinate of a straight line intersecting the axis of χ at the point χ = a. The constant c is the change of Y for an increase of χ equal to 1.

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  Quick Calculus

  A Self-Teaching Guide

  • Daniel Kleppner, Peter Dourmashkin, Norman Ramsey(Authors)
  • 2022(Publication Date)
  • Jossey-Bass

    (Publisher)

  CHAPTER TWO Differential Calculus In this chapter you will learn about • The concept of the limit of a function; • What is meant by the derivative of a function; • Interpreting derivatives graphically; • Shortcuts for finding derivatives; • How to recognize derivatives of some common functions; • Finding the maximum or minimum values of functions; • Applying differential calculus to a variety of problems. 2.1 The Limit of a Function 97 Before diving into differential calculus, it is essential to understand the concept of the limit of a function. The idea of a limit may be new to you, but it is at the heart of calculus, and it is essential to understand the material in this section before going on. Once you understand the concept of limits, you should be able to grasp the ideas of differential calculus quite readily. Limits are so important in calculus that we will discuss them from two different points of view. First, we will discuss limits from an intuitive point of view. Then, we will give a precise mathematical definition. Go to 98. 57 58 Differential Calculus Chap. 2 98 Here is a little bit of mathematical shorthand, which will be useful in this section. Suppose a variable x has values lying in an interval with the following properties: 1. The interval surrounds some number a. 2. The difference between x and a is less than another number B, where B is any number that you choose. 3. x does not take the particular value a. (We will see later why this point is excluded.) The above three statements can be summarized by the following: |x − a| > 0 (This statement means x cannot have the value a.) |x − a| < B (The magnitude of the difference between x and a is less than the arbitrary number B.) These relations can be combined in the single statement: 0 < |x − a| < B. (If you need to review the symbols used here, see frame 20.) The values of x which satisfy 0 < | x − a | < B are indicated by the interval along the x-axis shown in the figure.

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  Anton's Calculus

  Early Transcendentals

  • Howard Anton, Irl C. Bivens, Stephen Davis(Authors)
  • 2018(Publication Date)
  • Wiley

    (Publisher)

  172 4 In this chapter we will study various applications of the derivative. For example, we will use methods of calculus to analyze functions and their graphs. In the process, we will show how calculus and graphing utilities, working together, can provide most of the important information about the behavior of functions. Another important application of the derivative will be in the solution of optimization problems. For example, if time is the main consideration in a problem, we might be interested in finding the quickest way to perform a task, and if cost is the main consideration, we might be interested in finding the least expensive way to perform a task. Mathematically, optimization problems can be reduced to finding the largest or smallest value of a function on some interval, and determining where the largest or smallest value occurs. Using the derivative, we will develop the mathematical tools necessary for solving such problems. We will also use the derivative to study the motion of a particle moving along a line, and we will show how the derivative can help us to approximate solutions of equations. THE DERIVATIVE IN GRAPHING AND APPLICATIONS 4.1 ANALYSIS OF FUNCTIONS I: INCREASE, DECREASE, AND CONCAVITY Although graphing utilities are useful for determining the general shape of a graph, many problems require more precision than graphing utilities are capable of producing. The purpose of this section is to develop mathematical tools that can be used to determine the exact shape of a graph and the precise locations of its key features. INCREASING AND DECREASING FUNCTIONS The terms increasing, decreasing, and constant are used to describe the behavior of a func- tion as we travel left to right along its graph. For example, the function graphed in Fig- ure 4.1.1 can be described as increasing to the left of x = 0, decreasing from x = 0 to x = 2, increasing from x = 2 to x = 4, and constant to the right of x = 4.

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  Teaching Secondary School Mathematics

  Research and practice for the 21st century

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

    (Publisher)

  the attaining of a limit can lead to students simultaneously holding quite different conceptions in theoretical, as opposed to problem, situations (Juter, 2005; Williams, 2001). There are thus incompatibilities in seeing limits as a process and as an object. Many of the difficulties students experience with other concepts such as continuity, differentiability and integration can be related to their difficulties with limits, according to Tall (1992, 1996) and Williams (2001).

  Rate

  The traditional approach to the introduction of the concept of derivative assumes a sound understanding of rate and illustrates the derivative as the gradient of the tangent to the curve at a point, then moves quickly to emphasise symbolic manipulation. The symbolic representations of a function are manipulated to establish a symbolic expression for instantaneous rate by taking the limit of the average rate. Some students become competent in this manipulation and can accurately produce the symbolic representation of the derivative (delos Santos & Thomas, 2005), but may not appreciate its meaning and connection to other mathematical concepts studied in earlier years.

  Figure 12.15

  Rate diagrams

  Source: Anton et al. (2005, pp. 153–154).

  Many texts use speed as a prototypical example on which to build an understanding of rate where this particular rate is emphasised with a detailed discussion of displacement, velocity, average velocity and instantaneous velocity. The slope of a linear function may be described as a purely abstract definition as the change in the dependent variable resulting from a unit change in the independent variable. This definition may be connected to the graphical representation emphasising the unit rate by the diagram seen in the left of Figure 12.15 . Variable rate may be introduced through reference to the symbolic and graphic representations of the general function y = f(x), again emphasising a unit rate approach (see the right of Figure 12.15

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