Parametric Differentiation

3 Key excerpts on "Parametric Differentiation"

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  eBook - ePub

  The Learning and Teaching of Calculus

  Ideas, Insights and Activities

  • John Monaghan, Robert Ely, Márcia M.F. Pinto, Mike Thomas(Authors)
  • 2023(Publication Date)
  • Routledge

    (Publisher)

  educational issues related to offering students rules that cannot be justified at students’ current mathematical level. Edumatter In the consideration of differentiating implicit functions we introduced a rule ‘differentiate both sides of the equation and solve for d y d x ’. How do you justify this rule to yourself? How do you explain the rule to your students? But all the aforementioned problems with implicit functions are problems in standard analysis. If we take an infinitesimal or differential, then learning and teaching can be more straightforward; as Ely (2021, p. 597) says, “In a differentials-based approach the term does not even need to be used; nothing is different about it”. If y 2 = xy + x, then 2 ydy = xdy + ydx + 2 xdx. Our final considerations in this sub-section concern differentiating parametric functions. When we write a function in the form y = f (x), we regard the dependent variable y as a function of the independent variable x. An alternative way to consider the function is to consider both x and y as dependent variables of a third independent variable t, the so-called ‘parameter’. 14 There are at least two reasons why we might introduce parametric functions in an elementary calculus course. The first is that it is often convenient in kinematics. For example, if a stone is thrown horizontally, by a person on a cliff, with a velocity of 20 m/s, then the trajectory can be modelled using time (t): X = 20 t ; y = -5 t 2. NB by eliminating t in these equations we can see that the trajectory can be described in y = f (x) form, y = − 1 80 x 2. The second reason is that mathematics considers curves that cannot be 80 described by a single function. For example, x 2 + y 2 = 1 describes the unit circle, centre (0, 0). We require two single valued functions to describe this, y = 1 − x 2 and y = − 1 − x 2, but we only need one parametric function to describe this circle, x = cost; y = sin t

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  eBook - ePub

  Mathematics for Business Analysis

  • Paul Turner, Justine Wood(Authors)
  • 2023(Publication Date)
  • Mercury Learning and Information

    (Publisher)

  4

  DERIVATIVES AND DIFFERENTIATION

  The analysis of change is central to both Economics and Business. For example, we might be interested in how consumers adjust their spending plans as the relative price of commodities varies, or we might want to model how the level of output in the economy adjusts if the central bank alters the interest rate. The branch of mathematics which deals with the analysis of change is calculus . There are two main subfields of calculus which are known as differential calculus and integral calculus , respectively. You will need to become familiar with both in order to conduct economic and business analysis. In this chapter, we will begin by covering the basics of differential calculus.

  4.1 DIFFERENTIAL CALCULUS

  Differentiation is the process of finding the rate of change of one variable produced by changes in another variable. Differentiation provides an important mathematical tool in both economic and business theory. Although the theory of differentiation can initially appear quite daunting, the practical rules for its application are quite simple.

  Differential calculus is concerned with the process of finding the rate at which one quantity changes in response to changes in another related variable. Consider a function of the form , where the domain is some subset of the real numbers. Ideally, we would like to measure the instantaneous rate of change of y as x changes. As a first attempt, we can find an approximation for this as , where and . This is the slope of the straight line drawn between two points on the function. Differential calculus starts with an approximation of this form and then looks to determine what happens when the change in x is very small.

  Consider the example shown in Figure 4.1 . The graph shows the quadratic function , where the domain is the set of real numbers What does Figure 4.1 tell us about the gradient of this function? First, it is obvious that, unlike the case of the linear function, the gradient is not constant. Second, we can see that gradient varies systematically with the value of the x variable. When x is positive, the gradient is also positive, and, as the value of x increases, the gradient increases. If x is negative, then the gradient is negative and becomes larger (in absolute value) as x becomes more negative. This means that the relationship between the gradient and the value of x is itself a function of x

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  eBook - ePub

  Two and Three Dimensional Calculus

  with Applications in Science and Engineering

  • Phil Dyke(Author)
  • 2018(Publication Date)
  • Wiley

    (Publisher)

  Later when differential geometry is discussed, it will be seen that is a parameter that traces curves on the surface as it varies. There are lots of choices of course as the number of scribbles one can make on a given surface is infinite. The two choices just made for our particular are quite valid and correspond to the partial derivatives in the - and -direction. This will lead to directional derivatives, but that will have to wait until vectors have been covered (but see Exercise 2.3). Combining these two results by simply adding them defines what is called a total derivative 2.7 Adding a third variable would give Equation (2.1) divided by 2.8 and a generalisation to variables is possible: where, but will do most of the time in this text. First of all the above expression is easy to remember. It is as if each term captures the variation of due to each component then the total variation of is obtained by simply adding together all these contributions. This adding up of individual contribution implies linearity and is valid due to a limiting processes of the type involving and met in the last section; any non-linearity in is smoothed and ultimately everything is linearised. If is time, then Equation (2.8) will have applications in the physical world, notably mechanics where the time derivatives of and are written using the more Newtonian inspired notation as and and they denote the speed in each of the co-ordinate directions. Equation (2.8) can be extended to the case where, that is is also, in addition, an explicit variable for the function. In this case 2.9 and this has applications in fluid mechanics. Another useful notation introduced in Chapter 1 and reused in Example 2.4 is where the prime symbol denotes differentiation with respect to, in this instance,, but means differentiation with respect to the argument of the function whatever that argument might be. For example if, then so using the above results in so in this particular case

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