Graphs and Differentiation
Graphs in the context of differentiation represent the relationship between a function and its derivative. The graph of a function's derivative shows the rate of change of the original function at each point. Differentiation is the process of finding the derivative of a function, which measures how the function's output changes with respect to its input.
3 Key excerpts on "Graphs and Differentiation"
eBook - ePub- Mike Aitken, Bill Broadhurst, Stephen Hladky(Authors)
- 2009(Publication Date)
- Garland Science(Publisher)
...For example, seals make the transition between body temperature and their cooler environment gentler by coating themselves with layers of fat and fur. This chapter introduces calculus, the branch of mathematics developed to predict how changes in one quantity will alter another. Beginning with methods for determining the slopes of straight lines and simple curves, we will assemble the basic toolkit of techniques required for differentiation. The tools will then be put to work by showing how to analyze the shapes of different graphs, how to sketch the curve of an unknown function, how to make approximations, and how to handle small errors in experimental measurements. These mathematical details are needed to understand the way that metabolite concentrations change with time in reactions catalyzed by enzymes, to analyze the growth of bacteria, and to explain how a pH buffer solution works, which are all discussed in the next few chapters. The principles developed here also have applications in current research, from magnetic resonance imaging of the human head to generating computational models of the electrical activity of the heart. 5.1 The slope of a straight line Graphs provide an important way of conveying information in biology, as well as many other areas, such as economics. For example, Figure 5.3 shows the cumulative number of cattle infected with the foot and mouth virus each week during an outbreak of the disease. The detection and management of such an outbreak is interesting in terms of the biology of infection and the epidemiology of the spread of the disease, but also has serious financial consequences for farmers, not to mention the impact on the cattle themselves. To interpret the layers of information summarized by a graph of this sort, we need to start by revising some of the basics. In Section 1.5 we discussed the relationship between temperatures measured on the Fahrenheit (° F) and Celsius (° C) scales: the relationship is given in EQ1.26...
- F. Xavier Malcata(Author)
- 2020(Publication Date)
- Wiley(Publisher)
...(10.1). A graphical representation of differential is conveyed by Fig. 10.1, and the usefulness of differentials to approximate a function becomes clear from inspection thereof; after viewing dy as a small variation in the vertical direction, viz. (10.2) one may retrieve Eq. (10.1) to attain a relationship with the corresponding small variation in the horizontal direction, i.e. (10.3) The actual variation experienced by f { x }, i.e. df ≡ f { x + dx } − f { x }, becomes coincident with the variation predicted by the derivative, i.e. df / dx multiplied by dx, when dx is small enough. In other words, if the interval under scrutiny around a given point is sufficiently narrow, then any function will behave as if it were linear with slope equal to the underlying derivative – so its evolution will be driven by the straight line tangent to its graph at said point. Figure 10.1 Graphical representation of continuous function, with ordinate f { x } and tangent to its plot described by slope df / dx at abscissa x, and ordinate f { x + dx } when said abscissa undergoes increment of dx. If a bivariate function is considered, then the total differential should be formulated as (10.4) such that (10.5) the last term in Eq. (10.4) resembles the extra contribution to a hypothenuse of a right triangle, as per Pythagoras' theorem – see Eq. (2.431). Since ε is usually much smaller than the first two terms, one often simplifies Eq. (10.4) to (10.6) – or, in the general case, (10.7) pertaining to a multivariate function in n independent variables x 1, x 2, …, x n ; the latter formulation possesses the further advantage of requiring only partial derivatives as departing information. Inspection of Eq. (10.1) indicates that dy is a function of x – but only through df / dx, since dx represents an arbitrary change in the independent variable that is, in turn, independent of x itself; one may apply the concept of differential also to dy, according to (10.8) Application of operator, as per Eq...
eBook - ePub- Gregory Hill, Mel Friedman(Authors)
- 2013(Publication Date)
- Research & Education Association(Publisher)
...For example, The graph of and the line tangent at x = 2 are shown in Figure 3.4. Notice that the x - and y -intercepts of the line show the slope to be Figure 3.4 EXAMPLE 3.9 A particle is moving along the x -axis. Its position as a function of time is defined as Find an expression for the instantaneous rate of change of position at any moment t > 0. SOLUTION This is just a change of variables. Use Simplifying the limit above is done by a special process called “rationalizing the numerator” by multiplying the numerator and denominator by the conjugate of the numerator, This means the instantaneous rate of change of the position of the particle at any time, t, is the reciprocal of twice the square root of t. DIFFERENTIABILITY A term related to derivatives is differentiability. If the derivative of a function exists at a point of its domain, the function is said to be differentiable at that point. If the derivative is defined at all points of a given interval, then the function is said to be differentiable on that interval. In Example 3.9, It is clear to see that the derivative is defined only for positive values of t. Therefore p (t) is differentiable for t > 0; it is not differentiable for t ≼ 0 Remember that the derivative is defined by using a limit. If for any reason that limit fails to exist, then the derivative does not exist. Table 3.2 summarizes common reasons for the derivative to fail to exist at a given point x = a Table 3.2 Reason and sketch of f (x) at x = a Explanation Corner The limit of the slopes of the secants from the left and right sides of a are different. Cusp The limits of the slopes of the secants from the left and right are opposite. Discontinuity f (a) does not exist, so the limit cannot be evaluated...
