Derivatives and the Shape of a Graph
"Derivatives and the Shape of a Graph" explores the relationship between the derivative of a function and the shape of its graph. It delves into how the sign and value of the derivative can provide insights into the increasing, decreasing, concave up, and concave down nature of the function. Understanding these concepts is crucial for analyzing and interpreting the behavior of functions in mathematics.
7 Key excerpts on "Derivatives and the Shape of a Graph"
eBook - ePub- Arsen Melkumian(Author)
- 2012(Publication Date)
- Routledge(Publisher)
...4 Limits and derivatives In mathematics, differential calculus is a subfield of calculus that is concerned with the study of how quickly functions change over time. The primary concept in differential calculus is the derivative function. The derivative allows us to find the rate of change of economic variables over time. This chapter introduces the concept of a derivative and lays out the most important rules of differentiation. To properly introduce derivatives, one needs to consider the idea of a limit. We cover the concept of a limit in the first section. The chapter closes with growth rates of discrete and continuous variables. 4.1 Limits Consider a function g given by and shown in Figure 4.1. Clearly, the function is undefined for x = 0, since anything divided by zero is undefined. However, we can still ask what happens to g (x) when x is slightly above or below zero. Using a calculator we can find the values of g (x) in the neighborhood of x = 0, as shown in Table 4.1. As x approaches zero, g (x) takes values closer and closer to 2. So we can say that g(x) tends to 2 as x tends to zero. We write and say that the limit of g (x) as x approaches zero is equal to 2. Now that the idea of a limit is clear on an intuitive level, let us consider a formal definition of the right- and left-hand side limits. Let f be a function defined on some open interval (a, b). We say that L is the right-hand side limit of f (x) as x approaches a from the right and write if for every ε > 0 there is a δ > 0 such that Figure 4.1 Table 4.1 whenever As an example, let us consider the following function We want to show that Let us choose ε > 0. We need to show that there is a δ > 0 such that whenever Let us choose δ = (ε/ 2). Then, and therefore It follows immediately that whenever Now we have proved that the limit of as x approaches zero from the right is equal to 1. Now let us define a left-hand side limit. Let f be a function defined on some open interval (a, b)...
eBook - ePub- Gregory Hill, Mel Friedman(Authors)
- 2013(Publication Date)
- Research & Education Association(Publisher)
...CHAPTER 5 Applications of Differentiation CHAPTER 5 APPLICATIONS OF DIFFERENTIATION 5.1 INTRODUCTION As with all mathematical concepts, the power of differentiation comes from applying its concepts either to expanding our understanding of further mathematics or to explaining observed real-life situations. Derivatives can be used for both of these pursuits. Since they allow us to measure change—both instantaneous and average rates of change—and since functions and the world around us are not static, this chapter applies derivatives to the study of function behavior and real-world applications. 5.2 FUNCTIONS AND FIRST-DERIVATIVE APPLICATIONS LOCAL EXTREME VALUES One of the most important applications of calculus is determining maximum or minimum values for situations that involve many options. For example, manufacturers desire to maximize profits on items they produce. Profits will fluctuate depending on a variety of variables, such as overhead costs, selling price, or market conditions. In industry, the problem often is analyzed with numerical models, or by using multivariable calculus. The foundations of these processes begin with simple optimization, as experienced in this course. Set as goals the development of both an intuitive understanding of maxima and minima and a mastery of the analytic work behind determining them. Figure 5.1 shows examples of local maxima and minima and the various ways they can occur on the graph of a function. Figure 5.1 The abscissa is considered the location of a local maximum or minimum, and the ordinate is the actual maximum or minimum value. For example, a correct statement is, “The graph of h (x) has a local maximum of h (b) at x = b.” Informally, a local maximum is the highest point in a small interval, and a local minimum is the lowest point in a small interval. In Figure 5.1, local maxima exist at b, d, and f. Local minima occur at a, c, and e...
eBook - ePub- Mike Aitken, Bill Broadhurst, Stephen Hladky(Authors)
- 2009(Publication Date)
- Garland Science(Publisher)
...It is worth noting that the slope of the curve is zero at the maximum point, something that is true for any curve that possesses a feature of this sort. However, the converse is not always the case. A point with zero slope may also represent a minimum, as found at point (0, 1000) in Figure 5.13. Maxima and minima together are both referred to as turning points, because the gradient of the curve changes either from positive to negative or from negative to positive as you pass through the point. Box 5.1 Forms of notation for differentiation Throughout this book, derivatives are presented using Leibniz’s notation. The derivatives with respect to x for the function y = f (x) are written as: first derivative = d y d x = d d x (f (x)), n th derivative = d n y d x n. An advantage of using Leibniz’s notation is that the variable used in the differentiation process (x) is always mentioned explicitly in the denominator of the derivative. Other common approaches provide useful abbreviations in the right context, but their meaning is less obvious for newcomers. For example, Lagrange’s notation uses a ‘prime’ symbol to distinguish between a function and its derivative, with second and third derivatives marked by additional primes: for example f ′ (x) = d f d x, f ″ (x) = d 2 f d x 2, f ″ (x) = d 3 f d x 3, and f (n) (x) = d n f d x n. Higher derivatives can be represented by a superscript containing an Arabic number in brackets. This representation is compact and easy to typeset, so it is often used in mathematics and physics textbooks. However, it has a few disadvantages: the number of prime marks can easily be miscounted, and primes are used to denote many other things...
eBook - ePub- Adil H. Mouhammed(Author)
- 2015(Publication Date)
- Routledge(Publisher)
...CHAPTER TWO Derivatives and Applications Economic decisions are based on marginal analysis. For example, the monopolist's best level of output is determined by equating marginal cost and marginal revenue. To find the marginal cost and revenue, total cost and revenue functions must be differentiated with respect to the output level. Similarly, derivatives can be used in many applications in business and economics. For this reason, the rules of differentiation are outlined in this chapter, and many applications are provided. The Concept of Derivative The derivative of a function measures the rate of change of the dependent variable y with respect to the independent variable x--the slope of the function. That is, the derivative indicates the impact of a small change in x on y. For example, suppose the dependent variable y is the quantity supplied by a producer, and x is the price of that product. Mathematically, the function is written as y = f(x). Now, if the price x changes by a very small amount (dx), the quantity supplied will change by a very small amount (dy) as well. These small changes, dx and dy, are called the differential of x and y, respectively. After these changes, the new magnitude of the two variables becomes (y + dy) and (x + dx). And dy/dx is called the derivative of y with respect to x. In other words, dy/dx shows the changes in y per unit change in x. The process of finding the derivative is called the differentiation process. If a given function is a univariate function, such as the above, the following rules of differentiation (Glaister 1984; Chiang 1984; Ostrosky and Koch 1986) are applied: Rule 1: Derivative of a Constant function If y = f(x) = k, where k is a constant, then dy/dx = 0 Example 1: Differentiate y = f(x) = 30. Solution: dy/dx = 0 Example 2: If the fixed cost of a product q is FC = 20, then the derivative of the fixed cost with respect to q is d(FC)/dq = 0...
eBook - ePub- Adil H. Mouhammed(Author)
- 2020(Publication Date)
- Routledge(Publisher)
...Chapter Two Derivatives and Applications Economic decisions are based on marginal analysis. For example, the monopolist’s best level of output is determined by equating marginal cost and marginal revenue. To find the marginal cost and revenue, total cost and revenue functions must be differentiated with respect to the output level. Similarly, derivatives can be used in many applications in business and economics. For this reason, the rules of differentiation are outlined in this chapter, and many applications are provided. The Concept of Derivative The derivative of a function measures the rate of change of the dependent variable y with respect to the independent variable x--the slope of the function. That is, the derivative indicates the impact of a small change in x on y. For example, suppose the dependent variable y is the quantity supplied by a producer, and x is the price of that product. Mathematically, the function is written as y = f (x). Now, if the price x changes by a very small amount (dx), the quantity supplied will change by a very small amount (dy) as well. These small changes, dx and dy, are called the differential of x and y, respectively. After these changes, the new magnitude of the two variables becomes (y + dy) and (x + dx). And dy/dx is called the derivative of y with respect to x. In other words, dy/dx shows the changes in y per unit change in x. The process of finding the derivative is called the differentiation process. If a given function is a univariate function, such as the above, the following rules of differentiation (Glaister 1984; Chiang 1984; Ostrosky and Koch 1986) are applied: Rule 1: Derivative of a Constant Function If y = f(x) = k, where k is a constant, then dy/dx = 0 Example 1: Differentiate y = f(x)= 30. Solution: dy/dx = 0 Example 2: If the fixed cost of a product q is FC = 20, then the derivative of the fixed cost with respect to q is d(FC)/dq = 0...
- J. Rosebush, Stu Schwartz(Authors)
- 2016(Publication Date)
- Research & Education Association(Publisher)
...PART III DERIVATIVES Chapter 7 Derivatives I. DERIVATIVES A. Meaning of Derivative The derivative of a function is its slope. A linear function has a constant derivative since its slope is the same at every point. The derivative of a function at a point is the slope of its tangent line at that point. Non-linear functions have changing derivatives since their slopes (slope of their tangent line at each point) change from point to point. 1. Local linearity or linearization—when asked to find the linearization of a function at a given x -value or when asked to find an approximation to the value of a function at a given x -value using the tangent line, this means finding the equation of the tangent line at a “nice” x -value in the vicinity of the given x -value, substituting the given x -value into it and solving for y. i. For example, approximate using the equation of a tangent line to. We’ll find the equation of the tangent line to at x = 4 (this is the ‘nice’ x -value mentioned earlier). What makes it nice is that it is close to 4.02 and that. Since, so,. Also, f (4) = 2. Substituting these values into the equation of the tangent line, so the equation of the tangent line is. Substituting x = 4.02, y = 2.005. A more accurate answer (using the calculator) is. The linear approximation, 2.005, is very close to this answer. This works so well because the graph and its tangent line are very close at the point of tangency, thus making their y -values very close as well. If you use the tangent line to a function at x = 4 to approximate the function’s value at x = 9, you will get a very poor estimate because at x = 9, the tangent line’s y -values are no longer close to the function’s y -values. ii. The slope of the secant on (a, b), is often used to approximate the value of the slope at a point inside (a, b). For instance, given the table of values of f (x) below, and given that f (x) is continuous and differentiable, approximate f ′(3)...
- F. Xavier Malcata(Author)
- 2020(Publication Date)
- Wiley(Publisher)
...10 Differentials, Derivatives, and Partial Derivatives The concept of differential entails a small (tendentially negligible) variation in a variable x, denoted as dx, or a function f { x }, denoted as df { x }; the associated derivative of f { x } with regard to x is nothing but the ratio of said differentials, i.e. df / dx – usually known as Leibnitz’s formulation. In the case of a bivariate function, say, f { x,y }, differentials can be defined for both independent variables, i.e. dx and dy – so partial derivatives will similarly arise, i.e. ∂f / ∂x and ∂f / ∂y ; operator ∂ is equivalent to operator d, except that its use is exclusive to multivariate functions – in that it stresses existence of more than one independent variable. 10.1 Differential In calculus, the differential represents the principal part of the change of a function y =. f { x } – and its definition reads (10.1) where df / dx denotes the derivative of f { x } with regard to x ; it is normally finite, rather than infinitesimal or infinite – yet the precise meaning of variables dx and df depends on the context of application, and the required level of mathematical accuracy. The concept of differential was indeed introduced via an intuitive (or heuristic) definition by Gottfried W. Leibnitz, a German polymath and philosopher of the eighteenth century; its use was widely criticized until Cauchy defined it based on the derivative – which took the central role thereafter, and left dy free for given dx and df / dx as per Eq. (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...
