Application of Derivatives
The application of derivatives involves using the concept of derivatives from calculus to solve real-world problems. It includes finding maximum and minimum values, determining rates of change, and analyzing the behavior of functions. This application is widely used in various fields such as physics, economics, engineering, and biology to model and understand the behavior of systems and processes.
6 Key excerpts on "Application of Derivatives"
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- 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...
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- 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- Mike Aitken, Bill Broadhurst, Stephen Hladky(Authors)
- 2009(Publication Date)
- Garland Science(Publisher)
...In the previous chapter, we found that by starting with a function of a certain variable and following the rules for differentiation we could obtain a new function, the derivative. The derivative represents the instantaneous rate of change, measuring how rapidly the initial function will alter when the value of a particular variable is perturbed. However, in many situations it is easier to determine the rate at which a quantity changes than to measure how much of it has happened at a particular moment in time. For example, the population of a country is often difficult to monitor directly. It can be measured by a census, but citizens will only put up with intrusive questions once in each decade. Assuming that there is no immigration, the population between these data points can be determined by analyzing the birth rate and the death rate, which are easy to count if the law makes registering these events compulsory (Figure 6.6). Figure 6.6 The relationship between rates of population change, due to births and deaths, and the population size. Similarly, in a study of the ability of wetlands to trap nutrients it may be difficult to quantify the concentration of organic nitrogen species accumulating in a marsh. It is much less taxing to measure the rate at which water flows in from a polluted feeder stream and out through the river that drains the system. As in these examples, we often start out with information about the derivative of a function without prior knowledge of the equation from which the derivative is derived. The name given to the process of reconstructing an original function from its derivative is integration. In other words, integration is the operation that must be performed to undo the effects of differentiation...
- 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)...
