Derivative of Polynomial
The derivative of a polynomial is a fundamental concept in calculus that represents the rate of change of the polynomial function at any given point. It is calculated by finding the slope of the tangent line to the curve of the polynomial function at a specific point. The derivative of a polynomial can be used to analyze the behavior of the function and solve various engineering and technological problems.
5 Key excerpts on "Derivative of Polynomial"
- F. Xavier Malcata(Author)
- 2020(Publication Date)
- Wiley(Publisher)
...8 Limits, Derivatives, Integrals, and Differential Equations Most functions of practical interest – in that they can be used to simulate physicochemical phenomena, are intrinsically continuous; this means that they evolve smoothly along their independent variable, i.e. the value taken by the function at a given value of its independent variable coincides with what would be expected from the trends in the vicinity of said value for the latter variable. This calls for the concept of limit – based on realization that an infinite sequence of values taken by a function may be such that every element will never go beyond a finite threshold; said topic is particularly relevant when seeking asymptotic behaviors of functions – since they tend to take forms simpler than the original functions themselves; or when quantifying the tendency of evolution in the neighborhood of some point via the concept of derivative (or ratio of small variations of the function to its independent variable). Derivatives find their widest applicability when searching for optima – one of the most seminal goals of process engineers. In fact, local maxima of a given function are described by nil values of the corresponding derivative if not subjected to otherwise (externally) imposed physical constraint(s); decision on the type of optima would then come at the expense of higher order derivatives – thus extending the concept of differentiation of a function itself to differentiation of a previous derivative. Once a function (seen as an algebraic operator) is proposed, one may in principle define its inverse; this supports the common strategy to solve an algebraic equation. A similar rationale may be applied to the derivative – in which case the concept of integral arises...
eBook - ePub- Gregory Hill, Mel Friedman(Authors)
- 2013(Publication Date)
- Research & Education Association(Publisher)
...This same graphical comparison is now used on h (x) = 1 + | x | in Figures 3.7 and 3.8. In this case, however, no matter how closely one looks at the point (0, 1), the curve will never appear straight. Figure 3.7 Figure 3.8 In Section 3.3, the derivative is defined as the slope of the line tangent to a graph at a given point. The tangent line at any given point is the linear approximation of the function over relatively small intervals. For nonlinear functions, the approximation is accurate only over small intervals because any nonlinear function has a varying slope, whereas a line has a constant slope. EXAMPLE 3.11 Find the line tangent to at x = 4. SOLUTION An equation of a line can be written by using the slope of the line and any point on the line. The function provides the point, and the derivative of the function provides the slope. g (4) = 2 The point of tangency to g (x) is (4, 2). Applying the result of Example 3.9 to g, so. Thus the slope of the tangent at x = 4 is Using the point-slope form, the tangent line equation is Figure 3.9 shows the line tangent to g (x) at x = 4 Figure 3.9 Figure 3.9 reinforces the idea that near the point of tangency, the line provides reasonably good approximations of the function, but as the domain around x = 4 widens, the graphs diverge. 3.5 EXERCISES Work the following exercises without a calculator. Solutions follow this section. 1. In a sentence or two, describe the difference between the average rate of change of a function and the instantaneous rate of change of a function. 2. Find the average rate of change of h (x) = x 3 – 2 x on the domain [1,3]. 3. A circular metal plate was heated. When the temperature at the center reached 100 ° C, the heat source was removed. Table 3.3 shows the temperature at the center of the plate as it cooled over time...
- 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)...
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- 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...
