Antiderivatives
Antiderivatives, also known as indefinite integrals, are functions that reverse the process of differentiation. They represent a family of functions whose derivatives match a given function. Finding antiderivatives is a fundamental concept in calculus and is used to solve problems involving the accumulation of quantities over time.
7 Key excerpts on "Antiderivatives"
eBook - ePub- Arsen Melkumian(Author)
- 2012(Publication Date)
- Routledge(Publisher)
...9 Indefinite and definite integrals Integrals play a twofold role in calculus. The so-called indefinite integral is an operation inverse to differentiation: the integral of function f (x) is a function whose derivative gives us f (x). We have seen the importance of derivatives throughout the textbook, and naturally the inverse operation also plays an important role in mathematical and economic analysis. Later, in Chapter 12, we study relations between quantities and their rates of change – the so-called differential equations. Solving differential equations is impossible without indefinite integrals. The other type of integral is the definite integral. Geometrically a definite integral represents the area under a curve. But in applications the meaning of integration is totaling continuous quantities. For instance, to reconstruct profit from its rate of change we would integrate the latter function. Another application of integrals is in studying consumer–producer surplus. 9.1 Indefinite integrals If f (x) and F(x) are some functions of x such that then F(x) is called an antiderivative of f (x). For example, the function F 1 (x) = 3 x 2 + 5 is an antiderivative of f (x) = 6 x. Note that F 1 (x) = 3 x 2 + 5 is not the only antiderivative of f (x). In fact, the function f (x) = 6 x has infinitely many Antiderivatives and all Antiderivatives of f (x) are functions of the form where C is a constant. We will refer to F(x) = 3 x 2 + C as the general antiderivative of f (x) and we will write to indicate that. The symbol is referred to as the integral sign, 6 x is the integrand and C is the constant of integration. The expression is known as the indefinite integral of f (x) = 6 x. T HEOREM 9.1: Let f (x) be a differentiable function of x, and k, n and C be some constants...
eBook - ePub- Gregory Hill, Mel Friedman(Authors)
- 2013(Publication Date)
- Research & Education Association(Publisher)
...CHAPTER 6 Antidifferentiation and Definite Integrals CHAPTER 6 ANTIDIFFERENTIATION AND DEFINITE INTEGRALS 6.1 INTRODUCTION To this point the entire focus of the material has been on taking derivatives of functions and utilizing this process to achieve a better understanding of the behavior of functions as well as finding solutions to applied problems. The rest of an introductory calculus course reverses this process by using antidifferentiation and explores the multitude of practical situations for which this is useful. The connection between differential and integral calculus is one of the most amazing and beautiful achievements in mathematics. 6.2 CONCEPT OF THE ANTIDERIVATIVE Consider a simple function such as f (x) = 2 x. This function could very easily be the derivative of another function, such that From previous experience, it should be relatively easy to conclude that F (x) could be x 2. But could it also be F (x) = x 2 + 3? How about F (x) = x 2 – 10? Of course it could, since the derivative of a constant is 0. Antiderivative A function F (x) is an antiderivative of a function f (x) if All Antiderivatives of a given function differ by a constant. At this point, let’s examine a few examples just to get the idea of finding a simple antiderivative. It is wise to recognize that not every antiderivative is simple to find, and some functions have no symbolic antiderivative at all. This will be examined in greater detail later. EXAMPLE 6.1 Find an antiderivative of y ′ = 4 x 3. SOLUTION At this point, use a guess and check method. Since y ′ is a polynomial, its derivative came from reducing its power by 1, so guess y = x 4. Sure enough, if y = x 4 or y = x 4 + (any constant), EXAMPLE 6.2 Find a representation of all functions whose derivative is f (x) = sin(3 x). SOLUTION A first guess may be F (x) = cos(3 x) because the derivative of cosine involves sine...
eBook - ePub- Mike Aitken, Bill Broadhurst, Stephen Hladky(Authors)
- 2009(Publication Date)
- Garland Science(Publisher)
...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. We can start to unpack this concept by differentiating the function y = x 2, to achieve the familiar result that d y d x = 2 x. Integration is the reverse of this process, so alongside stating that ‘the derivative of x 2 is 2 x ’, it looks like we should also be able to say ‘the integral of 2 x is x 2 ’. Unfortunately, this picture is a little too simplistic. The true situation is more complex because an infinite number of functions can in fact be differentiated to get a result of 2 x. For example, we would get the same result if instead of y = x 2 we had chosen y = x 2 + 3 or even y = x 2 – 14. In each case, the derivatives of the second terms on the right-hand sides of the equations are zero, because they are all constants. Writing out these derivatives explicitly can emphasize the ambiguity of attempting to undo the effects of differentiation: d y d x (x 2) = 2 x ; d d x (x 2 + 3) = 2 x ; d y d x (x 2 − 14) = 2 x. In any attempt to reverse the process of differentiation, we have no idea what the value of the constant term in the original function may have been. Because of this, it is important to include an unknown constant in our answer for the integral of a function. The unknown constant, often written as c, is called the ‘constant of integration’. This means that the result for the integral of 2 x stated above was only partly correct...
- Stu Schwartz(Author)
- 2013(Publication Date)
- Research & Education Association(Publisher)
...All of differential calculus is based on this definition, so you need to know it. The relationship that we established early in this chapter can be expressed as: Our usual notation for the derivative can be one of a few. If the function is given in the form of “f (x) =”, the derivative will be written as f ′ (x) or f′. If the function is given in the form of “y =”, the derivative will be written as y or. These notations are interchangeable. So what you need to know is this: The informal definition of derivative is a formula for the slope of the tangent line to a curve at a point. The formal definition of a derivative is. A formal definition to find a derivative at a point is. EXAMPLE 5: Using the formal definition, find the derivative of f (x) = x 2 + 7 x − 2, and use it to find the slope of the tangent line to f at x = −2. SOLUTION: Using our technique of finding limits, realize that plugging in h = 0 will give a zero in the denominator. So we have to hope we can do some factoring and then canceling. In this type of problem, we always can. slope of tangent line at x = −2 = f ′(−2) = 2(−2) + 7 = 3 EXAMPLE 6: Using the secondary definition, find f ′ (5) if. SOLUTION: TEST TIP Very rarely will you ever be asked to take the derivative of a function by using the definition. There are better ways, covered in the following several sections. However, you must understand the definition so that you know that is equivalent to the slope of the tangent line to y = cos x at x = 1. Doing this by the definition is cumbersome. Using the derivatives rules below makes the problem quite simple. Also note that the usual h in the definition is replaced by ∆ x which means change in x. It makes no difference what variable is used to represent this change. Derivatives of Algebraic Functions Overview: Differential calculus requires you to be able to take derivatives quickly and efficiently...
- J. Rosebush, Stu Schwartz(Authors)
- 2016(Publication Date)
- Research & Education Association(Publisher)
...Separating the variables yields ∫ ydy = dx. Taking the antiderivative of both sides,. The two constants of integration get combined to yield, or simply,. There is no need to write the two constants; writing only one C is acceptable. To find the value of C apply the initial condition, y | x = 1 = 3. This yields:. The equation may be left as is or it may be algebraically manipulated into one of the various other forms: y 2 = 2 x + 7, or y 2 – 2 x = 7,, but not as the point (1, 3) doesn’t pass through. B. Slope fields are fields of slopes, literally. Given a family of differentiable functions, y = f (x) + C, imagine drawing a tiny tangent line at each point on these functions. The set of all of these tangent lines forms the slope field for the function. When given a differential equation, the original function can be obtained by drawing the slope field. 1. For example, given and a point on the original function,. Substituting some x and y values into helps us create the slope field, below. X y 0 1 0 0 2 0 0 –1 0 0 –2 0 1 0 – ∞ 1 1 –1 1 2 1 –1 1 1 –2 –1 0 ∞ –1 1 1 –1 2 –1 –1 –1 –1 –2 2 0 – ∞ 2 1 –2 2 –1 2 –2 0 ∞ –2 1 2 –2 –1 –2 This slope field suggests that the function whose derivative is given belongs to the family of circles with the center at the origin. Solving the differential equation by separating and integrating, yields The particular solution to the given differential equation is or. equivalently, x 2 + y 2 = 4, which represents a circle with the center at the origin and radius 2 units. VI. MOTION A. Rectilinear motion in the Cartesian system. Since the acceleration, velocity, and displacement of an object can be expressed as derivatives, that is, and, it follows that they may also be expressed as Antiderivatives. 1. The total displacement of an object moving in a straight line from t = t 1, to t = t 2 is represented by. The displacement equation is given by ∫ v (t) dt and requires an initial condition to be given...
- 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...
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...
