Calculus
Calculus is a branch of mathematics that focuses on the study of rates of change and accumulation. It includes two main branches: differential calculus, which deals with the concept of the derivative and its applications, and integral calculus, which focuses on the concept of the integral and its applications. Calculus is widely used in various fields such as physics, engineering, economics, and computer science.
5 Key excerpts on "Calculus"
- Magno Urbano(Author)
- 2019(Publication Date)
- Wiley(Publisher)
...2 Infinitesimal Calculus : A Brief Introduction 2.1 Introduction In this chapter we will do a brief introduction to infinitesimal Calculus or differential and integral, known as simply, Calculus. Wikipedia has as good definition about Calculus: Calculus is the mathematical study of continuous change, in the same way that geometry is the study of shape and algebra is the study of generalizations of arithmetic operations. It has two major branches, differential Calculus (concerning instantaneous rates of change and slopes of curves) and integral Calculus (concerning accumulation of quantities and the areas under and between curves). These two branches are related to each other by the fundamental theorem of Calculus. Both branches make use of the fundamental notions of convergence of infinite sequences and infinite series to a well‐defined limit. Generally, modern Calculus is considered to have been developed, independently, in the 17 th century by Isaac Newton and Gottfried Wilhelm Leibniz. Today, Calculus has widespread uses in science, engineering, and economics. Calculus is a part of modern mathematics education. 2.2 The Concept Behind Calculus Suppose we have a car traveling in a straight line for 1 h, at a constant speed of 100 km/h, and later reducing the speed in half and traveling for another hour. What is the average speed of that car? The answer is Now let us see a more complex problem. The car travels 15 min at a nonconstant speed of 15 km/h, stops for 5 min, travels for 5 km at a nonconstant speed of 8 km/h, stops again, and then travels a distance of 2 km at 15 km/h. What is the average speed now? The answer is not evident because the method we have cannot deal with variable entities...
- Stu Schwartz(Author)
- 2013(Publication Date)
- Research & Education Association(Publisher)
...AP Calculus AB Chapter 4 Basic Calculus Concepts and Limits Introduction On the AP exam, there is no requirement to know any of the fascinating evolution of Calculus. However, many students do not even know what Calculus is; they just know it is a math course. I think of Calculus as the study of change. In courses before Calculus, the value of x is fixed. You were told that x = 5, the hypotenuse of the triangle is 10, or the height of the water in the tank is 4 feet. You knew that, at least for that problem, those values would never change. But in Calculus, you might be told the value of x, but you are also told that x is changing. That opens up a brand new world. Think of all the aspects of yourself that are changing. You are getting taller, your hair and fingernails are growing, your weight might be increasing or decreasing, and blood is not stationary but moving throughout your body. And because so many things in the world are changing, Calculus is a course for which many real-life problems are pertinent, especially in science, economics, and engineering. There are two basic branches of Calculus. The first is called differential Calculus and the problem that it studies is finding the tangent line to a graph at a point. In the figure below, line L is tangent to the graph of f (x) at point P. Chapters 5 and 6 of this book are concerned with the issue of the tangent line problem and the doors this problem opens. The second branch of Calculus is called integral Calculus and the problem that it studies is finding the area under a curve between two vertical lines. In the figure below, the shaded area between the graph of f (x), the lines x = –2, x = 1, and the x -axis is 26.25. Chapters 7 and 8 are concerned with the issue of the area problem and the doors that this problem opens. DID YOU KNOW? Sir Isaac Newton (1642–1727) actually discovered Calculus between 1665 and 1667 after his university closed due to an outbreak of the plague...
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- Rose G Davies(Author)
- 2020(Publication Date)
- CRC Press(Publisher)
...1 Calculus Revision Calculus is a convenient tool in aerodynamics. It aids in explaining the characteristics of functions, which describe the airflow fields. It is assumed that students have gained the skills to differentiate basic functions from previous studies. Readers should have learnt the method to differentiate basic functions – for example, polynomials, logarithmic, trigonometric, and exponential functions. This chapter concentrates on explaining some applications of Calculus, including the meanings of derivatives of a function in real life, an analysis of the changes of a function with two or more variables, and simple concepts of integration. This chapter will use plain language as much as possible so that readers can understand the mathematic expressions when the concepts discussed are used in explaining aerodynamics principles. Differentiation In Calculus, you have learnt differentiation. It is assumed that readers are able to find the derivatives of a function. The following example shows how to revise the meanings of derivatives in physics. Assume a displacement (distance) y in m, of an object is a function of time t in s, and the function is continuous, as shown in Figure 1.1. The function is: FIGURE 1.1 A displacement function y (t). y = y (t) = t 3 − 2 t 2 − 2 t + 1 (1.1) where t is a variable and y is a function of t. Figure 1.1 illustrates this function, t ≥ 0. To differentiate y with respect to t produces the derivative y ′ = d y d t. The derivative y′ indicates the change rate of the original function y. The change rate y′ of the displacement function y is the velocity (speed), v, of the object. y′ is obtained by using (t n) ′ = n t n − 1 : v = y ′ = d y d t = 3 t 2 − 4 t − 2 (1.2) Figure 1.2 shows the velocity function v with respect to time t. FIGURE 1.2 The derivative function y′ (t) of y (t). We differentiate y′ to get y ″ = d 2 y d t 2. y″ is the change of y′, i.e. the change of velocity of the object...
eBook - ePub- Gregory Hill, Mel Friedman(Authors)
- 2013(Publication Date)
- Research & Education Association(Publisher)
...The result does not depend on whether a and b are positive or negative. The left base is 2 a – 1 and the right base is 2 b – 1. The height is b – a. The sum of the two results is b 3 – a 3 + b 2 – a 2 – b + a. 6.6 FUNDAMENTAL THEOREM OF Calculus One of the most impressive developments in Calculus was the connection between the differential and integral branches found in what came to be called the Fundamental Theorem of Calculus. One part of the Fundamental Theorem asserts that a definite integral of a continuous function is a function of its upper limit and is therefore itself differentiable. A second part of the Fundamental Theorem establishes how to analytically evaluate a definite integral. Both parts will be presented here without formal proof, but if it enhances your understanding, any online search for the proof of the Fundamental Theorem of Calculus will produce an abundance of sources to read and study. Fundamental Theorem of Calculus (Part 1) If f is a continuous function, a is a constant, and then This theorem essentially says, “The instantaneous rate of change of accumulation, or loss, of area between a function and the x- axis at any point, is equal to the function value of the integrand at that point.” Let’s take an intuitive way to explain what is happening, using a discrete approach to the concept of accumulating area under a curve. Think of a definite integral as a Riemann sum over a given interval. As the upper limit of the given interval changes, accumulated area increases or decreases by “adding” another infinitely thin rectangle to the previously summed rectangles. If the function values are “large” at a particular x value, then the next rectangle added will add more area than if the function values are small. It follows logically that the amount of change in accumulated area is related to the magnitude of the function being integrated. Notice that the upper limit is simply x...
