Derivatives and Continuity

9 Key excerpts on "Derivatives and Continuity"

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  eBook - PDF

  A Mathematics Course for Political and Social Research

  • Will H. Moore, David A. Siegel(Authors)
  • 2013(Publication Date)
  • Princeton University Press

    (Publisher)

  Part II Calculus in One Dimension Chapter Five Introduction to Calculus and the Derivative In our experience, calculus and all things calculus-related prove the most stress- ful of the topics in this book for those students who have not had prior calculus coursework. We conjecture that this is due to the foreignness of the subject. While probability and linear algebra certainly have some complex concepts one must internalize, much of the routine manipulations students perform in ap- plying these concepts use operations they are used to: addition, multiplication, etc. In contrast, calculus introduces two entirely new operators, the derivative and the integral, each with its own set of rules. Further, these operators are often taught as a lengthy set of rules, leading to stressful rote memorization and little true understanding of what are relatively straightforward concepts, at least as used in most of political science. 1 To try to avoid this, we’re going to take a little more time with the topic. In this chapter we will cover the basics of Calculus and the derivative in what we hope is an intuitive manner, saving the rules of its use for the next chapter. If you are working through this chap- ter as part of a course and are not sure of something, this is the time to ask questions—before you end up trying to take derivatives without having a clear understanding what they are. The first section below provides a brief overview of calculus. The second section introduces the derivative informally, and the third provides a formal definition and shows how it works with a few functions. 5.1 A BRIEF INTRODUCTION TO CALCULUS For our purposes, the primary use of calculus is that it allows us to deal with continuity in a consistent and productive manner. This is likely a useless claim at this point, so let us explain. As we discussed in Chapter 4, a continuous function is one that we can draw without lifting pencil from paper.

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  Calculus

  Early Transcendentals

  • Howard Anton, Irl C. Bivens, Stephen Davis(Authors)
  • 2016(Publication Date)
  • Wiley

    (Publisher)

  2.2 The Derivative Function 95 WARNING The converse of Theorem 2.2.3 is false; that is, a function may be continuous at a point but not differentiable at that point. This occurs, for example, at cor- ner points of continuous functions. For instance, f (x) = |x| is continuous at x = 0 but not differentiable there (Example 6). The relationship between continuity and differentiability was of great historical signif- icance in the development of calculus. In the early nineteenth century mathematicians believed that if a continuous function had many points of nondifferentiability, these points, like the tips of a sawblade, would have to be separated from one another and joined by smooth curve segments (Figure 2.2.12). This misconception was corrected by a series of discoveries beginning in 1834. In that year a Bohemian priest, philosopher, and mathe- matician named Bernhard Bolzano discovered a procedure for constructing a continuous function that is not differentiable at any point. Later, in 1860, the great German mathemati- cian Karl Weierstrass (biography on p. 32) produced the first formula for such a function. The graphs of such functions are impossible to draw; it is as if the corners are so numerous that any segment of the curve, when suitably enlarged, reveals more corners. The discovery of these functions was important in that it made mathematicians distrustful of their geo- metric intuition and more reliant on precise mathematical proof. Recently, such functions have started to play a fundamental role in the study of geometric objects called fractals. Fractals have revealed an order to natural phenomena that were previously dismissed as random and chaotic. Figure 2.2.12 DERIVATIVES AT THE ENDPOINTS OF AN INTERVAL If a function f is defined on a closed interval [a, b] but not outside that interval, then f  is not defined at the endpoints of the interval because derivatives are two-sided limits.

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  Calculus in 3D

  Geometry, Vectors, and Multivariate Calculus

  • Zbigniew Nitecki(Author)
  • 2018(Publication Date)
  • American Mathematical Society

    (Publisher)

  Conversely, we will in some cases concentrate on the case of two variables and if necessary indicate how to incorporate the third variable. In this chapter, we consider the definition and use of derivatives in this context. 3.1 Continuity and Limits Continuous Functions of Several Variables. Recall from § 2.3 that a sequence of vec-tors converges if it converges coordinatewise. Using this notion, we can define continuity of a real-valued function of three (or two) variables ? ( ⃗ ?) by analogy to the definition for real-valued functions ? (?) of one variable: 1 When the domain is a specified subset ? ⊂ ℝ 𝑛 we will write ?∶ ? → ℝ . 123 124 Chapter 3. Real-Valued Functions: Differentiation Definition 3.1.1. A real-valued function ? ( ⃗ ?) is continuous on a subset ? ⊂ ℝ 2 or 3 of its domain if whenever the inputs converge in ? (as points in ℝ 2 or 3 ) the corresponding outputs also converge (as numbers): ⃗ ? ? → ⃗ ? 0 ⇒ ? (⃗ ? ? ) → ? (⃗ ? 0 ) . It is easy, using this definition and basic properties of convergence for sequences of numbers, to verify the following analogues of properties of continuous functions of one variable. First, the composition of continuous functions is continuous (Exercise 5): Remark 3.1.2. Suppose ? ( ⃗ ?) is continuous on ? ⊂ ℝ 2 or 3 . (1) If ?∶ ℝ → ℝ is continuous on ? ⊂ ℝ and ? ( ⃗ ?) ∈ ? for every ⃗ ? = (?, ?, ?) ∈ ? , then the composition ? ∘ ?∶ ℝ 2 or 3 → ℝ , defined by (? ∘ ?)( ⃗ ?) = ? (? ( ⃗ ?)) , in other words (? ∘ ?)(?, ?, ?) = ? (? (?, ?, ?)) , is continuous on ? . (2) If ⃗ ?∶ ℝ → ℝ 3 is continuous on [𝑎, 𝑏] and ⃗ ? (?) ∈ ? for every ? ∈ [𝑎, 𝑏] , then ? ∘ ⃗ ?∶ ℝ → ℝ , defined by (? ∘ ⃗ ?)(?) = ? ( ⃗ ? (?) ) –i.e., (? ∘ ⃗ ?)(?) = ? (? 1 (?) , ? 2 (?) , ? 3 (?)) 2 – is continuous on [𝑎, 𝑏] . Second, functions defined by reasonable formulas are continuous where they are defined: Lemma 3.1.3.

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  Anton's Calculus

  Early Transcendentals

  • Howard Anton, Irl C. Bivens, Stephen Davis(Authors)
  • 2018(Publication Date)
  • Wiley

    (Publisher)

  In the early nineteenth century mathematicians be- lieved that if a continuous function had many points of nondifferentiability, these points, like the tips of a sawblade, would have to be separated from one another and joined by smooth curve segments (Figure 2.2.12). This misconception was corrected by a series of discoveries beginning in 1834. In that year a Bohemian priest, philosopher, and mathe- matician named Bernhard Bolzano discovered a procedure for constructing a continuous function that is not differentiable at any point. Later, in 1860, the great German mathemati- cian Karl Weierstrass (biography on p. 32) produced the first formula for such a function. The graphs of such functions are impossible to draw; it is as if the corners are so numerous that any segment of the curve, when suitably enlarged, reveals more corners. The discovery of these functions was important in that it made mathematicians distrustful of their geo- metric intuition and more reliant on precise mathematical proof. Recently, such functions have started to play a fundamental role in the study of geometric objects called fractals. Fractals have revealed an order to natural phenomena that were previously dismissed as random and chaotic. Figure 2.2.12 x y DERIVATIVES AT THE ENDPOINTS OF AN INTERVAL If a function f is defined on a closed interval [a, b] but not outside that interval, then f ′ is not defined at the endpoints of the interval because derivatives are two-sided limits. To deal with this we define left-hand derivatives and right-hand derivatives by f ′ − (x) = lim h →0 − f (x + h) − f (x) h and f ′ + (x) = lim h →0 + f (x + h) − f (x) h respectively. These are called one-sided derivatives. Geometrically, f ′ − (x) is the limit of the slopes of the secant lines as x is approached from the left and f ′ + (x) is the limit of the slopes of the secant lines as x is approached from the right.

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  Fast Start Differential Calculus

  • Daniel Ashlock(Author)
  • 2022(Publication Date)
  • Springer

    (Publisher)

  75 C H A P T E R 3 Limits, Derivatives, Rules, and the Meaning of the Derivative Traditional calculus courses begin with a detailed formal discussion of limits and continuity. This book departs from that tradition, with this chapter introducing limits only in an informal fashion so as to be able to get going with calculus. A formal discussion of limits and continuity appears in Chapter 6. The agenda for this chapter is to get you on board with a workable operational definition of limits; use this to give the formal definition of a derivative; develop the rules for taking derivatives; and end with a discussion of the physical meaning of the derivative. 3.1 LIMITS Suppose we are given a function definition like: f .x/ D x 2 4 x C 2 Then, as long as x ¤ 2, we can simplify as follows: f .x/ D x 2 4 x C 2 D .x 2/.x C 2/ .x C 2/ D .x 2/ ✘ ✘ ✘ ✘ .x C 2/ ✘ ✘ ✘ ✘ .x C 2/ D x 2 So, this function is a line—as long as x ¤ 2. What happens when x D 2? Technically, the function doesn’t exist. This is where the notion of a limit comes in handy. If we come up with a whole string of x values and look where they are going as we approach 2, they all seem to be going toward minus 4. The key phrase here is seem to be, and the rigorous, precise definition of this vague phrase is the meat of Chapter 6. For now, let’s examine a tabulation of the behavior of f .x/ near x D 2. 76 3. LIMITS, DERIVATIVES, RULES, AND THE MEANING OF THE DERIVATIVE From above From below x f .x/ x f .x/ -1 -3 -3 -5 -1.5 -3.5 -2.5 -4.5 -1.75 -3.75 2.25 -4.25 -1.8 -3.8 -2.2 -4.2 -1.9 -3.9 -2.1 -4.1 -1.95 -3.95 -2.05 -4.05 -1.99 -3.99 -2.01 -4.01 Heading for: -2 -4 -2 -4 Notice that this table approaches from above (numbers larger than x D 2) and below (num- bers smaller than x D 2). In a well-behaved function the approaches from above and below head for the same place, but there are functions where they don’t. We call these the limit from above and the limit from below.

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  The Calculus Lifesaver

  All the Tools You Need to Excel at Calculus

  • Adrian Banner(Author)
  • 2009(Publication Date)
  • Princeton University Press

    (Publisher)

  So there is no maximum; this illustrates that the interval of continuity has to be closed in order to guarantee that the Max-Min Theorem works. Of course, the conclusion of the theorem could still be true even if the interval isn’t closed. For example, the function in the third diagram above is only continuous on the open interval ( a, b ), but it still has a maximum at x = c and a minimum at x = d . This was just a lucky accident: you can only use the theorem to guarantee the existence of a maximum and minimum in an interval [ a, b ] if you know the function is continuous on the entire closed interval. 5.2 Differentiability We’ve spent a while looking at continuity. Now it’s time to look at another degree of smoothness that a function can have: differentiability. This essen-tially means that the function has a derivative. So, we’ll spend quite a bit of time looking at derivatives. One of the original inspirations for develop-ing calculus came from trying to understand the relationship between speed, distance, and time for moving objects. So let’s start there and work our way back to functions later on. 5.2.1 Average speed Imagine looking at a photo of a car on a highway. The exposure time was very short, so it’s not blurry—you can’t even tell whether the car was moving or not. Now, I ask you this: how fast was the car moving when the picture was taken? No problem, you say—just use the classic formula speed = distance time . The problem is that the photo conveys no sense of distance (the car hasn’t moved) or time (the photo essentially captures an instant of time). So you can’t answer my question. Ah, but what if I tell you that a minute after the picture was taken, the car had traveled one mile? Then you could use the above formula to see that the car was going at a mile a minute, or 60 mph. Still, how do you know that the car was going the same speed for that whole minute? It might have accelerated and decelerated many times during that minute.

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  Applied Calculus

  • Geoffrey Berresford, Andrew Rockett(Authors)
  • 2015(Publication Date)
  • Cengage Learning EMEA

    (Publisher)

  Due to electronic rights, some third party content may be suppressed from the eBook and/or eChapter(s). Editorial review has deemed that any suppressed content does not materially affect the overall learning experience. Cengage Learning reserves the right to remove additional content at any time if subsequent rights restrictions require it. 87 2.2 Rates of Change, Slopes, and Derivatives Introduction In this section we will define the derivative, one of the two most important con-cepts in all of calculus, which measures the rate of change of a function or, equival-ently, the slope of a curve.* We begin by discussing rates of change . Average and Instantaneous Rate of Change We often speak in terms of rates of change to express how one quantity changes with another. For example, in the morning the temperature might be “rising at the rate of 3 degrees per hour” and in the evening it might be “falling at the rate of 2 degrees per hour.” For simplicity, suppose that in some location the temperature at time x hours is ƒ( x ) 5 x 2 degrees. We shall calculate the average rate of change of temperature over various time intervals—the change in temperature divided by the change in time.

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  Mathematics for Electronic Technology

  Pergamon International Library of Science, Technology, Engineering and Social Studies

  • D. P. Howson(Author)
  • 2013(Publication Date)
  • Pergamon

    (Publisher)

  C H A P T E R I INTRODUCTORY CONCEPTS STUDENTS and engineers will come to a textbook such as this with widely differing mathematical training and skills. It therefore seemed useful to devote an introductory chapter to a range of topics, some of which may be familiar to the reader but all of which are indispensable if a proper grounding in the subject is to be ob-tained. As in the rest of the book, a number of examples of varying degrees of difficulty are included and solutions, at least in outline, are usually provided. The reader should make a point of working through these to improve the grasp of the subject, particularly as the text treatment of each topic is necessarily brief in a little book like this. Nevertheless, it has been thought worth while to com-mence with a formal grounding in differentiation and integration, partly to refresh the mind as to the precise meanings of these con-cepts, but also to allow comparison of the definitions with those used as the subject is developed in other sections of the work. The last part of the chapter is devoted to vector theory, commencing with elementary material, but proceeding to a relatively advanced level. 1.1 Differentiation The idea of the rate of change of a variable is basic to a study of many engineering problems, and it will be assumed that the student is familiar with this. Here only the salient points of the theory will be covered, and a summary provided of some of the most important results. The differential, or rate of change, of a function of x, f(x), with 1 2 M A T H E M A T I C S FOR E L E C T R O N I C T E C H N O L O G Y respect to an infinitesimal change in x, will be denoted by fx) or d//dx and defined at a point x 0 by idf = l jm f/(*o + Sx) - f(x 0 )} (1) f(x) will accordingly be said to be differentiable at x 0 if such a limit exists. This will occur if /(x) is continuous at x 0 , in other words if lim/(x)=/(x 0 ) X-+XO (2) independently of the way in which χ approaches x 0 .

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  Calculus, Metric Edition

  • James Stewart, Daniel K. Clegg, Saleem Watson, , James Stewart, James Stewart, Daniel K. Clegg, Saleem Watson(Authors)
  • 2020(Publication Date)
  • Cengage Learning EMEA

    (Publisher)

  107 We know that when an object is dropped from a height it falls faster and faster. Galileo discovered that the distance the object has fallen is proportional to the square of the time elapsed. Calculus enables us to calculate the precise speed of the object at any time. In Exercise 2.1.11 you are asked to determine the speed at which a cliff diver plunges into the ocean. Icealex / Shutterstock.com 2 Derivatives IN THIS CHAPTER WE BEGIN our study of differential calculus, which is concerned with how one quantity changes in relation to another quantity. The central concept of differential calculus is the derivative, which is an outgrowth of the velocities and slopes of tangents that we considered in Chapter 1. After learning how to calculate derivatives, we use them to solve problems involving rates of change and the approximation of functions. Copyright 2021 Cengage Learning. All Rights Reserved. May not be copied, scanned, or duplicated, in whole or in part. Due to electronic rights, some third party content may be suppressed from the eBook and/or eChapter(s). Editorial review has deemed that any suppressed content does not materially affect the overall learning experience. Cengage Learning reserves the right to remove additional content at any time if subsequent rights restrictions require it. 108 CHAPTER 2 Derivatives Derivatives and Rates of Change In Chapter 1 we defined limits and learned techniques for computing them. We now revisit the problems of finding tangent lines and velocities from Section 1.4. The special type of limit that occurs in both of these problems is called a derivative and we will see that it can be interpreted as a rate of change in any of the natural or social sciences or engineering.

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