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Calculus
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Calculus is important for first-year undergraduate students pursuing mathematics, physics, economics, engineering, and other disciplines where mathematics plays a significant role. The book provides a thorough reintroduction to calculus with an emphasis on logical development arising out of geometric intuition. The author has restructured the subject matter in the book by using Tarski's version of the completeness axiom, introducing integration before differentiation and limits, and emphasizing benefits of monotonicity before continuity. The standard transcendental functions are developed early in a rigorous manner and the monotonicity theorem is proved before the mean value theorem. Each concept is supported by diverse exercises which will help the reader to understand applications and take them nearer to real and complex analysis.
Frontmatter
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5 - Techniques of Integration
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- 16 February 2023, pp 173-216
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Summary
Our work on differentiation in the previous chapter has brought us to a fork in the road. We can pursue the implications of the first fundamental theorem to obtain techniques for computing integrals. Alternately, we can use the Fermat and monotonicity theorems to further develop the relationship between functions and their derivatives, leading to new techniques of calculating limits, approximation of functions by polynomials, use of integration to measure arc length, surface area and volume, and error estimates for numerical calculations of integrals.We have chosen to take up integration in this chapter. If you are more interested in the other applications of differentiation you can read Chapter 6 first.
The Second Fundamental Theorem
A function F is called an anti-derivative of f if F = f . Let us make some observations regarding the existence and uniqueness of anti-derivatives:
1. Not every function has an anti-derivative. By Darboux's theorem (Theorem 4.5.12), if f = F then f has the intermediate value property. Thus, a function with a jump discontinuity, like the Heaviside step function or the greatest integer function, cannot have an anti-derivative.
2. On the other hand, the first fundamental theorem shows that every continuous function on an interval has an anti-derivative.
3. A function's anti-derivative is not unique. For example, both sin x and 1 + sin x are anti-derivatives of cos x.
4. On the other hand, two anti-derivatives of the same function over an interval can differ only by a constant. Theorem 4.5.7 states that if F = G on an interval I, then F G is constant. Thus, every anti-derivative of cos x over an interval I has to have the form sin x + C, where C ∊ R.
5. Over non-overlapping intervals, two anti-derivatives of a function need not differ by the same constant. For example, the Heaviside step function and the zero function are anti-derivatives of the zero function over (-∞,0) ∪ (0,∞).
The first fundamental theorem established a connection between integration and differentiation: if we are able to calculate the definite integrals of a continuous function, then the first fundamental theorem gives us its anti-derivative. The next theorem uses that connection to provide an approach for evaluating definite integrals by using anti-derivatives.
1 - Real Numbers and Functions
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Calculus can be described as the study of how one quantity is affected by another, focusing on relationships that are smooth rather than erratic. This chapter sets up the basic language for describing quantities and the relationships between them. Quantities are represented by numbers and you would have seen different kinds of numbers: natural numbers, whole numbers, integers, rational numbers, real numbers, perhaps complex numbers. Of all these, real numbers provide the right setting for the techniques of calculus and so we begin by listing their properties and understanding what distinguishes them from other number systems. The key element here is the completeness axiom, without which calculus would lose its power.
The mathematical object that describes relationships is called “function.” We recall the definition of a function and then concentrate on functions that relate real numbers. Such functions are best visualized through their graphs, and this visualization is a key part of calculus. We make a small beginning with simple examples. A more thorough investigation of graphs can only be carried out after calculus has been developed to a certain level. Indeed, the more interesting functions, such as trigonometric functions, logarithms, and exponentials, require calculus for their very definition.
Field and Order Properties
We begin with a review of the set R of real numbers, which is also called the Euclidean line. It is a “review” in that we do not construct the set but just list its key attributes, and use them to derive others. For descriptions of how real numbers can be constructed from scratch, you can consult Hamilton and Landin [11], Mendelson [24], or most books on real analysis. The fundamental ideas underlying these constructions are easy to absorb, but the checking of details can be arduous. You would probably appreciate them more after reading this book.
What is the need for this review? Mainly, it is intended as a warm-up session before we begin calculus proper. Many intricate definitions and proofs lie in wait later, and we need to get ready for them by practising on easier material. If you are in a hurry and confident of your basic skills with numbers and proofs, you may skip ahead to the next section, although a patient reading of these few pages would also help in later encounters with linear algebra and abstract algebra.
8 - Taylor and Fourier Series
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The decimal expansion of a real number is an instance of an infinite series. For example, 3.14159 … can be viewed as the sum of the series created by its digits: . The convergence of such a series is established by a comparison with the geometric series å¥n=0 1/10n. Isaac Newton realized that by replacing the powers of 1/10 with powers of a variable x, one can create real functions that can be easily manipulated in analogy with the rules of decimal expansions. He developed rules for their differentiation and integration as well as a method for expanding inverse functions in this manner. Today's historians believe that for Newton, calculus consisted of working with these “power series.” (See Stillwell [32, pp 167–70].) In this approach, the main task is to express a given function as a power series, after which it becomes trivial to perform the operations of calculus on it. In the first two sections of this chapter we shall study the general properties of power series, and then the problem of expressing a given function as a power series.
A century after Newton, Joseph Fourier replaced the powers xn with the trigonometric functions sinnx and cosnx to create new ways of describing functions. The “Fourier series” could model much wilder behavior than power series, and forced mathematicians to revisit their notions of what is a function, and especially the definition of integration.We give a brief introduction to this topic in the third section.
In our final section, we introduce sequences and series of complex numbers. These bring further clarity to power and Fourier series, and even unify them through the famed identity of Euler: eix = cos x + i sin x.
Power Series
Let us recall our study of Taylor polynomials in §6.3. Given a function f that can be differentiated n times at x = a, we define the Taylor polynomial
Tn is intended to be an approximation for f near x = a, with the hope that the approximation improves when we increase n. These hopes are not always realized, but in many cases they are. (The remainder theorem gives us a way to assess them.) It is natural to make the jump from polynomials to series, and consider the expression
The questions that arise here are: (a) For which x does this series converge, and (b) When it converges, does it sum to f (x)? To tackle these questions, we initiate a general study of series of this form.
Acknowledgments
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Contents
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Preface
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Even now such resort to geometric intuition in a first presentation of the differential calculus, I regard as exceedingly useful, from the didactic standpoint, and indeed indispensable, if one does not wish to lose too much time. But that this form of introduction into the differential calculus can make no claim to being scientific, no one will deny.
— Richard Dedekind, 1872Calculus is a magical subject. A first encounter in school leads to a radical revision of one's ideas of what is mathematics. We are transported from a rather staid enterprise of counting and measuring to an adventure encompassing change, fluctuation, and a vastly increased ability to understand and predict the workings of the world. At the same time, the student encounters “magic” with both its connotations: awe and wonder on the one hand, mystery and a sense of trickery on the other. Calculus can appear to be a bag of tricks that are immensely useful, provided the apprentice wizard can perfectly remember the spells. As the student pursues mathematics further at university, her instructors may use courses in analysis to persuade her that calculus is a science rather than a mystical art. Alas, all too often the student perceives the new instruction as mere hair-splitting which gives no new powers and may even undermine her previous attainments. The first analysis course is for many an experience that makes them regret taking up higher mathematics.
This book is written to support students in this transition from the expectations of school to those of university. It is intended for students who are pursuing undergraduate studies in mathematics or in disciplines like physics and economics where formal mathematics plays a significant role. Its proper use is in a “calculus with proofs” course taught during the first year of university. The goal is to demonstrate to the student that attention to basic concepts and definitions is an investment that pays off in multiple ways. Old calculations can be done again with a fresh understanding that can not only be stimulating but also protects against error. More importantly, one begins to learn how knowledge can be extended to new domains by first questioning it in familiar terrain. For students majoring in mathematics, this book can serve as bridge to real analysis. For others, it can serve as a base from where they can make expeditions to various applications.
Introduction
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A calculus text written at this point of time can make its plea for existence on the novelty of its exposition and choice of content, rather than any originality in its mathematics. Let me state the case for this book. These explanations are primarily aimed at teachers, but students may also gain some perspective from them if they peruse them after going through at least the first two chapters.
The book aims to give a complete logical framework for calculus, with the proofs reaching the same levels of rigour as a text on real analysis. At the same time it eschews those aspects of analysis that are not essential to a presentation of calculus techniques. So it has the completeness axiom for real numbers, but not Cauchy sequences or the theorems of Heine–Borel and Bolzano–Weierstrass. At the other end of the spectrum, it omits the case for the importance of calculus through its applications to the natural sciences or to economics and finance.
The first chapter provides a description of real numbers and their properties, followed by functions and their graphs. For the most part, the material of this chapter would be known to students, but not in such an organized way. I typically use the first class of the course to ask students to share their thoughts on various issues. What are rational numbers? What are numbers? Which properties of numbers are theorems and which are axioms? What is the definition of a point or a line? What do we mean by a tangent line to a curve? These flow from one to another and from students’ responses. By the end of the hour, with many students firm in their beliefs but finding them opposed just as firmly by others, I have an opportunity to propose that we must carefully put down our axioms and ways of reasoning so that future discussions may be fruitful. One may still ask whether the abstract approach is overdone; is it necessary to introduce general concepts like field and ordered field? The reason for doing so is that it provides a context within which simple questions can be posed and the student can practice creating and writing small proofs as a warm-up to harder tasks that wait ahead.
3 - Limits and Continuity
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Integration can be seen as accumulation, the summing up of local changes to get a global result. In the previous chapter, we set up the general process for achieving this. We also saw how a specific kind of information—monotonicity—could be used to obtain global results. With its help we were able to formally define the natural logarithm and exponential functions, which are usually taken for granted in school mathematics.
Further progress requires a closer look at the local behavior of functions. The more we know of the local behavior, the better our chances of extracting global information. These considerations underlie our development of the notions of limit and continuity in this chapter. As applications, we will rigorously develop angles and their radian measures, followed by the trigonometric functions and their properties.
Limits
You have seen in school, the notation lim x!p f (x)= L, which is read as “the limit of f (x) at p is L” and is interpreted as “the values of f (x) approach L as the values of x approach p.” We need a clear definition of what we mean by “approaches.”
Example 3.1.1
Consider f (x) = 2x + 5. What happens if we take values of x that approach 0? Here are some calculations:
We see that as x gets closer to 0, f (x) appears to be getting closer to 5. Can we control this? Can we get the output f (x) close to 5 within any required accuracy level, simply by making the input x appropriately close to 0?
References
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Index
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4 - Differentiation
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In this chapter, we take a closer look at the idea that local information about functions should help us resolve integration problems.We already saw that continuity guarantees integrability. Another application of continuity combined with integration was in enabling the definition of the trigonometric functions. On the other hand, continuity did not give us new tools to calculate integrals. In this chapter we shall study the stronger property of differentiability, which will eventually give us techniques for calculating integrals. It also has its own significance, independent of integration. We shall use it for a better understanding of the shapes of graphs of functions.
Among the continuous functions, the ones that are easiest to integrate are the “piecewise linear” ones. Their graphs consist of line segments, such as in the example below:
This suggests that we try to locally approximate functions by straight line segments. If we can get good approximations of this type, we can use them to assess the integral. We shall give the name “differentiable” to functions that can be locally approximated by straight lines. Most of this chapter is devoted to identifying these functions and to calculating the corresponding straight lines. Then we make the first connection between the processes of differentiation and integration, the so-called first fundamental theorem of calculus. Finally, we see that differentiation has a life of its own, and we use it to explore the problems of finding the extreme values and sketching the graph of a function.
Derivative of a Function
Let us consider what happens if we zoom in for a closer look at the graph of a function such as y = x2, near a point such as (1,1).
We see that the graph of the function y = x2 looks more like the line y = x2 - 1 as we zoom in towards (1,1), and at some stage becomes indistinguishable from it.
This can happen even for functions with rapid oscillations. Let us look at the function defined by y = x2 cos(1/x) if x ≠ 0, and y = 0 if x = 0, near the origin.
No matter how much we zoom in, the function has infinitely many oscillations. Nevertheless, their amplitudes decrease and in that sense the function becomes closer to the line y = 0 as we zoom in.
Dedication
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2 - Integration
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Calculus has two parts: differential and integral. Integral calculus owes its origins to fundamental problems of measurement in geometry: length, area, and volume. It is by far the older branch. Nevertheless, it depends on differential calculus for its more difficult calculations, and so nowadays we typically teach differentiation before integration.
We shall revert to the historical sequence and begin our journey with integration. Our first reason is that it provides a direct application of the completeness axiom without needing the concept of limits. The second is that important functions such as the trigonometric, exponential, and logarithmic functions are most conveniently constructed through integration. Finally, the student should become aware that integration is not just an application of differentiation or a set of techniques of calculation.
Suppose we wish to find the area of a shape in the Cartesian plane.We can, at least, estimate it by comparing the shape with a standard area, that of a square.We cover the shape with a grid of unit squares and count how many squares touch it, and also how many squares are completely contained in it. This gives an upper and a lower estimate for the area.We can obtain better estimates by taking finer grids with smaller squares. The figures given immediately below illustrate this process of iteratively improving the estimates.
We have said that we are estimating area. But what is our definition of area? In school books you will find descriptions such as “Area is the measure of the part of a plane or region enclosed by the figure.” It should be evident that this is not a very useful prescription. It means nothing without a description of the measuring process. In fact, the estimation process described above could become the basis for a meaningful definition of area, by requiring it to be a number that lies above all the lower estimates produced by the process, and below all the upper estimates. Its existence would be guaranteed by the completeness axiom. This is a promising start, but the sceptic can raise various objections that would have to be answered:
1. Could there be a figure for which multiple numbers satisfy the definition of its area?
2. If we slightly shifted or rotated the grids, could that change our calculation? That is, could moving a figure change its area?
6 - Mean Value Theorems and Applications
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When calculus is applied to problems of subjects like physics and economics, it usually leads to equations involving the first and second derivatives of functions, and the task is to recover the original function from these equations. If the derivative is completely known and is continuous, we can use the second fundamental theorem
However, we usually have only a relation between the function and its derivatives rather than a full knowledge of the derivative. For example, we may know that f (x)= f (x)2 for every x. So, we need to find more ways of relating information about f with information about f .We already have two important instances: Fermat's theorem and the monotonicity theorem. In this chapter we will explore several consequences of these results. The payoffs will be new techniques of calculating limits (§6.2), approximation of functions by polynomials (§6.3), use of integration to measure arc length, surface area, and volume (§6.4), and error estimates for numerical calculations of integrals (§6.5). Sections 6.2 and 6.3 are required for the final two chapters on sequences and series, while sections 6.4 and 6.5 are important for the applications of calculus.
Darboux's theorem says that if a function f is differentiable on an interval then f will have the intermediate value property on that interval. Thus, f cannot have a jump discontinuity and it behaves like a continuous function in some ways. Nevertheless, it need not be continuous or even bounded.
As an example, consider the function defined by f (0) = 0 and f (x) = x2 sin(1/x) when x ≠ 0. This function is differentiable at non-zero points by the chain rule. It is also differentiable at zero by a direct calculation:
Thus, f is differentiable at every point. However, f is not continuous at zero: which does not exist.
We can modify the above example slightly to get a function that is differentiable but whose derivative is not bounded. Define g(0) = 0 and g(x) = x3/2 sin(1/x) when x ≠ 0. We have
We say f is continuously differentiable on an interval I if it is differentiable on I and f is continuous on I.
Appendix: Solutions to Odd-Numbered Exercises
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Appendix: Solutions to Odd-Numbered Exercises
7 - Sequences and Series
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In calculus, we mainly study continuous change. However, there are situations where discrete changes have to be considered. For example, when we try to describe a number such as _ by its decimal representation, we actually create an iterative process of successively better approximations: 3, 3.1, 3.14, 3.141, 3.1415, 3.14159, and so on. A similar situation arises when we work with the Taylor polynomials of a function— we successively approximate a function by polynomials of increasing degree. What is common to the two examples is that there is a first stage, a second stage, a third stage, and we are interested in what happens as we keep going. Clearly, we need to develop a theory of limits for this context.We shall do so in this chapter. Further, we shall work out in detail the situation when discrete changes accumulate and we are interested in the total. This will have many similarities as well as a direct relation with integration.
As an example, let us consider a geometry problem that leads to an iterative method for approximating square roots by fractions. It is named Heron's method after a Greek mathematician, but the evidence is strong that this kind of reasoning was carried out earlier in ancient Iraq and India, three to four thousand years ago. The statement of the problem is: “Given a rectangle, construct a square with the same area.” Now, if the rectangle has sides a and b, the square needs to have side √ab. To us, this may be a triviality, but what if the only numbers you know are the fractions? Then the problem will, in general, have only approximate solutions. How do we find good fractional approximations to √ab? Consider the following steps.
The final square is obviously a bit too big. Nevertheless, its side of (a + b)/2 is visibly better than the initial sides of a and b. If we have ab = N, we can repeat the process with a rectangle whose sides are a = (a + b)/2 and b0 = N/a. This will lead to a new and further improved square with side (a + b)/2.