无极限微积分Calculus Without Limits

在不使用限制的情况下，将可区分性作为本地属性引入。 该课程是为具有微积分背景的生命科学专业设计的，他们的主要兴趣在于微积分的应用。
Contents Preface 0 A Preview 1 1 Some Background Material 7 1.1 Lines 1. 2 Parabolas and Higher Degree Polynomials 13 1. 3 The Exponential and Logarithm Functions 20 1.4 Use of graphing Utilities 31 2 The Derivative 35 2. 1 Definition of the derivative 12 2.2 Differentiability as a Local Property 47 2.3 Derivatives of some basic functions 48 2.4 Slopes of Secant Lines and Rates of Change.......... 57 2.5 Upper and Lower Parabolas 6 2.6 Other Notations for the derivative 69 2.7 Exponential Growth and Decay .,70 2.8 More Exponential Growth and Decay 76 2.9 Dierentiability Implies Continuity 81 2.10 Bcing Closc Versus Looking Like a Linc 2.11 Rules of differentiation 84 2. 11.1 Linearily of the Derivative 2.11.2 Product and Quotient rule 87 2.11.3 Chain Rule 91 2.11, 4 Derivatives of inverse functions .,101 2. 12 Implicit Differentiation 111 2. 13 Related Rates 115 2.14 Numerical methods 118 2. 14.1 Approximation by differentials 118 2.14.2 Newton's Method 120 2.143 Euler’ s Method 125 2.15 Summary 131 3 Applications of the derivative 137 3.1 Differentiability on Closed Intervals 3.2 Cauchy's Mean Value Theorem .138 3.3 The First Derivative and Monotonicity 144 3.4 The Second and Higher Derivatives ,.,,154 3.5 The Second Derivative and Concavity .156 3.6 Local extrema and Inflection Points 164 3. 7 The First Derivative Test 166 3. 8 The Second derivative Test 3.9 Extrema of Functions 173 3.10 Detection of Inflection points 183 3.11 Optimization Problems 187 3.12 Sketching Graphs 198 4 Integration 203 4.1 Upper and Lower Sums ,,,,,,,,,,207 4.2 Integrability and Areas 213 4.3 Some elementary observations 1. Integrable Funct 4.5 Antiderivatiy 224 4.6 The Fundamental theorem of calculus 226 4.7 Substituting 234 4.8 Areas between Graphs 242 1.9 Numerical Integration 244 4.10 Applications of the Integra 251 4.11 The Exponential and Logarithm Functions 5 Prerequisites from Precalculus 263 5.1 The Real numbers 263 5.2 Inequalities and Absolute Value 5.3 Functions. Definition and notation 268 5.4 Graphing Equations 274 5.5 Trigonometric Functions 276 5.6I unctions 5.7 Ncw Functions From old oncs Preface These notes are written for a onesemester calculus course which meets three times a week and is, preferably, supported by a computer lab. The course is designed for life science majors who have a precalculus back ground, and whose primary interest lics in thc applications of calculus. Wc try to focus on those topics which are of greatest importance to them and use life science examples to illustrate them. At the same time, we try of stay mathemat ically coherent without becoming technical. To make this feasible, we are willing to sacrifice generality. There is less of an emphasis on by hand cal culations. Instead, more complex and demanding problems find their place in a computer lab. In this sense, we are trying to adopt severa. ideas from calculus reform. Among them is a more visual and less analytic approach We typically explore new ideas in examples before we give formal definitions In one more way we depart radically from the traditional approach to calculus. We introduce differentiability as a local property without using linits. The philosophy behind this idea is Chal linits are the a big stu bling block for most students who see calculus for the first time, and they take up a substantial part of the first semester. Though mathematically rigorous, our approach to the derivative makes no use of limits, allowing the students to get quickly and without unresolved problems to this con cept. It is true Chat our delinition is nore restrictive Chan Che ordinary one, and fewer functions are differentiable in this manuscript than in a standard text. But the functions which we do not recognize as being differentiable are not particularly important for students who will take only one semester of calculus. In addition, in our opinion the underlying geometric idea of the derivative is at least as clear in our approach as it is in the one using linits More technically speaking, instead of the traditional notion of differen tiability, we use a notion modeled on a lipschitz condition. Instead of an Ed definition we use an explicit local (or global)estimate. For a function to be differentiable at a point ro one requires that the difference between the function and the tangent line satisfies a Lipschitz condition of order 2 in . ro for all in an open interval around o: instead of assuming that this difference is o(c This approach, which should be to easy to follow for anyone with a back ground in analysis, has been used previously in teaching calculus. The au hor learned about il when he was leaching assistant (Ubungsgruppenleiter) for a course taught by Dr. Bernd Schmidt in Bonn about 20 years ago There this approach was taken for the same reason to find a less technica. and efficient approach to the derivative. Dr. Schmidt followed suggestions Thich were promoted and carried out by professor H. Karcher as innovations for a reformed high school as well as undergraduate curriculu. Prolessor Karcher had learned calculus this way from his teacher, Heinz Schwarze There are german language college level textbooks by Kitting and moller and a high school level book by muller which use this approach Calculus was developed by Sir Isaac Newton(16421727)and Gottfried Wilhelm Leibnitz(16461716) in the 17th century. The emphasis was on differentiation and integration, and these techniques were developed in the quest for solving real life problems. Among the great achievements are the explanation of Keplers laws, the development of classical mechanics, and the solutions of many important diffcrcntial cquations. Though very suc cessful, the treatment of calculus in those days is not rigorous by nowadays mathematical standards In the 19th century a revolution took place in the development of calcu lus, foremost through the work of AugustinLouis Cauchy(17891857)and Karl Weierstrass(18151897), when the modern idea of a function was intro d and the definitions of limits and continuous functions were developed This elevated calculus to a mature, well rounded,. mathematically satisfying theory. This also made calculus much more demanding A considerable mathematically challenging setup is required(limits) before one comes to the central ideas of differentiation and integration A second revolution took place in the first half of the 20th century with the introduction of generalized functions(distributions). This was stimu lated by the development of quantum mechanics in the 1920ies and found is final mathematical form in the work of laurent schwartz in the 1950ies What are we really interested in? We want to introduce the concepts of differentiation and integration. The functions to which we like to apply these techniques are those of the first period. In this sense, we do not see page 42 of: A. Zygmund, Trigonometric Series, Vol l, Cambridge University Press 1959, reprint.ed with corrections and some additions 1968 need the powerful machine developed in the 19th century. Still, we like to be mathematically rigorous because this is the way mathematics is done nowadays. This is possible through the use of the slightly restrictive notion of differentiability which avoids the abstraction and the delicate, technically demanding notions of the second period To support the student's learning we rely extensively on examples and graphics. Often times we accept computer generated graphics without hav. ing developed the background to deduce their correctness from mathematical principles Calculus was developed together with its applications. Sometimes the applications were ahead, and sometimes the mathematical theory was. We incorporate applications for the purpose of illustrating the theory and to motivate it. But then we cannot assume that the students know already the subjects in which calculus is applied, and it is also not our goal to teach them. For this reason the application have to be rather easy or simplified PREFACE Chapter o a Preview In this introductory course about calculus you will learn about two principal concepts, differentiation and integration. We would like to explain them in an intuitive manner using examples. In Figure l you see the graph of a function. Suppose it represents a function which describes the size of a Figure 1: Yeast population as a function of time population of live yeast bacteria in a bun of pizza dough. Abbreviating CHAPTER O. A PREVIEW time by t (say measured in hours )and the size of the population by P(say measured in millions of bacteria), we denote this function by P(t). You like to know at what rate the population is changing at some fixed time, say at time to=1. For a straight line, the rate of change is its slope We like to apply the idea of rale of change or slope also lo the function P(l) although its graph is certainly not a straight line What can we do? Let us try to replace the function P(t)by a line L(t) at least for values of t near to. The distance between the points (t P(t) and(t, L(t)) on the respective graphs is E()=P()L(0) This is the error which we make by using L(t) instead of P(t) at time t. W will require that this error is "small"in a sense which we will precise soon If a line L(t) can be found so that the error is small for all t in some open interval around to, then we call L(t) the tangent line to the graph of P at lo. The slope of the line L(C) will be called the slope of the graph of P(c)al the point(to, P(to)), or the rate of change of P(t)at the time t=to Figure 2: Zoom in on a point Figure 3: Graph tangent line Let us make an experiment. Put the graph under a microscope or on your graphing calculator, zoom in on the point(4, P(4) on the graph This process works for the given example and most other functions treated in these notes. You see the zoom picture in Figure 2. Only under close
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