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Über dieses Buch

The goal of this text is to help students learn to use calculus intelligently for solving a wide variety of mathematical and physical problems. This book is an outgrowth of our teaching of calculus at Berkeley, and the present edition incorporates many improvements based on our use of the first edition. We list below some of the key features of the book. Examples and Exercises The exercise sets have been carefully constructed to be of maximum use to the students. With few exceptions we adhere to the following policies. • The section exercises are graded into three consecutive groups: (a) The first exercises are routine, modelled almost exactly on the exam­ ples; these are intended to give students confidence. (b) Next come exercises that are still based directly on the examples and text but which may have variations of wording or which combine different ideas; these are intended to train students to think for themselves. (c) The last exercises in each set are difficult. These are marked with a star (*) and some will challenge even the best students. Difficult does not necessarily mean theoretical; often a starred problem is an interesting application that requires insight into what calculus is really about. • The exercises come in groups of two and often four similar ones.

Inhaltsverzeichnis

Frontmatter

Introduction

Abstract
Calculus has earned a reputation for being an essential tool in the sciences. Our aim in this introduction is to give the reader an idea of what calculus is all about and why it is useful.
Jerrold Marsden, Alan Weinstein

Orientation Quizzes

Without Abstract
Jerrold Marsden, Alan Weinstein

Chapter R. Review of Fundamentals

Abstract
Success in the study of calculus depends upon a solid understanding of algebra and analytic geometry. In this chapter, we review topics from these subjects which are particularly important for calculus.
Jerrold Marsden, Alan Weinstein

Chapter 1. Derivatives and Limits

Abstract
Differential calculus describes and analyzes change. The position of a moving object, the population of a city or a bacterial colony, the height of the sun in the sky, and the price of cheese all change with time. Altitude can change with position along a road; the pressure inside a balloon changes with temperature. To measure the rate of change in all these situations, we introduce in this chapter the operation of differentiation.
Jerrold Marsden, Alan Weinstein

Chapter 2. Rates of Change and the Chain Rule

Abstract
In Chapter 1, we learned how to differentiate algebraic functions and, thereby, to find velocities and slopes. In this chapter, we will learn some applications involving rates of change. We will also develop a new rule of differential calculus called the chain rule. This rule is important for our study of related rates in this chapter and will be indispensable when we come to use trigonometric and exponential functions.
Jerrold Marsden, Alan Weinstein

Chapter 3. Graphing and Maximum-Minimum Problems

Abstract
Now that we know how to differentiate, we can use this information to assist us in plotting graphs. The signs of the derivative and the second derivative of a function will tell us which way the graph of the function is “leaning” and “bending.”
Jerrold Marsden, Alan Weinstein

Chapter 4. The Integral

Abstract
In everyday language, the word integration refers to putting things together, while differentiation refers to separating, or distinguishing, things.
Jerrold Marsden, Alan Weinstein

Chapter 5. Trigonometric Functions

Abstract
Many problems involving angles, circles, and periodic motion lead to trigonometric functions. In this chapter, we study the calculus of these functions, and we apply our knowledge to solve new problems.
Jerrold Marsden, Alan Weinstein

Chapter 6. Exponentials and Logarithms

Abstract
In our work so far, we have studied integer powers (b n ) and rational powers (b m/n ) as functions of a variable base, i.e., y = x m or y = x m/n . In this chapter, we will study powers as functions of a variable exponent, i.e., y = b x : To do this, we must first define b x when x is not a rational number. This we do in Section 6.1; the rest of the chapter is devoted to the differential and integral calculus of the exponential functions y = b x and their inverses, the logarithms. The special value b = e = 2.7182818285... leads to especially simple formulas.
Jerrold Marsden, Alan Weinstein

Backmatter

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