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The goal of this text is to help students leam 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, modelIed almost exactly on the exam­ pIes; 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

Chapter 7. Basic Methods of Integration

Abstract
In this chapter, we first collect in a more systematic way some of the integration formulas derived in Chapters 4–6. We then present the two most important general techniques: integration by substitution and integration by parts. As the techniques for evaluating integrals are developed, you will see that integration is a more subtle process than differentiation and that it takes practice to learn which method should be used in a given problem.
Jerrold Marsden, Alan Weinstein

Chapter 8. Differential Equations

Abstract
In the first two sections of this chapter, we study two of the simplest and most important differential equations, which, describe oscillations, growth, and decay. A variation of these equations leads to the hyperbolic functions, which are important for integration and other applications. To end the chapter, we study two general classes of differential equations whose solutions can be expressed in terms of integrals. These equations, called separable and linear equations, occur in a number of interesting geometrical and physical examples. We shall continue our study of differential equations in Chapter 12 after we have learned more calculus.
Jerrold Marsden, Alan Weinstein

Chapter 9. Applications of Integration

Abstract
Our applications of integration in Chapter 4 were limited to area, distance-velocity, and rate problems. In this chapter, we will see how to use integrals to set up problems involving volumes, averages, centers of mass, work, energy, and power. The techniques developed in Chapter 7 make it possible to solve many of these problems completely.
Jerrold Marsden, Alan Weinstein

Chapter 10. Further Techniques and Applications of Integration

Abstract
Besides the basic methods of integration associated with reversing the differentiation rules, there are special methods for integrands of particular forms. Using these methods, we can solve some interesting length and area problems.
Jerrold Marsden, Alan Weinstein

Chapter 11. Limits, L’Hôpital’s Rule, and Numerical Methods

Abstract
Our treatment of limits up to this point has been rather casual. Now, having learned some differential and integral calculus, you should be prepared to appreciate a more detailed study of limits.
Jerrold Marsden, Alan Weinstein

Chapter 12. Infinite Series

Abstract
The decimal expansion \(\tfrac{1}{3} = 0.3333 \ldots\) is a representation of \(\tfrac{1}{3}\) as an infinite sum \(\tfrac{3}{{10}} + \tfrac{3}{{100}} + \tfrac{3}{{1000}} + \tfrac{3}{{10,000}} + \cdots\). In this chapter, we will see how to represent numbers as infinite sums and to represent functions of x by infinite sums whose terms are monomials in x. For example, we will see that
$$\ln 2 = 1 - \frac{1}{2} + \frac{1}{3} - \frac{1}{4} + \cdots$$
and
$$\sin x = x - \frac{{{{x}^{3}}}}{{1 \cdot 2 \cdot 3}} + \frac{{{{x}^{5}}}}{{1 \cdot 2 \cdot 3 \cdot 4 \cdot 5}} - \cdots .$$
Jerrold Marsden, Alan Weinstein

Backmatter

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