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## Über dieses Buch

This is a new type of calculus book: Students who master this text will be well versed in calculus and, in addition, possess a useful working knowledge of one of the most important mathematical software systems, namely, MACSYMA. This will equip them with the mathematical competence they need for science and engi­ neering and the competitive workplace. The choice of MACSYMA is not essential for the didactic goal of the book. In fact, any of the other major mathematical software systems, e. g. , AXIOM, MATHEMATICA, MAPLE, DERIVE, or REDUCE, could have been taken for the examples and for acquiring the skill in using these systems for doing mathematics on computers. The symbolic and numerical calcu­ lations described in this book will be easily performed in any of these systems by slight modification of the syntax as soon as the student understands and masters the MACSYMA examples in this book. What is important, however, is that the student gets all the information necessary to design and execute the calculations in at least one concrete implementation language as this is done in this book and also that the use of the mathematical software system is completely integrated with the text. In these times of globalization, firms which are unable to hire adequately trained technology experts will not prosper. For corporations which depend heavily on sci­ ence and engineering, remaining competitive in the global economy will require hiring employees having had a traditionally rigorous mathematical education.

## Inhaltsverzeichnis

### 1. Functions

Abstract
The concept of function is central to mathematics. A function is a rule assigning values to certain objects. If a function is called f, the value it assigns to x is denoted by f (x). A well-defined function f assigns to each such x a single value f (x). However, several objects x1, x2,. . . , x n may get the same value f (x1) = f (x2) = . . . =f(x n ). Here are some examples.

### 2. Elementary functions used in calculus

Abstract
This chapter is a brief introduction to the exponential, trigonometric, and hyperbolic functions, and their inverses. These functions, together with the polynomial and rational functions of Chap. 1, are used throughout calculus.

### 3. Limits and continuity

Abstract
The most basic concept in calculus is the limit. It is used in the study of continuity, derivatives, integrals, and all other important topics in calculus. Indeed, one cannot use calculus intelligently without understanding limits.

### 4. Differentiation

Abstract
Two functions f and g are tangent at a point x0 if their graphs almost coincide, near x0, in the sense of Definition 4.1 below. A function f can have, at a given point x0, at most one tangent which is a linear function, say l (x) = f (x0) + m(xx0) , in which case
• l is called the tangent line, or simply the tangent, of f at x0,
• the slope m of l is called the slope, or derivative, of f at x0.

### 5. Differentiation rules

Abstract
If f is differentiable, its derivative f′ can be computed using the limit (4.11),
$$f'\left( x \right) = \mathop {\lim }\limits_{\xi \to x} {\rm{ }}{{f\left( \xi \right) - f\left( x \right)} \over {\xi - x}},$$
which is often difficult. However, sometimes f has a special structure that allows differentiating it without evaluating the limit (4.11). For example, if u and v are differentiable functions, and if f is their product f = uv, then the derivative f′ can be easily computed from the derivatives u′ and v′ . This situation is covered by a differentiation rule called the product rule (Theorem 5.1). Other rules given in this chapter are the quotient rule (Theorem 5.5) and the chain rule (Theorem 5.11).

### 6. Extremum problems

Abstract
The following example is an illustration of the problems studied in this chapter. Example 6.1. You are the president of the XYZ Widget Company, which is in the business of producing and selling widgets. You must decide how many widgets, say x, to produce in the coming season. The information available to you is:
• the company can produce no more than 10 000 widgets,
• the company can sell any quantity it produces at the price of p : = 9 [$/widget], • there is a fixed cost of 6 000 [$] (a cost such as rent, which does not depend on the decision x),
• the variable cost of producing x widgets is 2x + 0.001 x2 [\$].
As president your job is to maximize the profit, which is revenue minus costs.

### 7. Mean value theorem

Abstract
The derivative of a function f at a point ξ
$$f'\left( \xi \right) = \mathop {\lim }\limits_{\Delta x \to 0} {\rm{ }}{{f\left( {\xi + \Delta x} \right) - f\left( \xi \right)} \over {\Delta x}},$$
is the slope of the line tangent to the graph of f at the point P = (ξ ,f (ξ)). Restricting to Δx > 0 we see that f′(ξ) is the limit of the slopes of secants PQ, as QP from the right (Fig. 7.1 a).

### 8. Definite integrals

Abstract
Calculus is based on the concept of limit and on two limiting operations:
• integration, computing integrals which are limits of appropriate sums;
• differentiation, computing derivatives which are limits of appropriate differences.

### 9. Fundamental theorem of calculus

Abstract
In this chapter we study functions of the form
$$F\left( x \right) = \int\limits_a^x {f\left( t \right)} {\rm{ d}}t$$
called indefinite integrals of f. If f. is continuous, then F is an antiderivative of f, see Theorem 9.13. Indefinite integrals allow an easy computation of definite integrals as follows,
$$\int\limits_a^b {f\left( x \right)} {\rm{ d}}x = F\left( b \right) - F\left( a \right),$$
see Theorem 9.15. These two theorems are known jointly as the fundamental theorem of calculus. An application to physics is given in Sect. 9.2.

### 10. Integration techniques

Abstract
We study methods for computing antiderivatives, methods known collectively as integration techniques. Three such methods are covered here:
• the change of variables (or substitution) method (Sect. 10.1),
• integration by parts (Sect. 10.2),
• the partial fractions expansion method (Sect. 10.3).
The first two are more than techniques. They are an essential part of calculus, and at the same time they will enable you to compute most integrals you are likely to encounter in practice.

### 11. Applications of integrals

Abstract
What do area, length, volume, work, and hydrostatic force have in common? All of these (and many other important concepts in science and engineering) can be modelled as Riemann sums (8.6)
$$\sum\limits_{k = 1}^n {f\left( {{\xi _k}} \right)} {\rm{ }}\Delta {x_k},$$
and computed as integrals (8.28),
$$\int\limits_{a}^{b} {f(x)dx: = \mathop{{\lim }}\limits_{{\parallel \mathcal{P}\parallel \to 0}} } \sum\limits_{{k = 1}}^{n} {f({{\xi }_{k}})\Delta {{x}_{k}}.}$$
In this chapter integrals are applied to problems of computing areas (Sects. 11.1, 11.2, and 11.6), arc lengths (Sect. 11.3), volumes (Sects. 11.4 and 11.5), moments and centroids (Sects. 11.7 and 11.8), work (Sect. 11.9), and hydrostatic force (Sect. 11.10).

### 12. Sequences and series

Abstract
This chapter is devoted to the study of convergence of sequences (a0, a1, a2, . . . ) and series ∑ n=0 an of real numbers an. We use the notation N0 := {0, 1, 2, 3,. . .} and N := {1, 2, 3,. . .} throughout.
$$\sum\limits_{k = 0}^\infty {{u_k}\left( x \right)}$$