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## Über dieses Buch

We believe that to understand and be comfortable with mathematical con­ cepts and methods, it is necessary to do mathematics and, traditionally, doing meant with a pencil and paper. Now, a modest home computer can provide a platform for a computer algebra package like Maple, which can perform all of the operations encountered in secondary school mathematics and beyond, and pro­ vide graphical representations of functions, including animations. The capability of rendering accurate graphics for mathematical functions greatly enhances the learning experience, and helps intuition work in new situations, before beginning to do the algebra and calculus needed to solve a problem. For example, if you can see that a function has a clear minimum from its graph, then you are more likely to be able to identify its location precisely by analysis. This book is designed to support the interactive Maple worksheets that we have developed for the two senior secondary school years and made available via the Internet at the anonymous ftp site ftp. ut irc. utoronto . ca (/pub/ednet/maths/maple). This takes you from basic algebra, functions and sequences, to calculus and its applications. Thus, the book is a hardcopy version of the worksheets, with additional explanatory text, cross-referencing, appen­ dices containing worked solutions to all exercises, and indexing.

## Inhaltsverzeichnis

### Chapter 1. Introduction to Maple

Abstract
Welcome to Maple. This chapter will help you become familiar with some of the most commonly used commands in the subsequent chapters.
Christopher T. J. Dodson, Elizabeth A. Gonzalez

### Chapter 2. Functions

Abstract
A relation is a set of ordered pairs; the set of first elements in each ordered pair is called the domain, and the set of second elements is called the range. For example, in a family, the property ‘being uncle of’ is expressed as a relation among the set of family members; A is uncle of B can have more than one B for each choice of A. A function is a relation for which each value in the domain corresponds to a unique value in the range. In other words, a vertical line drawn anywhere from the x-axis will intersect the graph of a function at one and only one point. Mathematicians also use the words map or mapping to mean a function. The concept of a function as a well-defined, that is unambiguous, rule is one of the most important in mathematics.
Christopher T. J. Dodson, Elizabeth A. Gonzalez

Abstract
Many important functions in mathematics, and especially in its applications, are continuous functions. These have the property that small changes in input result in small changes in output. Linear functions have this property. The simplest non-linear continuous functions are the quadratics. The general form of a quadratic function is
$$f\left( x \right) = a{x^2} + bx + c,where{\mkern 1mu} \,a{\mkern 1mu} \, \ne 0.$$
Christopher T. J. Dodson, Elizabeth A. Gonzalez

### Chapter 4. Solving Quadratic Equations

Abstract
In this chapter, a new form of the solve command will be introduced. When we solve equations with more than one solution, we cannot use the assign command as we have in the past.
Christopher T. J. Dodson, Elizabeth A. Gonzalez

### Chapter 5. Polynomial Functions

Abstract
The general form of a polynomial function is
$$f\left( x \right) = a{x^n} + b{x^{n - 1}} + c{x^{n - 2}} + ... + k$$
where a, b, c,…, k are constants. The order, or degree, of a polynomial is the value of the highest exponent on the independent variable, x. We say that the function above is ‘nth order’, or ‘of degree n’ The higher the order of a polynomial function, the greater the curvature and the more curvature changes in general. We shall study curvature changes in more detail in Chapter 18.
Christopher T. J. Dodson, Elizabeth A. Gonzalez

### Chapter 6. Exponential Functions

Without Abstract
Christopher T. J. Dodson, Elizabeth A. Gonzalez

### Chapter 7. Logarithmic Functions

Abstract
The natural logarithm function, represented by ln(x) or log e (x) or just log(x), is the inverse of the natural exponential function so exp(ln(x)) = x = ln(exp(x)). In Maple, the base for the log function is e, however, you can ask for base 10 in Maple with log10(x), as we shall see later. (Be careful when using e in Maple. You must input a capital E, but Maple will return a lower case e.) When mathematicians say log they always mean to base e.
Christopher T. J. Dodson, Elizabeth A. Gonzalez

### Chapter 8. Circular Functions

Abstract
Consider a circle of radius r, and a point P(x,y) on its circumference at angle t radians (recall that π radians is 180°) above the x-axis, then the primary trigonometric functions are defined as:
$$\sin (t) = \frac{y}{r},\cos (t) = \frac{x}{r},and \tan (t) = \frac{y}{x}.$$
Christopher T. J. Dodson, Elizabeth A. Gonzalez

### Chapter 9. Trigonometry

Abstract
You should commit to memory the sides and angles of two simple triangles:
1.
half of a unit square, formed by a diagonal line, gives a 45°, 45°, 90° triangle with sides 1,1,$$\sqrt 2$$

2.
half of an equilateral triangle with side length 2 gives a 30°, 60°, 90° triangle with sides 1,2,$$\sqrt 3$$

Christopher T. J. Dodson, Elizabeth A. Gonzalez

### Chapter 10. Similar Figures

Abstract
Two polygons are said to be similar if corresponding angles are the same and all of the ratios of the lengths of the corresponding sides are equal.
Christopher T. J. Dodson, Elizabeth A. Gonzalez

### Chapter 11. Circles and Spheres

Abstract
The equation of a circle of radius r, centred at x = y = 0, is x2 + y2 = r2. The coordinates of a point P(x,y) on the circle x2 + y2 = r2 and the radius of the the circle are the circle are related by Pythagoras’ Theorem. See 11.1(a)
Christopher T. J. Dodson, Elizabeth A. Gonzalez

### Chapter 12. Loci

Abstract
A locus is the set of all points satisfying a certain condition. For example, a circle centred at the origin with radius r can be described by the set of all points P in the x,y plane located r units from the origin, the defining equation of which is x2 + y2 = r2. A sphere is the set of all points located at a fixed distance from a given point in space.
Christopher T. J. Dodson, Elizabeth A. Gonzalez

### Chapter 13. Sequences and Series

Abstract
A sequence is an ordered set, usually ordered by the natural numbers (N = {1,2,3,…}). Each element in the range of the sequence is a term of the sequence.
Christopher T. J. Dodson, Elizabeth A. Gonzalez

### Chapter 14. Statistics and Probability

Abstract
Before we can draw any conclusions from data that we collect, we must be able to present it in a useful way. The data we shall be examining is a set of numbers assigned to raw_data. Actually, we chose this set by hitting keys at random.
Christopher T. J. Dodson, Elizabeth A. Gonzalez

### Chapter 15. Secants and Tangents

Abstract
The general equation of a line is y = mx + b, where m is the slope of the line and b is the y-intercept. For example, find the slope of the line f(x) = 2x − 1, shown in Figure 15.1(a).
Christopher T. J. Dodson, Elizabeth A. Gonzalez

### Chapter 16. Sequences and Limits

Abstract
As we learned in Chapter 13, a sequence is an ordered set, usually ordered by the set of natural numbers.
Christopher T. J. Dodson, Elizabeth A. Gonzalez

### Chapter 17. Derivatives of Functions

Abstract
The derivative of a function y = f(x) is defined as follows:
• with(student):
• Limit $$({\text{slope([x + delta, f(x + delta)],[x,f(x)]), delta = 0);}}$$;$$\mathop {\lim }\limits_{\delta \to 0} \frac{{{\text{f(}}x + \delta {\text{) - f(}}x{\text{)}}}}{\delta }$$
Christopher T. J. Dodson, Elizabeth A. Gonzalez

### Chapter 18. Functions and Graphs

Abstract
With the help of Maple, or another computer algebra system or plotting package, it is very easy to plot any function without knowing anything about its shape. But there is a lot you can learn about a function by performing a few mathematical manipulations. In this chapter, we shall be studying the function f(x) = 4x3 − 6x2 + 1. It will not be plotted until the end of the chapter.
Christopher T. J. Dodson, Elizabeth A. Gonzalez

### Chapter 19. Rates

Abstract
The velocity of a particle is the rate of change of its position; the rate of change of its velocity is its acceleration.
Christopher T. J. Dodson, Elizabeth A. Gonzalez

### Chapter 20. Integration

Abstract
We can approximate the area under a curve by partitioning the region into rectangular sections and calculating the area of the rectangles (Figure 20.1(a)). The greater the number of rectangles, the more accurate the approximation of the area (Figure 20.1(b)). By inspection of the curve, we see that the area under it must be about (4×2) − 2 = 6.
Christopher T. J. Dodson, Elizabeth A. Gonzalez

### Chapter 21. Trigonometry

Abstract
From their graphs, the trigonometric functions are not linear, so we are not surprised by the following. The sine (or any other trigonometric function) of a sum of angles is not equal to the sum of the sines. For instance, consider the sine of $$\frac{\pi }{6} + \frac{\pi }{4}$$.
Christopher T. J. Dodson, Elizabeth A. Gonzalez

### Chapter 22. Exponents and Logarithms

Abstract
The base of the natural logarithm and exponential, e, is the limit as n approaches infinity of (1 + $$\frac{1}{n}$$)n, shown in Figure 22.1. Like π, e is an irrational number. This seems a most unnatural choice at first, but later studies show that e is indeed the best base. It gives rise to the most important function in all mathematics.
Christopher T. J. Dodson, Elizabeth A. Gonzalez

### Chapter 23. Polar Coordinates

Abstract
A point, P, can be described in Cartesian coordinates as P(x,y), or in polar coordinates as P(r, θ), where r is the radius and θ is the angle from the terminal arm (the positive section of the x-axis). From Chapter 8, the expressions which relate x and y to r and θ are x = r cos(θ) and y = r sin(θ).
Christopher T. J. Dodson, Elizabeth A. Gonzalez

### Backmatter

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