About this title Calculus Made Easy has long been the most popular calculus primer, and this major revision of the classic math text makes the subject at hand still more comprehensible to readers of all levels. With a new introduction, three new chapters, modernized language and methods throughout, and an appendix of challenging and enjoyable practice problems, Calculus Made Easy has been thoroughly updated for the modern reader. To a non-mathematician, its simplicity and clarity reveals the mathematical genius of Newton, Leibniz, and Thompson himself. Martin Gardner deserves huge thanks for renewing this great book. Clark, Massachusetts Institute of Technology Praise for Martin Gardner "For more than half a century, Martin Gardner has been the single brightest beacon defending rationality and good science He is one of the most brilliant men and gracious writers I have known.
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No concept in mathematics, especially in calculus, is more fundamental than the concept of a function. The term was first used in a letter written by Gottfried Wilhelm Leibniz, the German mathematician and philosopher who invented calculus independently of Isaac Newton.
Since then the term has undergone a gradual extension of meaning. In traditional calculus a function is defined as a relation between two terms called variables because their values vary. Call the terms x and y. If every value of x is associated with exactly one value of y, then y is said to be a function of x. It is customary to use x for what is called the independent variable, and y for what is called the dependent variable because its value depends on the value of x.
As Thompson explains in Chapter 3, letters at the end of the alphabet are traditionally applied to variables, and letters elsewhere in the alphabet usually first letters such as a,b,c Constants are terms in an equation that have a fixed value. They remain the same as x and y vary. In this case the function is called a one-to-one function because the dependency goes both ways. We know from the Pythagorean theorem that the square of the hypotenuse equals the sum of the squares of the other two sides.
In this case the sides are equal. The most common way to denote a function is to replace y, the dependent variable, by f x — f being the first letter of "function. This means that y, a function of x, depends on the value of x in the expression 2x — 7. In this form the expression is called an explicit function of x. It is easily obtained from the equation by rearranging terms. Instead of f x , other symbols are often used. If the dependent variable is a function of a single independent variable, the function is called a function of one variable.
Familiar examples, all one-to-one functions, are: The circumference or area of a circle in relation to its radius. The surface or volume of a sphere in relation to its radius. The log of a number in relation to the number. Sines, cosines, tangents, and secants are called trigonometric functions. Logs are logarithmic functions. Functions can depend on more than one variable. Again, there are endless examples. The hypotenuse of a right triangle depends on its two sides, not necessarily equal.
The function of course involves three variables, but it is called a two-variable function because it has two independent variables. Note that this is not a one-to-one function. Knowing x and y gives z a unique value, but knowing z does not yield unique values for x and y. Two other familiar examples of a two-variable function, neither one-to-one, are the area of a triangle as a function of its altitude and base, and the area of a right circular cylinder as a function of its radius and height.
Functions of one and two variables are ubiquitous in physics. The period of a pendulum is a function of its length. The distance covered by a dropped stone and its velocity are each functions of the elapsed time since it was dropped. Atmospheric pressure is a function of altitude. The electrical resistance of a wire depends on the length of the wire and the diameter of its circular cross section. Functions can have any number of independent variables. A simple instance of a three-variable function is the volume of a rectangular room.
The volume of a four-dimensional hyper-room is a function of four variables. A beginning student of calculus must be familiar with how equations with two variables can be modeled by curves on the Cartesian plane. Values of the independent variable are represented by points along the horizontal x axis.
Values of the dependent variable are represented by points along the vertical y axis. Points on the plane signify an ordered pair of x and y numbers. Points along each axis represent real numbers rational and irrational , positive on the right side of the x axis, negative on the left; positive at the top of the y axis, negative at the bottom.
If x is the side of a square, we assume it is neither zero nor negative, so the relevant curve would be only the right side of the parabola.
Move vertically up from 3 on the x axis to the curve, then go left to the y axis where you find that the square of 3 is 9. I apologize to readers for whom all this is old hat. If a function involves three independent variables, the Cartesian graph must be extended to a three-dimensional space with axes x, y, and z. I once heard about a professor, whose name I no longer recall, who liked to dramatize this space to his students by running back and forth while he exclaimed "This is the x axis!
Unfortunately, a professor cannot dramatize axes higher than three by running or jumping. Note the labels "domain" and "range" in Figure 1. In recent decades it has become fashionable to generalize the definition of function. Values that can be taken by the dependent variable are called the range. On the Cartesian plane the domain consists of numbers along the horizontal x axis. The range consists of numbers along the vertical y axis. Domains and ranges can be infinite sets, such as the set of real numbers, or the set of integers; or either one can be a finite set such as a portion of real numbers.
The numbers on a thermometer, for instance, represent a finite interval of real numbers. If used to measure the temperature of water, the numbers represent an interval between the temperatures at which water freezes and boils.
In modern set theory this way of defining a function can be extended to completely arbitrary sets of numbers for a function that is described not by an equation but by a set of rules. The simplest way to specify the rules is by a table. For example, the table in Figure 2 shows a set of arbitrary numbers that constitute the domain on the left. The corresponding set of arbitrary numbers in the range is on the right. The rules that govern this function are indicated by arrows.
These arrows show that every number in the domain correlates to a single number on the right. As you can see, more than one number on the left can lead to the same number on the right, but not vice versa. Another example of such a function is shown in Figure 3, along with its graph, consisting of 6 isolated points in the plane. Because every number on the left leads to exactly one number on the right, we can say that the numbers on the right are a function of those on the left. Some writers call the numbers on the right "images" of those on the left.
The arrows are said to furnish a "mapping" of domain to range. Some call the arrows "correspondence rules" that define the function. For most of the functions encountered in calculus, the domain consists of a single interval of real numbers.
We call such a function "continuous" if its graph can be drawn without lifting the pencil from the paper, and "discontinuous" otherwise. The complete definition of continuity, which is also applicable to functions with more complicated domains, is beyond the scope of this book. For example, the three functions just mentioned are all continuous. Figure 4 shows an example of a discontinuous function. In this book we will be concerned almost entirely with continuous functions.
Note that if a vertical line from the x axis intersects more than one point on a curve, the curve cannot represent a function because it maps an x number to more than one y number. Figure 5 is a graph that clearly is not a function because vertical lines, such as the one shown dotted, intersect the graph at three spots. It should be noted that Thompson did not use the modern definition of "function.
In this generalized definition of function, a one-variable function is any set of ordered pairs of numbers such that every number in one set is paired with exactly one number of the other set. Put differently, in the ordered pairs no x number can be repeated though a y number can be. In this broad way of viewing functions, the arbitrary combination of a safe or the sequence of buttons to be pushed to open a door, are functions of counting numbers.
To open a safe you must turn the knob back and forth to a random set of integers. They represent the order in which numbers must be taken to open the safe, or the order in which buttons must be pushed to open a door. In recent years mathematicians have widened the notion of function even further to include things that are not numbers. Indeed, they can be anything at all that are elements of a set. A function is simply the correlation of each element in one set to exactly one element of another set.
This leads to all sorts of uses of the word function that seem absurd. Positions of towns on a map are a function of their positions on the earth. The number of toes in a normal family is a function of the number of persons in the family. Different persons can have the same mother, but no person has more than one mother. This allows one to say that mothers are a function of persons.
Elephant mothers are a function of elephants, but not grandmothers because an elephant can have two grandmothers. As one mathematician recently put it, functions have been generalized "up to the sky and down into the ground.
Any element in a domain, numbers or otherwise, is put into the box. Out will pop a single element in the range. The machinery inside the box magically provides the correlations by using whatever correspondence rules govern the function. In calculus the inputs and outputs are almost always real numbers, and the machinery in the black box operates on rules provided by equations. Because the generalized definition of a function leads to bizarre extremes, many educators today, especially those with engineering backgrounds, think it is confusing and unnecessary to introduce such a broad definition of functions to beginning calculus students.
Changing the size of the type does not increase the size of any mathematical notation. Aug 26, Allyn rated it it was amazing. Thompson is one of msde individuals represented on the Engineers Walk in Bristol, England. I reread the text a few times and worked out most of the problems and feel I now understand calculus well enough to appreciate its significance and genius. Good introduction book to the concepts of calculus. Calculus Made Easy thompwon I do know that I will buy this book for all my children when they are taking Calculus.
Calculus made easy - A very simple introduction to differential and integral calculus
Calculus Made Easy