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This book fills an educational void by adapting unique classroom-tested techniques that students find most congenial...that strip the shroud of mystery from an esoteric subject...that prepare students for applications of calculus in later courses.
"...fills an educational void by adapting unique, classroom- tested techniques that will prepare students for the applications of calculus in later courses."
We begin calculus with a chapter on functions because virtually all problems in calculus involve functions. We discuss functions in general, and then concentrate on the special functions which will be used repeatedly throughout the course.
A function may be thought of as an input-output machine. Given a particular input, there is a corresponding output. This process may be represented by various schemes, such as a table or a mapping diagram listing inputs and outputs (Fig. 1). Functions will usually be denoted by single letters, the most common being f and g. If the function g produces the output 3 when the input is 2, we write g(2) = 3.
Often functions are described with formulas. If f(x) = [x.sup.2] + x then f(3) = 9 + 3 = 12, f(a) = [a.sup.2] + a, f(a + b) = [(a + b).sup.2] + (a + b) = [a.sup.2] + 2ab + [b.sup.2] + a + b. We might refer to "the function [x.sup.2] + x" without using a special name such as f.
For example, if f(x) = 2x - 9 then
f(3) = 6 - 9 = - 3 ƒ(0) = -9 f(a) = 2a - 9 f(a + b) = 2(a + b) - 9 = 2a + 2b - 9 f(a) + f(b) = 2a - 9 + 2b - 9 = 2a + 2b - 18 f(3a) = 2(3a) - 9 = 6a - 9 3f([a) = 3(2a - 9) = 6a - 27 f([a.sup.2]) = 2[a.sup.2] - 9 [(f(a)).sup.2] = [(2a - 9).sup.2] = 4[a.sup.2] - 36a + 81 f(-a) = 2(-a) - 9 = -2a - 9 - f(a) = -(2a - 9) = -2a + 9.
The input of a function f is called the independent variable, while the output is the dependent variable. We say that the function f maps x to ƒ(x), and call ƒ(x) the value of the function at x. The set of inputs is called the domain of f, and the set of outputs is the range.
A function ƒ(x) is not allowed to send one input to more than one output. Figure 2 illustrates a correspondence that is not a function. For example, it is illegal to write g(x) = ± [square root of (2[x.sup.2] + 3), since each value of x produces two outputs. It certainly is legal to write and use the expression ± [square root of 2[x.sup.2] + 3], but it cannot be named g(x) and called a function.
Functions often arise when a problem is translated into mathematical terms. The solution to the problem may then involve operating on the functions with calculus. Before continuing with functions in more detail we'll give an example of a function emerging in practice. Suppose a pigeon is flying from point A over water to point B on the beach (Fig. 3), and the energy required to fly is 60 calories per mile over water but only 40 calories per mile over land. (The effect of cold air dropping makes flying over water more taxing.) The problem is to find the path that requires minimum energy. The direct path from A to B is shortest, but it has the disadvantage of being entirely over water. The path ACB is longer, but it has the advantage of being mostly over land. In general, suppose the bird first flies from A to a point P on the beach x miles from ITLITL, and then travels the remaining 10 - x miles to B. The value x = 0 corresponds to the path ACB, and x = 10 corresponds to the path AB. The total energy E used in flight can be calculated as follows:
E = energy expended over water + energy expended over land (1)
= calories per water mile × water miles + calories per land mile × land miles
= 60 [bar.AP] + 40 [bar.PB]
= 60 [square root of (36 + [x.sup.2]) + 40(10 - x), 0 [less than or equal to] x [less than or equal to] 10.
Thus the energy is a function of x. Calculus will be used in Section 4.2 to finish the problem and find the value of x that minimizes E.
In deriving (1), we restricted x so that 0 [less than or equal to] x [less than or equal to] 10 since we assumed that to minimize energy the bird should fly to a point P between C and B as indicated in Fig. 3. Since problems often restrict the independent variable in a similar fashion, certain notation and terminology has become standard.
The set of all x such that a [less than or equal to] x [less than or equal to] b is denoted by [a, b] and called a closed interval (Fig. 4). With this notation, the variable x in (1) lies in the interval. The set of all x such that a < x < b is denoted by (a, b) and called an open interval. Similarly we use [a, b) for the set of x where a [less than or equal to] x < b,(a, b] for a < x [less than or equal to] b, [a, [infinity]) for x [greater than or equal to] a, (a, [infinity]) for x > a, (-[infinity], a] for x [less than or equal to] a, and (-[infinity], a) for x < a. In general, the square bracket, and the solid dot in Fig. 4, means that the endpoint belongs to the set; a parenthesis, and the small circle in Fig. 4, means that the endpoint does not belong to the set. The notation (-[infinity], [infinity]) refers to the set of all real numbers.
As another example of a function, consider the greatest integer function: Int x is defined as the largest integer that is less than or equal to x. Equivalently, Int x is the first integer at or to the left of x on the number line. For example, Int 5.3 = 5, Int 5.4 = 5, Int 7 = 7, Int(-6.3) = - 7 . Note that for positive inputs, Int simply chops away the decimal part. The domain of Int is the set of all (real) numbers. (Elementary calculus uses only the real number system and excludes nonreal complex numbers such as 3i and 4 + 2i.) The range of Int is the set of integers. Frequently, Int x is denoted by [x]. Many computers have an internal Int operation available. To illustrate one of its uses, suppose that a computer obtains a numerical result, such as x = 2.1679843, and is instructed to keep only the first 4 digits. The computer multiplies by 1000 to obtain 2167.9843, applies Int to get 2167, and then divides by 1000 to obtain the desired result 2.167 or, in our functional notation, 1/1000 Int(1000 x).
Most work in calculus involves a few basic functions, which (amazingly) have proved sufficient to describe a large number of physical phenomena. As a preview, and for reference, we list these functions now, but it will take most of the chapter to discuss them carefully. The material is important preparation for the rest of the course, since the basic functions dominate calculus.
Problems for Section 1.1
1. Let ƒ(x) = 2 - [x.sup.2] and g(x) = [(x - 3).sup.2]. Find
(a) f(0) (d) g(0) (g) [(g(b)).sup.3] (b) f (1) (e) g(1) (h) f(2a + b) (c) f([b.sup.3]) (f) g([b.sup.3]) (i) the range of f and of g, if the domain is (-[infinity, [infinity])
2. Let f(x) = |x|/x.
(a) Find f(-7)and ƒ(3)
(b) For what values of x is the function defined?
(c) With the domain from part (b), find the range of f.
(d) Does f(2 + 3) equal f(2) + f(3)?
(e) Does f(-2 + 6) equal f(-2) + f(6)?
(f) Does f(a + b) ever equal f(a) + f(b)?
3. The number [x.sub.0] is called a fixed point of the function f if f([x.sub.0]) = [x.sub.0]; i.e., a fixed point is a number that maps to itself. Find the fixed points of the following functions: (a) |x|/x (b) Int x (c) [x.sup.2] (d) [x.sup.2] + 4.
4. Let f(x) = 2x + 1. Does f([a.sup.2]) ever equal [(f(a)).sup.2]?
5. If f(x) = 2x + 3 then f(f(x)) = f(2x + 3) = 2(2x + 3) + 3 = 4x + 9.
(a) Find f (f(x)) if f(x) = [x.sup.3]. (b) Find Int(Int x). (c) If f(x) = -x + 1, Find f (f(x)), f(f(f(x))), and so on, until you seethe pattern.
6. A charter aircraft has 350 seats and will not fly unless at least 200 of those seats are filled. When there are 200 passengers, a ticket costs $300, but each ticket is reduced by $1 for every passenger over 200. Express the total amount A collected by the charter company as a function of the number p of passengers.
1.2 The Graph of a Function
Information can usually be perceived more easily from a diagram than from a set of statistics or a formula. Similarly, the behavior of a function can often be better understood from its graph, which is drawn in a rectangular coordinate system by using the inputs as x-coordinates and the outputs as y-coordinates; i.e., the graph of ƒ is the graph of the equation y = f(x). In sketching a graph it may be useful to make a table of values of the input x and the corresponding output y.
The graph of the function f(x) = -2x + 3 is the line with equation y = -2x + 3 (Fig. 1). It has slope -2 and passes through the point (0,3).
The graph of Int x is shown in Fig. 2 along with a partial table of values used to help plot the graph. The graph shows for instance that as x increases from 2 toward 3, Int x, the y-coordinate in the picture, remains 2; when x reaches 3, Int x suddenly jumps to 3.
Example 1 The graph of a function g is given in Fig. 3. Various values of g can be read from the picture: since the point (0,6) is on the graph, we have g(0) = 6; similarly, g(4) = 11, g(10) = 4. Since P is lower than Q, we can tell that g(2) < g(3). If g(x) represents the final height of a tree when it is planted with x units of fertilizer, then using no fertilizer results in a 6-foot tree, using 10 units of fertilizer overdoses the tree and it grows to only 4 feet, while 4 units of fertilizer produces an 11-foot tree, the maximum possible height according to the data.
The vertical line test Not every curve can be the graph of a function. The curve in Fig. 4 is disqualified because one x is paired with several y's, and a function cannot map one input to more than one output. In general, a curve is the graph of a function if and only if no vertical line ever intersects the curve more than once. In other words, if a vertical line intersects the curve at all, it does so only once.
Equations versus functions The hyperbola in Fig. 5 is the graph of the equation xy = 1. It is also (solve for y) the graph of the function f(x) = 1/x. The hyperbola in Fig. 6 is the graph of the equation [y.sup.2] - 2[x.sup.2] = 6. It is not the graph of a function because it fails the vertical line test. However, the upper branch of the hyperbola is the graph of the function [square root of (2[x.sup.2] + 6)] (solve for y and choose the positive square root since y > 0 on the upper branch), and the lower branch is the graph of the function - [square root of (2[x.sup.2] + 6)].
Continuity If the graph of f breaks at x = [x.sub.0], so that you must lift the pencil off the paper before continuing, then f is said to be discontinuous at x = [x.sub.0]. If the graph doesn't break at x = [x.sub.0], then f is continuous at [x.sub.0].
The function -2x + 3 (Fig. 1) is continuous (everywhere). On the other hand, Int x (Fig. 2) is discontinuous when x is an integer, and 1/x (Fig. 5) is discontinuous at x = 0.
Many physical quantities are continuous functions. If h(t) is your height at time t, then h is continuous since your height cannot jump.
One-to-one functions, non-one-to-one functions and nonfunctions A function is not allowed to map one input to more than one output (Fig. 7). But a function can map more than one input to the same output (Fig. 8), in which case the function is said to be non-one-to-one. A one-to-one function maps different inputs to different outputs (Fig. 9).
The function [x.sup.2] is not one-to-one because, for instance, inputs 2 and -2 both produce the output 4. The function [x.sup.3] is one-to-one since two different numbers always produce two different cubes.
A curve that passes the vertical line test, and thus is the graph of a function, will further be the graph of a one-to-one function if and only if no horizontal line intersects the curve more than once (horizontal line test). The function in Fig. 10 fails the horizontal line test and is not one-to-one because [x.sub.1] and [x.sub.2] produce the same value of y.
Constant functions If, for example, f(x) = 3 for all x, then f is called a constant function. The graph of a constant function is a horizontal line (Fig. 11). The constant functions are among the basic functions of calculus, listed in the table in Section 1.1.
Power functions Another group of basic functions consists of the power functions x' such as
[x.sup.2] = x x [x.sup.-1] = 1/x [x.sup.½] = [square root of (x)] (the positive square root of x) [x.sup.-1/3] = 1/3[square root of (x)] [x.sup.7/4] = 4[square root of ([x.sup.7)] = [(4[square root of (x)]).sup.7] [x.sup.2.6] = [x.sup.26/10] = 10[square root of ([x.sup.26])].
To sketch the graph of [x.sup.3], we make a table of values and plot a few points. When the pattern seems clear, we connect the points to obtain the final graph (Fig. 12). The connecting process assumes that [x.sup.3] is continuous, something that seems reasonable and can be proved formally. In general, x' is continuous wherever it is defined. If r is negative then x' is not defined at x = 0 and is discontinuous there; the graph of 1/x, that is, the graph of [x.sup.-1], is shown in Fig. 5 with a discontinuity at the origin. Figure 13 gives the graph of [x.sup.2] (a parabola) and of [x.sup.4]. For -1 < x < 1, the graph of [x.sup.4] lies below the graph of [x.sup.2] since the fourth power of a number between -1 and 1 is smaller than its square; otherwise [x.sup.4] lies above [x.sup.2]. Figure 14 gives the graph of y = [square root of (x)], the upper half of the parabola x = [y.sup.2].
Increasing and decreasing functions Suppose that whenever a > b, we have f(a) > f(b); that is, as x increases, f(x) increases also. In this case, f is said to be increasing. The graph of an increasing function rises to the right (Figs. 12 and 14).
Suppose that whenever a > b, we have f(a) < f(b); that is, as x increases, f(x) decreases. In this case, f is decreasing. The graph of a decreasing function falls to the right (Fig. 1).
The functions [x.sup.2] and [x.sup.4] (Fig. 13) decrease on the interval (-[infinity], 0] and increase on [0, [infinity]); overall, on (-[infinity], [infinity]), they are neither increasing nor decreasing. The function 1/x (Fig. 5) decreases on the intervals (-[infinity], 0) and (0, [infinity]) but is neither decreasing nor increasing on the interval (-[infinity, [infinity]).
Excerpted from The Calculus Tutoring Book by Carol Ash Robert B. Ash Excerpted by permission.
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The Graph of a Function.
The Trigonometric Functions.
Inverse Functions and the Inverse Trigonometric Functions.
Exponential and Logarithm Functions.
Solving Inequalities Involving Elementary Functions.
Graphs of Translations, Reflections, Expansions and Sums.
Review of Problems for Chapter 1.
Finding Limits of Combinations of Functions.
Review Problems for Chapter 2.
THE DERIVATIVE PART I.
Definition and Some Applications of the Derivative.
Derivatives of the Basic Functions.
Derivatives of Constant Multiples, Sums, Products and Quotients.
The Derivative of a Composition.
Implicit Differentiation and Logarithmic Differentiation.
Review Problems for Chapter 3.
THE DERIVATIVE PART II.
Relative Maxima and Minima.
Absolute Maxima and Minima.
L'Hôpital's Rule and Orders of Magnitude.
Indeterminate Products, Differences and Exponential Forms.
Drawing Graphs of Functions.
Separable Differential Equations.
Review Problems for Chapter 4.
THE INTEGRAL PART I.
Definition and Some Applications of the Integral.
The Fundamental Theorem of Calculus.
Review Problems for Chapter 5.
THE INTEGRAL PART II.
Further Applications of the Integral.
The Centroid of a Solid Hemisphere.
Area and Arc Length.
The Surface Area of a Cone and a Sphere.
Integrals with a Variable Upper Limit.
Review Problems for Chapter 6.
Pre-Table Algebra I.
Pre-Table Algebra II: Partial Faction Decomposition.
Integration by Parts.
Choosing a Method.
Combining Techniques of Antidifferentiation with the Fundamental Theorem.
Review Problems for Chapter 7.
Convergence Tests for Positive Series I.
Convergence Tests for Positive Series II.
Power Series Functions.
Power Series Representations for Elementary Functions I.
Power Series Representations for Elementary Functions II (Maclaurin Series).
The Taylor Reminder Formula and an Estimate for the Number e.
Power Series in Powers of x - b (Taylor Series).
Review Problems for Chapter 8.
Vector Addition, Subtraction, Scalar Multiplication and Norms.
The Dot Product.
The Cross Product.
The Scalar Triple Product.
The Velocity Vector.
The Acceleration Vector.
Review Problems for Chapter 9.
TOPICS IN THREE-DIMENSIONAL ANALYTIC GEOMETRY.
Cylindrical and Quadric Surfaces.
Cylindrical and Spherical Coordinates.
Review Problems for Chapter 10.
Graphs and Level Sets.
Chain Rules for First-Order Partial Derivatives.
Chain Rules for Second-Order Partial Derivatives.
Maxima and Minima.
Differentials and Exact Differential Equations.
Review Problems for Chapter 11.
Definition and Some Applications of the Double Integral.
Computing Double Integrals.
Double Integration in Polar Coordinates.
Area and Volume.
Further Applications of the Double Integral.
Triple Integration in Spherical Coordinates.
Center of Mass.
Review Problems for Chapter 12.
A1: Distance and Slope.
A2: Equations of Lines.
A3: Circles, Ellipses, Hyperbolas and Parabolas.
A4: The Binomial Theorem.
A6: Polar Coordinates.
Solutions to the Problems.
Abbreviations Used in the Solutions.
List of Symbols.