VARIABLES AND FUNCTIONS - Functions, Rates, and Limits - The Calculus Primer

The Calculus Primer (2011)

Part I. Functions, Rates, and Limits


1—1. The Calculus. Emerson once said that it didn’t matter much where a man was, so long as you knew the direction in which he was moving. In somewhat the same way, the significance of a graph or curve often lies not so much in what height a point on the curve has reached, as it does in how fast its height is changing, and whether it is increasing or decreasing.

These observations suggest, in a crude way to be sure, the keynote of the Calculus. In the study of Analytic Geometry our concern is with the interrelations between equations and loci; we examined the properties of “static” curves, and studied the relations between the variables at any given time. In the Calculus, on the other hand, we shall be concerned chiefly with change rather than with “frozen figures.” Speaking rather broadly, the Calculus enables us to understand the nature of changes, their magnitude and their rate; we focus the searchlight on the anatomy of change, revealing the behavior of related variables.

1—2. Variables and Constants. Consider the statement y = 2x + 5. This is an open sentence; that is, it is neither true nor false. Clearly we can substitute many numbers for the letter x: for example, when x is assigned the number 6, the number that then corresponds to y is 17; when x is −4, y then becomes −3; etc. When symbols are used in this way, we call them variables. Usually letters in the latter part of the alphabet, such as s, t, u, v, w, x, y, and z, and also the Greek letters ρ, θ, and ω are used for this purpose. A variable, then, is a symbol used to represent any one of a given set of numbers.

A constant, on the other hand, is a symbol denoting one particular number of a set. The constant may be represented by a specific numeral, such as the “2” and the “5” in the sentence y = 2x + 5; or the constant may be represented by a “general numeral,” i.e., a letter, such as the “m” and the “b” in the sentence y = mx + 6. In any event, the “value” or meaning of a constant is regarded as not changing throughout a given discussion. Constants are usually represented by letters in the early part of the alphabet, as, for example, a, b, c, h, k, m, n, p, or by Greek letters such as α, β, γ, images, λ, and μ.

An absolute constant is a constant whose value never changes; examples are 8, −100, images, images, log 2, π, e. An arbitrary constant is one whose value changes from one problem to another, but does not change during the course of any given discussion, as in the equations x2 + y2 = r2, y = mx + b, y = a sin bx, and y = loga x, where a, b, m, and r are constants. The absolute value of a constant or a variable is its arithmetical value, irrespective of its algebraic sign, and is designated as |a|. Thus, |−5| = 5; |3| = 3; |k| = images.

1—3. Meaning of Function. If two variables x and y are so related that, whenever a value is assigned to x, there is automatically assigned, by some rule or correspondence, a value to y, we call y a single-valued function of x. Some relations admit of assigning two (or more) values to y whenever we assign a value to x. For example, consider the relation y2 = 2x; when x = 2, y = +2, and also −2; when x = 12images, y = +5 as well as −5; when x = 50, y = ±10; when x = 10, y = ±images; etc. Such a relation is sometimes called a multiple-valued function, or more properly, simply a relation between variables. In any event, the variable x, to which values are assigned at will, is known as the independent variable; the variable y, whose corresponding values depend upon the value chosen for x, is known as the dependent variable. The permissible values that may be assigned to x constitute the domain of the function, or relation; the set of corresponding values taken on by y constitutes the range of the function, or relation.

It should be noted, in connection with functions, that corresponding values of the variables may be regarded as ordered pairs. Any value of the independent variable, say x1, is written first, and the value of the dependent variable which corresponds to it, say y1, is written second. Thus we have an ordered pair of numbers: (x1,y1).

A function, then, consists of two things:

(1)A collection of numbers, called the domain of definition of the function.

(2)A rule, verbal or symbolic, that assigns to each number in the domain of definition one and only one number.

The range of values of a function is the collection of all the numbers which the rule of the function assigns to the numbers in the domain of definition of the function. Note that in a function, when the first element of an ordered pair is known, there is no ambiguity as to what the second element is, since only one ordered pair in a given function has the specified first element. The fact that different first elements may have the same second element is immaterial. In other words, a function is a special kind of relation in which the first element of every ordered pair has a unique second element.

These ideas will become clearer from the following. In the relation y = x2 − 4, we say that “y is a function of x,” since to any x selected there corresponds one and only one y. (The fact that two different x’s may yield the same y-value doesn’t matter.) On the other hand, in the relation y2 = x + 3, we do not say that y is a function of x, since to any x selected there correspond two y-values.


1—4. Explicit and Implicit Functions. Let us look at these two relations a little more closely. The relation y = x2 − 4 has been “solved” for one variable in terms of the other. We generally refer to this kind of a relation as an explicit function. The relation of y to x is explicitly stated; for every x there is a unique y. Other examples of explicit functions are:

(1)y = 3 sin x; y is an explicit function of x.

(2)v = t2 + 5t − 3; v is an explicit function of t.

(3)x = images; x has an explicit relation to y.

(4)z = a + x cos2 y; z has an explicit relation to x and y.

Statements (1) and (2) are examples of functions; (3) and (4) are examples of relations (or multiple-valued “functions”). In all four statements the relation has been solved explicitly for the dependent variable.

Now let us look at y2 = x + 3 again. As it stands, it has not been solved for either variable. If we solve for y, we get


where y is not a function of x, but it is stated as having an explicit relation to x. If we solve it for x, we get

x = y2 − 3,

and we see that x is a function of y. Thus:

written x = y2 − 3, x is an explicit function of y;

written y2 = x + 3, x is an implicit function of y.

In other words, an equation involving x and y may not express y explicitly in terms of x, as for example:

(1)xy = 10

(2)y2 = x

(3)x2 + y2 = 9

(4)x2xy + y2 = 4

However, each of these equations defines y in the sense that for any given value of x, there correspond one or more values of y. Such equations are examples of implicit functions, or implicit relations; equation (1) is an implicit function, and the remaining three equations are implicit relations (or multiple-valued “functions”).

The graph of a function consists of all points on the graph of the equation for which the x-coordinate is in the domain of the function.


1. If v = 3t2 − 4, and the domain of t is the set of natural numbers, find the first four corresponding values of v.

2. If the set of all positive odd integers is denoted by 2x + 1, what is the domain of the variable x?

3. What is the set of integers which leave a remainder of 3 when divided by 4? If this set of integers is represented by y, where y = 4x + 3, what is the domain of the variable x? What is the range of y?

4. Any three-digit number may be represented by 100h + 10t + u. What is the domain of the variable u? of t? of h?

5. The relation A = P(1 + ni) gives the amount A of a given principal P at various annual rates of interest i for various numbers of years n.

(a) What is the domain of the variable i if the interest rate is taken by steps of images and may not exceed 4%?

(b) What is the domain of P if no interest is allowed on fractional parts of a dollar?

(c) What is the domain of n if interest is computed by months? by days?

6. Find two or three values for the function y = x2 − 3x + 5 when the domain of x is (a) the set of natural numbers; (b) the set of positive rational numbers; (c) the set of positive irrationals.

7. Which of these graphs represent functions of x, where the domain of x is represented along the X-axis?








1—5. Functional Notation. There are many kinds of functions in mathematics. The functions y = 4x + 5 and y = ax + b are polynomial functions of the first degree in x. As they are of the first degree in x, they are called linearfunctions. The “5” and the “b” are called the absolute term in their respective equations since neither involves a variable, that is, they are constants. Or again, the function y = ax2 + bx + c, where a, b, and c are constants and a ≠ 0, is a polynomial function of the second degree in x. In general, a polynomial function in x may be written in the form

y = a0xn + a1xn−1 + a2xn−2 + ··· + an−1x + an,

ory = xn + p1xn−1 + p2xn−2 + ··· + pn−1x + pn,

where the a’s and the p’s are constants, and an and pn are the absolute terms.

The product of two polynomials in x is always another polynomial in x. Thus if u and v are polynomials in x, then uv is a polynomial in x; also, if u is a polynomial in x and n is a positive integer, then un is also a polynomial in x. However, the ratio images of two polynomials is, in general, not a polynomial in x. Instead, it is called a rational function of x. Here the word “rational” derives its meaning from the word ratio, just as a rational number is defined as the ratio of two integers.

Since a functional relation is essentially a collection of ordered pairs of numbers which associate each element x of its domain with exactly one element y of its range, we often use the symbols f(x) or g(x) to designate the second element of the ordered pair whose first element is x. The symbol “f(x)” is read “f of x”; it means the “value of the dependent variable when any of a given set of values has been assigned to x”. Thus the “f” symbolizes the particular rule which makes the association between the variables. The reader is warned not to think of “f(x)” as meaning “f” times “x.” The “f( )” is the rule indicating the association; a specific number when inserted in the parenthesis yields a corresponding value for y. Thus

y = f(x)” means “y is the value functionally associated with x.”

For example, if the function is 4x + 10, we may write

f(x) = 4x + 10,

ory = 4x + 10,

ory = f(x).

For this particular function, we note that

iff(x) = 4x + 10,

thenf(1) = 4(1) + 10 = 14

f(−3) = 4(−3) + 10 = −2

f(images) = 4(images) + 10 = 13

f(0) = 4(0) + 10 = 10

Or again:

ifF(x) = x3 − 4x2 + 2x − 8

F (3) = (3)3 − 4(3)2 + 2(3) − 8 = −11

F(−1) = (−1)3 − 4(−1)2 + 2(−1) − 8 = −15

F(0) = (0)3 − 4(0)2 + 2(0) − 8 = −8

F(10) = (10)3 − 4(10)2 + 2(10) − 8 = 612

1—6. Continuity of a Function. If in the equation y = f(x), y is defined for every possible value assigned to x, then as x varies continuously from a to b, if y varies continuously from f(a) to f(b), we say that the function is continuous for all values of x. If, however, there is some value assigned to x such that there is no corresponding value of y, that is, for which y is not defined, then the function is said to be discontinuous at that point, or for that value of x. For example, in the function xy = k, or y = images, y is not defined when x = 0, since images is an impossible operation; hence the function is discontinuous at x = 0. Again, in the function x2y2 = 36, or y = images y is not defined for absolute values of x < 6, that is, when −6 < x < +6; hence the function is discontinuous in the interval from x = − 6 to x = +6.

Intuitively, a continuous function may be said to have an “unbroken” graph. If there is a break of some kind in the graph, the function is discontinuous. This crude “definition” is, of course, merely a very rough description. A more precise definition of the continuity of a function will be given later.


1. Write, in function notation, the general rational integral function of the fifth degree in x.

2. Write the general form of the polynomial of the ninth degree which contains only the odd powers of the variable.

3. If f(x) = ax2bx + c, write f(m); f(5).

4. If F(x) = 3x3 + 4x2 + 2x − 6, what is the value of F(−l)? of F(0)?

5. If f(x) = 2x2 + 5x − 3, what function does f(x + h) represent? Write this function as a general polynomial in x; what is the absolute term of this latter polynomial?

6. Given f(x) = 3x−2; find f(0); f(1); f(2); f(−1).

7. Given ϕ(x) = x3 − 4x2 + 8x − 8; find ϕ(x + 1); ϕ(xh).

8. Given Q(x) = images; find Q(x − 1) − Q(x).

9. Given ϕ(x) = ax2 + bx + c; find ϕ(x + h) − ϕ(h).

10. Given F(z) = log images; find F(u) − F(v).

11. Given the function f(x) = x2 + 4x − 2; using the set of real numbers as the domain of definition, draw the graph of f(x), and state the range of f(x), or y.

12. Given the domain of definition for each of the following functions as the set {0,1, 2, 3, 4, 5, 6}, draw the graph, and in each case state the range of the function:

(a) f(x) = 3x + 2

(b) g(x) = 2x2

(c) F(x) = 3x