Chapter 1
The Mathematical Toolbox: A Summary

1.2 Linear Functions

An expression such as represents a linear equation (function), where is the Y-intercept (it gives the value of Y when X = 0) and is the slope (often referred to as rise/run). Here Y is the dependent variable and X is the independent variable. Note that both and are constants.

1.3.1 Solving Two Simultaneous Linear Equations

At times you will need to obtain a solution to a set of simultaneous linear equations, that is, to a set of equations of the general form:

A system such as this is said to be consistent if it has at least one solution. Moreover, if , then this equation system is consistent. For instance, the equation system

is consistent since (1) (-2) - (3) (-1) = 1 ? 0. In fact, to obtain the (unique) solution, we can multiply (Equation 1.3) by -3 so as to obtain -3X + 3Y = -18, and then add this multiple to (Equation 1.4) to get Y = -14. If we then substitute Y = -14 into (Equation 1.3), we obtain X = -8. How do we know that we have generated the correct solution to this equation system?

Answer: Substitute X = -8 and Y = -14 back into, say, (Equation 1.4) and show that equality holds.

It is easily demonstrated that the equation system

is inconsistent or dependent in that it has no solution. Here (4) (8) - (16) (2) = 0. Clearly, these two equations represent parallel lines-they do not intersect.

1.4 Summation Notation

The operation of addition of a set of n values is readily carried out by using the "sigma" notation. In this regard, the left-hand side of the expression

reads: "the sum of all values as i goes from 1 to n." The right-hand side shows that the operation of addition has been executed. Some useful summation rules are as follows:

1. Rule 1:
2. Rule 2: , where c is a constant.
3. Rule 3: where c is a constant.

Note also that

if is the sample mean, then

The Pearson sample correlation coefficient can be written as either

where and are the sample means of X and Y, respectively; or as

Given a collection of data points we can fit a linear equation of the form Y = a + bX through them, where a is the Y-intercept and b is the slope. Here,

or

and .

1.5 Sets

We know that a set is a collection or grouping of items (called elements) without regard to structure or order and that a set is usually defined by listing its elements. A set containing no elements is called the null set or the empty set and is denoted as ?. A set containing all elements under a particular discussion or in a given context is termed the universal set and is denoted as U. Given a set A, its complement, denoted , is the set containing all the elements within U that lie outside of A.

The intersection of two sets A and B is the set of elements common to both A and B. It is denoted as . The union of two sets A and B contains the elements in A, or in B, or in both A and B. It is denoted as . A moments reflection reveals the following:

1.6 Functions and Graphs

A function f is a rule or law of correspondence that associates with each value of a variable x a unique value of a variable y. Here y is termed the image of x under rule f. This "rule" is written as y = f (x) and is read: "y is a function of x." Think of a function as a recipe for getting unique y values from x values. Here x is termed the independent variable and y is called the dependent variable. The set of all admissible x values is called the domain of the function; the collection of y values, which are the image of at least one x, is termed the range of the function. For instance, a function may appear as . What is the rule that is operative here? The rule is: Select a value of x, multiply it by itself, and then add 2 to the result to get y. So, if x = 2, the image of x = 2 via rule f is

It is important to note that for each x value there is one and only one resulting value of y. However, a y value may be the image of more than one x value. However, if implies , then f is said to be a one-to-one function.

A useful device for illustrating a function is its graph. Here y is plotted on the vertical axis and x is plotted on the horizontal axis. Then, via the rule f, a set of x values are chosen and their corresponding y values are determined. These ordered (x, y) combinations are then connected to form the graph of the function in the x-y-plane (see Figure 1.1).

Figure 1.1 The graph of the function y = f(x) = x2.

Given a particular graph, how can we determine whether or not it represents a function?

Answer: We can employ the vertical line test: For a given value of x, find y by drawing a vertical line at x. If the line cuts the graph at only a single value of y, then the graph indeed represents a function.

1.7 Working with Functions

At times it is important to be able to graph a linear function in a quick and efficient fashion. To this end, let us consider the following approach:

1. Intercept method: Since two points are enough to draw a unique straight line, find both the horizontal intercept (set y = 0 and solve for the x-intercept) and the vertical intercept (set x = 0 and solve for the y-intercept) and connect the resulting two points, which are called basic points. For instance, suppose y = 2x - 6. Then,
   1. if y = 0, then 2x - 6 = 0 or x = 3; and
   2. if x = 0, then y = -6.

   Hence, the two basic points are (3, 0) and (0, -6). These points are then connected to obtain the desired graph.

An alternative method for graphing a function (which is useful for both linear and nonlinear equations) is the point method:

1. Point method: Given an admissible set of x values, we use the rule f to get the corresponding y values and then we connect each of the resulting ordered pairs (x, y) (see Figure 1.1).

1.8 Differentiation and Integration

We may think of a function as being continuous if its graph is without breaks. Additionally, the derivative of a function y = f(x) with respect to x is defined as

Here dy/dx is the instantaneous rate of change in y per unit change in x as ?x gets smaller and smaller. Note that if f has a derivative at a point x = a, then it is also continuous at x = a. However, if f is continuous at a point x = a, it does not follow that f has a derivative there.

The derivative is also interpreted as the slope of the function y = f(x) at a given point, that is, for x = a, is the slope of f at this x value. Additionally, depicts the marginal change in y with respect to x.

Rules of Differentiation:

1. Power rule: If , then . (If .)
2. The derivative of a constant is zero.
3. Coefficient rule: If , then . (If , then . Here, k = 6 and .)
4. Sum (difference) rule: If , then . (If , then .)
5. Product rule: If , then . (If , then .)
6. Quotient rule: If , then

   (If , then )

7. Derivatives of exponential and logarithmic functions:
   1. If .
   2. If , where k is a constant.
   3. If .

Higher Order Derivatives:

1. Given that the first derivative of f with respect to x appears as , the second derivative of f with respect to x is written as and is calculated as the derivative of the first derivative.

   (If .)

   Geometrically, the second derivative of f is the rate of change of the slope of f at a specific point on the graph of f. For instance, at a point x = a, if , then f is said to be concave downward at x = a; and if , then f is termed concave upward of x = a.

2. The third derivative of f with respect to x is the derivative of the second derivative. (Given above, .)
3. In general, the nth order derivative of f with respect to x is the derivative of the (n - 1)th order derivative, for example, the sixth derivative of f is the derivative of the fifth derivative of f.

There are two distinct ways to view the concept of an integral. On the one hand, we have the definite integral-an integral that is the limit of an...