14.1.1 Definitions and Formulas

Quotient

A quotient is simply the result of dividing one quantity by another quantity.

Average slope

The average slope between two points on a function is what you get when you start with a function (f), evaluate it at two points (say x₁ and x₂) and then take the difference of these values, f(x₂) - f(x₁) and divide it by the distance between the two x-values (x₂ - x₁). Thus,

Note that the order is important! If you start with x₂ first in the numerator, you must also start with x₂ in the denominator. The graph below shows the basic idea and illustrates why it’s called average slope and not the actual slope. The dashed line between the two points represents the average slope of the function (the curved line) between those two points. In between the two points, though, notice that there are places where the curve has a more negative slope than the average slope and places where the slope is even positive!

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Figure 14.4: Average slope between two points.

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Difference quotient

The difference quotient is another way of writing the average slope. Instead of looking at the average slope between x₁ and x₂, we look at the average slope between x₁ and x₁ + h, where we think of h as a small number. So x₁ + h is another way of writing x₂. This form of x₂ allows us to focus on how the function changes at x₁. Using x₁ + h in place of x₂ changes the denominator of the average slope formula. Instead of x₂ - x₁, we have (x₁ + h) - x1 = h. So, the average slope formula takes on a new name and a new look:

Consider the line passing through the point (x₁,f(x₁)) and having the same slope as the difference quotient, with a fixed value of h, say 1. If we look at this line for smaller and smaller values of h (say 0.1, 0.01, 0.001, etc.) we see that the line eventually becomes ”parallel” with the function at the point (x₁,f(x₁)). This visual process of watching the line become parallel can be carried out mathematically through a limit.

Marginal Analysis

This is a financial/business term for the process of finding the instantaneous rate of change of a function at a point. Essentially, this is a difference quotient, and it is useful for answering the question ”If my independent variable increases by 1 unit, how much will my dependent variable increase (or decrease)?” Another way to think of this is:

How much bang do I get for each additional buck that I spend?

Marginal Cost

Basically, when the word ”marginal” is followed by a term like ”cost”, it means that you are looking at the instantaneous rate of change of the cost function, which is just its derivative.

Marginal Profit

Instantaneous rate of change of the profit function.

Marginal Revenue

Instantaneous rate of change of the revenue function.

Derivative function

The derivative function is a function derived from the slopes of another function. Basically, at each point (x,f(x)) the function has a slope, usually denoted by f′(x). If we collect all these slopes into a new function, so that plugging in a value of the independent variable, x, results in the slope of f at that point, then we have the derivative function. The derivative of a function at a point is also denoted by the notation which indicates that we are interested in the slope of f in the x-direction. Thus, a positive number for the derivative means that as x increases (we always move to the right) the value of f is increasing. Likewise, a negative value of the derivative indicates that the function is decreasing at that point. Officially, the derivative of a function at a point is computed by taking the difference quotient and letting h go to zero. This is noted mathematically by the ”limit of the difference quotient”:

Second derivative

Since the first derivative of a function is (usually) a function itself, we can take its derivative. We refer to the derivative of the derivative of a function as the second derivative. It is denoted by f′′ or . Since the derivative tells how fast the function is changing, the second derivative tells us how fast the first derivative is changing. Thus, it measures the rate of change of the slope, which is called concavity. In a graph, concavity is easy to see: it refers to the direction and steepness of the way the function bends. If it bends up (looks like a cup) then the concavity is positive. If it bends down (looks like a frown) then the concavity is negative. If the function is almost flat, then the concavity is close to zero.

In this graph, there are five points marked A - E. The function and its derivatives are described at each of these points below.

1. Here the function is negative, the slope is negative (it is a decreasing graph) and the second derivative (concavity) is zero, since the graph is basically flat. Thus, f(A) < 0, f′(A) < 0, f′′(A) = 0.
2. Here we have the function negative (it is below the x-axis, the line y = 0). The slope is zero, since the graph is horizontal at this point. The concavity is positive since the graph is curving upward. Thus, f(B) < 0, f′(B) = 0, f′′(A) > 0.
3. Here the function is positive, the slope is positive (it is an increasing graph) and the second derivative (concavity) is zero, since the graph is basically flat. Thus, f(C) > 0, f′(C) > 0, f′′(C) = 0.
4. Here we have the function positive (above the x-axis, y = 0), the slope is zero, since the graph is horizontal at this point, and the concavity is negative since the graph is curving downward. Thus, f(D) > 0, f′(D) = 0, f′′(D) < 0.
5. Here the function is positive, the slope is negative (it is a decreasing graph) and the second derivative (concavity) is zero, since the graph is basically flat. Thus, f(E) > 0, f′(E) < 0, f′′(E) = 0. In addition, since the function much steeper at point E than at point A, we know that the slope at E is more negative. Thus, we can also say that f′(E) < f′(A).