What Is Differential Calculus?

 

Lines and Slopes

Although many people don't think of it this way, a line is a first-degree polynomial curve because the highest power is one. The two forms of the following line equation are equivalent. The first one fits the form shown in Equation 1.1 and the second is the form introduced in early math classes.

(1.3) Two forms of the equation for a line. 

The first coefficient (m in the second form) is the slope of the line. It defines how steeply the line angles as the values of the variable increase or decrease. The second coefficient is an offset. It defines a constant offset value for the line. Without the offset, all lines would pass through the point (0, 0) regardless of the slope. Figure 1.1 shows how the coefficients affect the line.

Figure 1.1: Examples of lines.

To put it a different way, the slope represents the ratio of the change in the result of the function to the change in the variable. This ratio is shown in Equation 1.4.

(1.4) The slope as the ratio of y to x. 

In most cases, the slope of a line is much more interesting and important than the offset because it describes the direction of the line. The slope is the same for any point on the line, which makes it easier to visualize. For higher-degree polynomials , it isn't quite this simple. In the next section, I'll show you functions for which the slope is constantly changing. Throughout this book, you'll see ways in which the slope is very important.

Before I move to polynomials of a higher degree, I would like to remind you that mathematically, there is no difference between a polynomial with a degree of one and a polynomial with a higher degree. I have set them apart here so that you could begin to think about slopes in familiar terms, but you'll soon see that they all behave the same way.