Slopes of Functions

Consider a line passing through points (x₁, y₁) and (x₂, y₂). The slope of the line is defined as the ratio of change in y to change in x:

 

Example 1: Find the slope of a line passing through points (3, -4) and (-7, 15). Slope = (15 - (-4))/(-7 - 3) = -19/10

The slope of a straight line is constant, so the slope defined by known points (x₁, y₁) and (x₂, y₂) equals the slope defined by an indefinite point (x, y), and either (x₁, y₁) or (x₂, y₂):

If we equate this definition of slope with the known value of the slope, then we have a complete description of the line.

 

Example 2. The slope of a line passing through (3, -4) and (-7, 15) is -19/10 (see Example 1). We can define the slope of this line using an indefinite point (x, y) and either (3, -4) or (-7, 15): Slope = (y - (-4))/(x - 3) = (y + 4)/(x - 3) = -19/10 Slope = (y - 15)/(x - (-7)) = (y - 15)/(x + 7) = -19/10 Both of these equations are valid descriptions of the same line.

There are infinite ways to describe the same line, but the preferred way is the "slope-intercept" equation, y = slope*x + intercept. This equation immediately reveals two important properties of the line: the slope and the y-intercept.

 

Example 3. We can "rearrange" both of the equations found in Example 2 into the slope-intercept equation of the line. Starting with the first equation, (y + 4)/(x-3) = -19/10, we multiply both sides by (x-3) and then subtract 4 from each side to get y = (-19/10)x + 17/10. Starting with the second equation, (y-15)/(x + 7) = -19/10, we multiply both sides by (x + 7) and then add 15 to each side to get the same slope-intercept equation, y = (-19/10)x + 17/10. This equation tells us that the slope of the line is -19/10, and the y-intercept is 17/10.

The slope-intercept equation presents x as the independent variable and y as the dependent variable. Dependent variable y is a function of independent variable x, and this relationship is often shown by writing y(x) or F(x) instead of y.

Here are some characteristics of slopes and lines:

• Positive slope: y(x) moves up-hill as x increases
• Negative slope: y(x) moves down-hill as x increases
• Horizontal line: The slope equals zero, and the equation of the line is y = constant.
• Vertical line: The slope is infinite (and undefined, because it could equal (+) infinity or (-) infinity). Also, there is no y-intercept. The equation of the line is x = constant.
• Broken line (y(x) is not continuous) or Corner (y(x) is continuous but bends sharply): The slope has two values when approaching the break or corner from opposite directions, and is undefined at the break or corner. Two equations per break or corner are needed to describe the function y(x).

Suppose that (x, F(x)) and (u, F(u)) are points on the same curve F. The slope of the line connecting these points, [F(u) - F(x)]/[u - x], approximates the slope of the tangent at (x, F(x)):

The smaller the distance between x and u, the better the approximation. If we slide u toward x, then the line connecting (x, F(x)) and (u, F(u)) becomes the tangent at (x, F(x)). This is called "the limit that u approaches x," and is written in shorthand "limit u --> x" or "lim u --> x." The slope of F at x is defined

This definition of slope is useful for deriving an approximation equation, but not for finding a function S(x) given a function F(x).

We can improve our definition of slope S(x) by defining [u - x] as the increment Dx (pronounced delta-x), so that u = x + Dx and F(u) = F(x + Dx). As we slide u toward x, the increment Dx approaches zero. Slope S(x) is defined

S(x) is called the derivative of F(x), and it is usually written dF/dx or F'(x). We can easily find the derivative if F(x) is a polynomial.

 

Example 4. Let F(x) = 5 x2 + 3 x + 1. Then F(x + Dx) = 5(x + Dx)2 + 3(x + Dx) + 1 = 5x2 + 10xDx + 5Dx2 + 3x + 3Dx + 1. The slope of the line connecting (x, F(x)) and (x + Dx, F(x + Dx)) is Slope = [F(x + Dx) - F(x)]/[(x + Dx) - x] = [5x2 + 10xDx + 5Dx2 + 3x + 3Dx + 1 - (5x2 + 3x + 1)]/Dx = [10xDx + 5Dx2 + 3Dx]/Dx = 10x + 5Dx + 3 As Dx approaches zero, the slope of the line connecting (x, F(x)) and (x + Dx, F(x + Dx)) becomes the slope of the tangent at (x, F(x)): S(x) = limit Dx --> 0 [10x + 5Dx + 3] = 10x + 3

When F(x) is not a polynomial, then finding derivative S(x) is not so easy. Fortunately, the derivatives of other functions have already been found for us. Below is a Table of Derivative with derivative-formulas for many common functions. More complicated derivatives can be found in any calculus or math-reference book.

In this table of derivatives, z is the variable, F(z) and G(z) are functions of z, and k and m are constants independent of z. The derivative of F(z) is defined as

dF(z)/dz = limit as Dz --> 0 [F(z + Dz) - F(z)]/Dz

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