Slope

[1] Often denoted by the letter m, slope is calculated as the ratio of the vertical change to the horizontal change ("rise over run") between two distinct points on the line, giving the same number for any choice of points.

The line may be physical – as set by a road surveyor, pictorial as in a diagram of a road or roof, or abstract.

An application of the mathematical concept is found in the grade or gradient in geography and civil engineering.

The steepness, incline, or grade of a line is the absolute value of its slope: greater absolute value indicates a steeper line.

The line trend is defined as follows: Special directions are: If two points of a road have altitudes y1 and y2, the rise is the difference (y2 − y1) = Δy.

Neglecting the Earth's curvature, if the two points have horizontal distance x1 and x2 from a fixed point, the run is (x2 − x1) = Δx.

Generalizing this, differential calculus defines the slope of a plane curve at a point as the slope of its tangent line at that point.

When the curve is approximated by a series of points, the slope of the curve may be approximated by the slope of the secant line between two nearby points.

When the curve is given as the graph of an algebraic expression, calculus gives formulas for the slope at each point.

Slope is thus one of the central ideas of calculus and its applications to design.

There seems to be no clear answer as to why the letter m is used for slope, but it first appears in English in O'Brien (1844)[2] who introduced the equation of a line as "y = mx + b", and it can also be found in Todhunter (1888)[3] who wrote "y = mx + c".

[4] The slope of a line in the plane containing the x and y axes is generally represented by the letter m,[5] and is defined as the change in the y coordinate divided by the corresponding change in the x coordinate, between two distinct points on the line.

This is described by the following equation: (The Greek letter delta, Δ, is commonly used in mathematics to mean "difference" or "change".)

Substituting both quantities into the above equation generates the formula: The formula fails for a vertical line, parallel to the

Suppose a line runs through two points: P = (1, 2) and Q = (13, 8).

In statistics, the gradient of the least-squares regression best-fitting line for a given sample of data may be written as: This quantity m is called as the regression slope for the line

This may also be written as a ratio of covariances:[6] The concept of a slope is central to differential calculus.

For non-linear functions, the rate of change varies along the curve.

If we let Δx and Δy be the distances (along the x and y axes, respectively) between two points on a curve, then the slope given by the above definition, is the slope of a secant line to the curve.

(The slope of the tangent at x = 3⁄2 is also 3 − a consequence of the mean value theorem.)

By moving the two points closer together so that Δy and Δx decrease, the secant line more closely approximates a tangent line to the curve, and as such the slope of the secant approaches that of the tangent.

Using differential calculus, we can determine the limit, or the value that Δy/Δx approaches as Δy and Δx get closer to zero; it follows that this limit is the exact slope of the tangent.

If y is dependent on x, then it is sufficient to take the limit where only Δx approaches zero.

Therefore, the slope of the tangent is the limit of Δy/Δx as Δx approaches zero, or dy/dx.

The value of the derivative at a specific point on the function provides us with the slope of the tangent at that precise location.

An extension of the idea of angle follows from the difference of slopes.

[7][8] The slope of a roof, traditionally and commonly called the roof pitch, in carpentry and architecture in the US is commonly described in terms of integer fractions of one foot (geometric tangent, rise over run), a legacy of British imperial measure.

Other units are in use in other locales, with similar conventions.

There are two common ways to describe the steepness of a road or railroad.

The concept of a slope or gradient is also used as a basis for developing other applications in mathematics: