|
In mathematics, the slope or gradient of a line is a number that describes both the ''direction'' and the ''steepness'' of the line. Slope is often denoted by the letter ''m''. * The ''direction'' of a line is either increasing, decreasing, horizontal or vertical. * *A line is increasing if it goes up from left to right. The slope is positive, i.e. . * *A line is decreasing if it goes down from left to right. The slope is negative, i.e. . * *If a line is horizontal the slope is zero. This is a constant function. * *If a line is vertical the slope is ''undefined'' (see below). * The ''steepness'', incline, or grade of a line is measured by the absolute value of the slope. A slope with a greater absolute value indicates a steeper line Slope is calculated by finding the ratio of the "vertical change" to the "horizontal change" between (any) two distinct points on a line. Sometimes the ratio is expressed as a quotient ("rise over run"), giving the same number for every two distinct points on the same line. A line that is decreasing has a negative "rise". The line may be practical - as set by a road surveyor, or in a diagram that models a road or a roof either as a description or as a plan. The rise of a road between two points is the difference between the altitude of the road at those two points, say ''y''1 and ''y''2, or in other words, the rise is (''y''2 − ''y''1) = Δ''y''. For relatively short distances - where the earth's curvature may be neglected, the run is the difference in distance from a fixed point measured along a level, horizontal line, or in other words, the run is (''x''2 − ''x''1) = Δ''x''. Here the slope of the road between the two points is simply described as the ratio of the altitude change to the horizontal distance between any two points on the line. In mathematical language, the slope ''m'' of the line is : The concept of slope applies directly to grades or gradients in geography and civil engineering. Through trigonometry, the grade ''m'' of a road is related to its angle of incline ''θ'' by the tangent function : Thus, a 45° rising line has a slope of +1 and a 45° falling line has a slope of −1. As a generalization of this practical description, the mathematics of differential calculus defines the slope of a curve at a point as the slope of the tangent line at that point. When the curve given by a series of points in a diagram or in a list of the coordinates of points, the slope may be calculated not at a point but between any two given points. When the curve is given as a continuous function, perhaps as an algebraic formula, then the differential calculus provides rules giving a formula for the slope of the curve at any point in the middle of the curve. This generalization of the concept of slope allows very complex constructions to be planned and built that go well beyond static structures that are either horizontals or verticals, but can change in time, move in curves, and change depending on the rate of change of other factors. Thereby, the simple idea of slope becomes one of the main basis of the modern world in terms of both technology and the built environment. == Definition == The slope of a line in the plane containing the ''x'' and ''y'' axes is generally represented by the letter ''m'', 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".) Given two points (''x''1,''y''1) and (''x''2,''y''2), the change in ''x'' from one to the other is (''run''), while the change in ''y'' is (''rise''). Substituting both quantities into the above equation generates the formula: : The formula fails for a vertical line, parallel to the ''y'' axis (see Division by zero), where the slope can be taken as infinite, so the slope of a vertical line is considered undefined. 抄文引用元・出典: フリー百科事典『 ウィキペディア（Wikipedia）』 ■ウィキペディアで「slope」の詳細全文を読む スポンサード リンク
|