Slope is used to describe the steepness, incline, gradient, or grade of a straight line. The grade (or gradient or pitch or slope) of any physical feature such as a Hill, Stream, Roof, railroad, or A higher slope value indicates a steeper incline. The slope is defined as the ratio of the "rise" divided by the "run" between two points on a line, or in other words, the ratio of the altitude change to the horizontal distance between any two points on the line. It is also always the same thing as how many rises in one run.

Using calculus, one can calculate the slope of the tangent to a curve at a point. Calculus ( Latin, calculus, a small stone used for counting is a branch of Mathematics that includes the study of limits, Derivatives For the tangent function see Trigonometric functions. For other uses see Tangent (disambiguation. In Mathematics, the concept of a curve tries to capture the intuitive idea of a geometrical one-dimensional and continuous object

The concept of slope, and much of this article, applies directly to grades or gradients in geography and civil engineering. The grade (or gradient or pitch or slope) of any physical feature such as a Hill, Stream, Roof, railroad, or In Vector calculus, the gradient of a Scalar field is a Vector field which points in the direction of the greatest rate of increase of the scalar Geography (from Greek γεωγραφία - geografia) is the study of the Earth and its lands features inhabitants and phenomena Civil engineering is a professional engineering discipline that deals with the design construction and maintenance of the physical and naturally built

Contents 1 Definition 2 Examples 3 Geometry 3.1 Slope of a road 4 Algebra 5 Calculus 6 See also

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 delta symbol, "Δ", is commonly used in mathematics to mean "difference" or "change". This is a listing of common symbols found within all branches of the science of Mathematics. Delta (uppercase Δ, lowercase δ; Δέλτα Thelta is the fourth letter of the Greek alphabet. )

Given two points (x₁, y₁) and (x₂, y₂), the change in x from one to the other is x₂ - x₁, while the change in y is y₂ - y₁. Substituting both quantities into the above equation obtains the following:

Scientific Definition: The rate at which an object accelerates on a distance versus time graph is shown. Calculated by Slope = Rise / Run of a graph. Since the y-axis is vertical and the x-axis is horizontal by convention, the above equation is often memorized as "rise over run", where Δy is the "rise" and Δx is the "run". Therefore, by convention, m is equal to the change in y, the vertical coordinate, divided by the change in x, the horizontal coordinate; that is, m is the ratio of the changes. This concept is fundamental to algebra, analytic geometry, trigonometry, and calculus. Algebra is a branch of Mathematics concerning the study of structure, relation, and Quantity. Analytic geometry, also called coordinate geometry and earlier referred to as Cartesian geometry or analytical geometry, is the study of Geometry Circle-trig6svg|300px|thumb|right|All of the Trigonometric functions of an angle θ can be constructed geometrically in terms of a unit circle centered at O. Calculus ( Latin, calculus, a small stone used for counting is a branch of Mathematics that includes the study of limits, Derivatives

Note that the way the points are chosen on the line and their order does not matter; the slope will be the same in each case. Other curves have "accelerating" slopes and one can use calculus to determine such slopes. In Mathematics, the concept of a curve tries to capture the intuitive idea of a geometrical one-dimensional and continuous object Calculus ( Latin, calculus, a small stone used for counting is a branch of Mathematics that includes the study of limits, Derivatives

Examples

Suppose a line runs through two points: P(1, 2) and Q(13, 8). By dividing the difference in y-coordinates by the difference in x-coordinates, one can obtain the slope of the line:

The slope is 1/2 = 0. 5.

As another example, consider a line which runs through the points (4, 15) and (3, 21). Then, the slope of the line is

Geometry

The larger the absolute value of a slope, the steeper the line. A horizontal line has slope 0, a 45° rising line has a slope of +1, and a 45° falling line has a slope of -1. A vertical line's slope is undefined meaning it has "no slope. "

The angle θ a line makes with the positive x axis is closely related to the slope m via the tangent function:

and

(see trigonometry). The Mathematical concept of a function expresses dependence between two quantities one of which is given (the independent variable, argument of the function Circle-trig6svg|300px|thumb|right|All of the Trigonometric functions of an angle θ can be constructed geometrically in terms of a unit circle centered at O.

Two lines are parallel if and only if their slopes are equal and they are not coincident or if they both are vertical and therefore have undefined slopes. Two lines are perpendicular if and only if the product of their slopes is -1 or one has a slope of 0 (a horizontal line) and the other has an undefined slope (a vertical line). In Geometry, two lines or planes (or a line and a plane are considered perpendicular (or orthogonal) to each other if they form congruent

Slope of a road

Main articles: Grade (slope), Grade separation

There are two common ways to describe how steep a road or railroad is. The grade (or gradient or pitch or slope) of any physical feature such as a Hill, Stream, Roof, railroad, or Grade separation is the process of aligning a junction of two or more transport axes at different heights ( A road is an identifiable route, way or path between two or more places. One is by the angle in degrees, and the other is by the slope in a percentage. See also mountain railway. A mountain railway is a Railway that ascends and descends a Mountain slope that has a steep grade. The formulae for converting a slope as a percentage into an angle in degrees and vice versa are:

and

where angle is in degrees and the trigonometry functions operate in degrees. For example, a 100% slope is 45°.

A third way is to give one unit of rise in say 10, 20, 50 or 100 horizontal units, e. g. 1:10. 1:20, 1:50 or 1:100 (etc. ).

Slope warning sign in the Netherlands

Slope warning sign in Poland

A 1371-meter distance of a railroad with a 20‰ slope. The Netherlands ( Dutch:, ˈnedərlɑnt is the European part of the Kingdom of the Netherlands, which consists of the Netherlands the Netherlands Poland (Polska officially the Republic of Poland A per mil or per mille (also spelled permil, per mill or promille) ( Latin, literally meaning 'for (every thousand' is a tenth Czech Republic

Steam-age railway gradient post indicating a slope in both directions at Meols railway station, United Kingdom

Algebra

If y is a linear function of x, then the coefficient of x is the slope of the line created by plotting the function. The Czech Republic ( ˈt͡ʃɛskaː ˈrɛpuˌblɪka short form in Česko ˈt͡ʃɛskɔ also called Czechia, Meols railway station is situated in Meols, Wirral, Merseyside, England. The United Kingdom of Great Britain and Northern Ireland, commonly known as the United Kingdom, the UK or Britain,is a Sovereign state located In Mathematics, the term linear function can refer to either of two different but related concepts Therefore, if the equation of the line is given in the form

then m is the slope. This form of a line's equation is called the slope-intercept form, because b can be interpreted as the y-intercept of the line, the y-coordinate where the line intersects the y-axis. In Coordinate geometry, the y -intercept is the y-value of the point where the Graph of a function or relation intercepts the y -axis

If the slope m of a line and a point (x₀, y₀) on the line are both known, then the equation of the line can be found using the point-slope formula:

For example, consider a line running through the points (2, 8) and (3, 20). A linear equation is an Algebraic equation in which each term is either a Constant or the product of a constant and (the first power of a single Variable This line has a slope, m, of

.

One can then write the line's equation, in point-slope form:

or:

.

The slope of a linear equation in the general form:

is given by the formula:

. A linear equation is an Algebraic equation in which each term is either a Constant or the product of a constant and (the first power of a single Variable

Calculus

The concept of a slope is central to differential calculus. Differential Calculus, a field in Mathematics, is the study of how functions change when their inputs change For non-linear functions, the rate of change varies along the curve. The derivative of the function at a point is the slope of the line tangent to the curve at the point, and is thus equal to the rate of change of the function at that point. In Calculus, a branch of mathematics the derivative is a measurement of how a function changes when the values of its inputs change For the tangent function see Trigonometric functions. For other uses see Tangent (disambiguation.

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. A secant line of a Curve is a line that (locally intersects two points on the curve For a line, the secant between any two points is the line itself, but this is not the case for any other type of curve.

For example, the slope of the secant intersecting y = x² at (0,0) and (3,9) is m = (9 - 0) / (3 - 0) = 3 (which happens to be the slope of the tangent at, and only at, x = 1. 5, a consequence of the mean value theorem). In Calculus, the mean value theorem states roughly that given a section of a smooth curve there is at least one point on that section at which the Derivative

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. Differential Calculus, a field in Mathematics, is the study of how functions change when their inputs change In Mathematics, the limit of a function is a fundamental concept in Calculus and analysis concerning the behavior of that function near a particular 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. We call this limit the derivative. In Calculus, a branch of mathematics the derivative is a measurement of how a function changes when the values of its inputs change

See also

• The gradient is a generalization of the concept of slope for functions of more than one variable. In Vector calculus, the gradient of a Scalar field is a Vector field which points in the direction of the greatest rate of increase of the scalar
• Slope definitions

© 2009 citizendia.org; parts available under the terms of GNU Free Documentation License, from http://en.wikipedia.org
Dapyx Software network: MP3 Explorer | Ebook Manager | Zenithic