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Slope


Slope

A quantity which gives the inclination of a curve or line with respect to another curve or line. For a line in the xy-plane making an angle theta with the x-axis, the slope m is a constant given by

 m=(Deltay)/(Deltax)=tantheta,
(1)

where Deltax and Deltay are changes in the two coordinates over some distance.

For a plane curve specified as y(x), the slope is

 m(x)=(dy)/(dx),
(2)

for a curve specified parametrically as (f(t),g(t)), the slope is

 m(t)=(g^'(t))/(f^'(t))
(3)

where f^'(t)=df/dt and g^'(t)=dg/dt, for a curve specified as U(x,y)=0, the slope is

 m(x,y)=-((partialU)/(partialx))/((partialU)/(partialy)),
(4)

and for a curve given in polar coordinates as r(theta), the slope is

 m(theta)=(tantheta(dr)/(dtheta)+r)/((dr)/(dtheta)-rtantheta)
(5)

(Lawrence 1972, pp. 8-9).

It is meaningless to talk about the slope of a curve in three-dimensional space unless the slope with respect to what is specified.

J. Miller has undertaken a detailed study of the origin of the symbol m to denote slope. The consensus seems to be that it is not known why the letter m was chosen. One high school algebra textbook says the reason for m is unknown, but remarks that it is interesting that the French word for "to climb" is "monter." However, there is no evidence to make any such connection. In fact, Descartes, who was French, did not use m (Miller). Eves (1972) suggests "it just happened."

The earliest known example of the symbol m appearing in print is O'Brien (1844). Salmon (1960) subsequently used the symbols commonly employed today to give the slope-intercept form of a line

 y=mx+b
(6)

in his famous treatise published in several editions beginning in 1848. Todhunter (1888) also employed the symbol m, writing the slope-intercept form

 y=mx+c.
(7)

However, Webster's New International Dictionary (1909) gives the "slope form" as

 y=sx+b.
(8)

(Miller).

In Swedish textbooks, the slope-intercept equation is usually written as

 y=kx+m,
(9)

where k may derive from "koefficient" in the Swedish word for slope, "riktningskoefficient." In the Netherlands, the equation is commonly written as one of

y=ax+b
(10)
y=px+q
(11)
y=mx+n.
(12)

In Austria, k is used for the slope, and d for the y-intercept (Miller).


See also

Gradient, Line, Slope Field, x-Intercept, y-Intercept

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References

Eves, H. W. Mathematical Circles Revisited: A Second Collection of Mathematical Stories and Anecdotes. Prindle, Weber, and Schmidt, 1972.Lawrence, J. D. A Catalog of Special Plane Curves. New York: Dover, 1972.Miller, J. "Earliest Uses of Symbols from Geometry." http://members.aol.com/jeff570/geometry.html.O'Brien, M. A Treatise on Plane Co-Ordinate Geometry, or, The Application of the Method of Co-Ordinates to the Solution of Problems in Plane Geometry. Cambridge, England: Deightons, 1844.Salmon, G. Conic Sections, 6th ed. New York: Chelsea, 1960.Todhunter, I. Treatise on Plane Co-Ordinate Geometry as Applied to the Straight Line and the Conic Sections. London: Macmillan, 1888.

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Slope

Cite this as:

Weisstein, Eric W. "Slope." From MathWorld--A Wolfram Web Resource. https://mathworld.wolfram.com/Slope.html

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