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Slope illustrated for y = (3/2)x − 1.Click on to enlarge Slope of a line in coordinates system, from f(x) = −12x + 2 to f(x) = 12x + 2. 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.
[a] The slope of a curve at a particular point is equal to the slope of the tangent to that point. For example, y = x 2 {\displaystyle y=x^{2}} has a slope of 4 {\displaystyle 4} at x = 2 {\displaystyle x=2} because the slope of the tangent line to that point is equal to 4 {\displaystyle 4} :
The slope field can be defined for the following type of differential equations ′ = (,), which can be interpreted geometrically as giving the slope of the tangent to the graph of the differential equation's solution (integral curve) at each point (x, y) as a function of the point coordinates. [3]
The derivative f′(x) of a curve at a point is the slope of the line tangent to that curve at that point. This slope is determined by considering the limiting value of the slopes of the second lines. Here the function involved (drawn in red) is f(x) = x 3 − x. The tangent line (in green) which passes through the point (−3/2, −15/8) has a ...
In projective geometry and related contexts, an asymptote of a curve is a line which is tangent to the curve at a point at infinity. [ 1 ] [ 2 ] The word asymptote is derived from the Greek ἀσύμπτωτος ( asumptōtos ) which means "not falling together", from ἀ priv. + σύν "together" + πτωτ-ός "fallen". [ 3 ]
Here, a differential equation can be thought of as a formula by which the slope of the tangent line to the curve can be computed at any point on the curve, once the position of that point has been calculated. The idea is that while the curve is initially unknown, its starting point, which we denote by , is known (see Figure 1).
The line with equation ax + by + c = 0 has slope -a/b, so any line perpendicular to it will have slope b/a (the negative reciprocal). Let (m, n) be the point of intersection of the line ax + by + c = 0 and the line perpendicular to it which passes through the point (x 0, y 0). The line through these two points is perpendicular to the original ...
Then the slope of a line tangent to the curve at that point is the value of the marginal function at that point. The slope of a ray drawn from the origin through the point is the value of the average function.