Slope

In mathematics, particularly in calculus, the slope is ( also called slope ) a measure of the slope of a straight line or a curve.

The problem of determining the slope, this arises not only in geometrical problems, but also for example in physics or in economics. So is approximately the slope in a time -distance diagram of the speed or incline in a time -charge diagram of the current.

Slope of a line

Definition and calculation

The slope of a line is often referred to by the letter. If one uses Cartesian coordinates, so has the straight line defined by two points, the slope

(read: delta x) in this context means the difference of the x - values ​​corresponding to the difference of the associated y-values.

For the pictured line through the points and, for example, gives the slope:

It does not matter which of points of the straight line is used, the coordinates in the formula. Taking for example, and so we get:

Rises (as seen in the positive x - direction, ie from left to right) Right, so its slope is positive. For a descending straight line, the slope is negative. Gradient of 0 means that the straight line extends horizontally, ie parallel to the x- axis.

Has the straight line, the slope and the y- axis intersects it at the point, so it has the equation

Note: The parallel to the y- axis lines are not graphs of functions and therefore also have no slope value. You can give them the pitch "infinity" (∞ ) award.

Road

The slope of a line also plays a role in traffic. The traffic sign for the road gradient of a road based on the same slope term, however, it is usually expressed in percent. The inclusion of a 12% gradient for example, means that for every 100 m, the height increases by 12 m in the horizontal direction. According to the definition given above, one has to divide 12 m by 100 m, which leads to the result 0.12 (in percent notation 12%).

The steepest street in the world is Baldwin Street in New Zealand. The maximum slope of the 200 meters long street is 1:2,86 ( 19.3 ° or about 35%).

A ratio indicating how 1:2,86 is another way, inclinations or slopes to define. They are, as a percentage figure, the height difference per horizontal distance of 1 m to 2.86 m distance = 1/2, 86 = 0.34965 = 35 % ( = 35 m to 100 m distance). Even banks are given. The ratio often used artificial slopes of 1:1.5 (depending on the material, etc. ) returns 1 meter difference in height to 1.5 meters horizontal distance. This represents a slope of 66.7 %, and a pitch angle of arctan ( 1/1, 5) = 33.7 °.

Slope or angle of inclination

From the slope of a line can be related to the corresponding pitch or angle of inclination of the line with the help of tangent and arctangent function calculated on the positive axis:

A correlation from the Trigonometry states that in a right triangle, the tangent of either acute angle is equal to the quotient of the respective counter and the adjacent side, so it is clear that the slope of both the tangent of the slope angle ( in degrees) compared with the positive - axis is:

When specified as a percentage (%) is to be noted that pitch and pitch angles are not proportional to each other, so it is not possible to convert gradients and slope angle using a simple three-set each other. Corresponds for example a slope of 1 ( = 100%), a pitch angle of 45 °, a pitch of 2 (= 200% ), however, only one angle of approximately 63.4 °, and finally, for a helix angle of 90 ° would have the slope to infinity grow.

Approximate proportionality of pitch and pitch angle, however, is given only for small slope angles up to about 5 ° - thus corresponds to a slope of ± 0.01 and ± 1%, a pitch angle of approximately ± 0.57 °, and conversely, a pitch angle of ± 1 ° a slope of approximately ± 0.0175 and 1.75% respectively.

For larger pitch angle, however, or if their size should be precisely determined, one needs the inverse of the tangent, that is, the arc tangent function:

In the above example we calculated:

With negative slopes to note here is that - due to the point of symmetry of the arc tangent function, - then the angle of inclination can be negative.

Cutting angle

The gradient term also provides a convenient method to determine the angle of intersection of two straight lines with given slopes and:

Two lines are parallel if and only (= 0 ° ) if their slopes match. They are then exactly perpendicular to each other ( = 90 °) when their gradients satisfy the orthogonality condition · = -1.

Slope of threads

For metric threads, the slope indicates the pitch, ie the distance between two threaded stages along the thread axis, in other words the axial path, which is completed by one turn of the thread.

For imperial threads, however, the number of threads is specified on the route of an inch as the value.

Generalization: the slope of a curve

One of the fundamental problems of analysis is to find the slope of a curve at a given point on the curve. The discussed above formula is no longer usable, since only one point is given. If you select the second point arbitrarily, we get no clear result or, if both points are chosen to be identical, the result is undefined because division by 0.

One defines the slope of the graph of a function at a point of the graph, therefore the slope of the curve tangent at this point. The calculus provides the concept of derivative as a tool to calculate such slope values.

For example, the slope of the curve point and the corresponding inclination angle should be calculated for the graph of the function.

First, we determine the equation of the derivative function:

Now, the x-coordinate of the given point is used:

From the value of the slope gives the angle of inclination:

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