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Slope
In mathematics, the slope or gradient of a Line (mathematics), line is a number that describes the direction (geometry), direction of the line on a plane (geometry), plane. Often denoted by the letter ''m'', slope is calculated as the ratio of the vertical change to the horizontal change ("rise over run") between two distinct points on the line, giving the same number for any choice of points. The line may be physical – as set by a Surveying, road surveyor, pictorial as in a diagram of a road or roof, or Pure mathematics, abstract. An application of the mathematical concept is found in the grade (slope), grade or gradient in geography and civil engineering. The ''steepness'', incline, or grade of a line is the absolute value of its slope: greater absolute value indicates a steeper line. The line trend is defined as follows: *An "increasing" or "ascending" line goes from left to right and has positive slope: m>0. *A "decreasing" or "descending" line goes from left to right ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Grade (slope)
The grade (US) or gradient (UK) (also called stepth, slope, incline, mainfall, pitch or rise) of a physical feature, landform or constructed line is either the elevation angle of that surface to the horizontal plane, horizontal or its tangent. It is a special case of the slope, where zero indicates horizontal plane, horizontality. A larger number indicates higher or steeper degree of "tilt". Often slope is calculated as a ratio of "rise" to "run", or as a fraction ("rise over run") in which ''run'' is the horizontal distance (not the distance along the slope) and ''rise'' is the vertical distance. Slopes of existing physical features such as canyons and hillsides, bank (geography), stream and river banks, and stream bed, beds are often described as grades, but typically the word "grade" is used for human-made surfaces such as roads, landscape grading, roof pitches, rail tracks, railroads, aqueduct (watercourse), aqueducts, and pedestrian or bicycle routes. The grade may refer to ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Differential Calculus
In mathematics, differential calculus is a subfield of calculus that studies the rates at which quantities change. It is one of the two traditional divisions of calculus, the other being integral calculus—the study of the area beneath a curve. The primary objects of study in differential calculus are the derivative of a Function (mathematics), function, related notions such as the Differential of a function, differential, and their applications. The derivative of a function at a chosen input value describes the Rate (mathematics)#Of_change, rate of change of the function near that input value. The process of finding a derivative is called differentiation. Geometrically, the derivative at a point is the slope of the tangent, tangent line to the graph of a function, graph of the function at that point, provided that the derivative exists and is defined at that point. For a real-valued function of a single real variable, the derivative of a function at a point generally determines ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Tangent
In geometry, the tangent line (or simply tangent) to a plane curve at a given point is, intuitively, the straight line that "just touches" the curve at that point. Leibniz defined it as the line through a pair of infinitely close points on the curve. More precisely, a straight line is tangent to the curve at a point if the line passes through the point on the curve and has slope , where ''f'' is the derivative of ''f''. A similar definition applies to space curves and curves in ''n''-dimensional Euclidean space. The point where the tangent line and the curve meet or intersect is called the ''point of tangency''. The tangent line is said to be "going in the same direction" as the curve, and is thus the best straight-line approximation to the curve at that point. The tangent line to a point on a differentiable curve can also be thought of as a '' tangent line approximation'', the graph of the affine function that best approximates the original function at the given point ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Gradient
In vector calculus, the gradient of a scalar-valued differentiable function f of several variables is the vector field (or vector-valued function) \nabla f whose value at a point p gives the direction and the rate of fastest increase. The gradient transforms like a vector under change of basis of the space of variables of f. If the gradient of a function is non-zero at a point p, the direction of the gradient is the direction in which the function increases most quickly from p, and the magnitude of the gradient is the rate of increase in that direction, the greatest absolute directional derivative. Further, a point where the gradient is the zero vector is known as a stationary point. The gradient thus plays a fundamental role in optimization theory, where it is used to minimize a function by gradient descent. In coordinate-free terms, the gradient of a function f(\mathbf) may be defined by: df=\nabla f \cdot d\mathbf where df is the total infinitesimal change in f for a ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Gradient Of A Line In Coordinates From -12x+2 To +12x+2
In vector calculus, the gradient of a scalar-valued differentiable function f of several variables is the vector field (or vector-valued function) \nabla f whose value at a point p gives the direction and the rate of fastest increase. The gradient transforms like a vector under change of basis of the space of variables of f. If the gradient of a function is non-zero at a point p, the direction of the gradient is the direction in which the function increases most quickly from p, and the magnitude of the gradient is the rate of increase in that direction, the greatest absolute directional derivative. Further, a point where the gradient is the zero vector is known as a stationary point. The gradient thus plays a fundamental role in optimization theory, where it is used to minimize a function by gradient descent. In coordinate-free terms, the gradient of a function f(\mathbf) may be defined by: df=\nabla f \cdot d\mathbf where df is the total infinitesimal change in f for an ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Line (mathematics)
In geometry, a straight line, usually abbreviated line, is an infinitely long object with no width, depth, or curvature, an idealization of such physical objects as a straightedge, a taut string, or a ray of light. Lines are spaces of dimension one, which may be embedded in spaces of dimension two, three, or higher. The word ''line'' may also refer, in everyday life, to a line segment, which is a part of a line delimited by two points (its ''endpoints''). Euclid's ''Elements'' defines a straight line as a "breadthless length" that "lies evenly with respect to the points on itself", and introduced several postulates as basic unprovable properties on which the rest of geometry was established. ''Euclidean line'' and ''Euclidean geometry'' are terms introduced to avoid confusion with generalizations introduced since the end of the 19th century, such as non-Euclidean, projective, and affine geometry. Properties In the Greek deductive geometry of Euclid's ''Elements'', ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Derivative
In mathematics, the derivative is a fundamental tool that quantifies the sensitivity to change of a function's output with respect to its input. The derivative of a function of a single variable at a chosen input value, when it exists, is the slope of the tangent line to the graph of the function at that point. The tangent line is the best linear approximation of the function near that input value. For this reason, the derivative is often described as the instantaneous rate of change, the ratio of the instantaneous change in the dependent variable to that of the independent variable. The process of finding a derivative is called differentiation. There are multiple different notations for differentiation. '' Leibniz notation'', named after Gottfried Wilhelm Leibniz, is represented as the ratio of two differentials, whereas ''prime notation'' is written by adding a prime mark. Higher order notations represent repeated differentiation, and they are usually denoted in Leib ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Division By Zero
In mathematics, division by zero, division (mathematics), division where the divisor (denominator) is 0, zero, is a unique and problematic special case. Using fraction notation, the general example can be written as \tfrac a0, where a is the dividend (numerator). The usual definition of the quotient in elementary arithmetic is the number which yields the dividend when multiplication, multiplied by the divisor. That is, c = \tfrac ab is equivalent to c \cdot b = a. By this definition, the quotient q = \tfrac is nonsensical, as the product q \cdot 0 is always 0 rather than some other number a. Following the ordinary rules of elementary algebra while allowing division by zero can create a mathematical fallacy, a subtle mistake leading to absurd results. To prevent this, the arithmetic of real numbers and more general numerical structures called field (mathematics), fields leaves division by zero undefined (mathematics), undefined, and situations where division by zero might occur m ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Angle
In Euclidean geometry, an angle can refer to a number of concepts relating to the intersection of two straight Line (geometry), lines at a Point (geometry), point. Formally, an angle is a figure lying in a Euclidean plane, plane formed by two Ray (geometry), rays, called the ''Side (plane geometry), sides'' of the angle, sharing a common endpoint, called the ''vertex (geometry), vertex'' of the angle. More generally angles are also formed wherever two lines, rays or line segments come together, such as at the corners of triangles and other polygons. An angle can be considered as the region of the plane bounded by the sides. Angles can also be formed by the intersection of two planes or by two intersecting curves, in which case the rays lying tangent to each curve at the point of intersection define the angle. The term ''angle'' is also used for the size, magnitude (mathematics), magnitude or Physical quantity, quantity of these types of geometric figures and in this context an a ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Surveying
Surveying or land surveying is the technique, profession, art, and science of determining the land, terrestrial Plane (mathematics), two-dimensional or Three-dimensional space#In Euclidean geometry, three-dimensional positions of Point (geometry), points and the Euclidean distance, distances and angles between them. These points are usually on the surface of the Earth, and they are often used to establish maps and boundaries for ownership, locations, such as the designated positions of structural components for construction or the surface location of subsurface features, or other purposes required by government or civil law, such as property sales. A professional in land surveying is called a land surveyor. Surveyors work with elements of geodesy, geometry, trigonometry, regression analysis, physics, engineering, metrology, programming languages, and the law. They use equipment, such as total stations, robotic total stations, theodolites, Satellite navigation, GNSS receivers, ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Secant Line
In geometry, a secant is a line (geometry), line that intersects a curve at a minimum of two distinct Point (geometry), points.. The word ''secant'' comes from the Latin word ''secare'', meaning ''to cut''. In the case of a circle, a secant intersects the circle at exactly two points. A Chord (geometry), chord is the line segment determined by the two points, that is, the interval (mathematics), interval on the secant whose ends are the two points. Circles A straight line can intersect a circle at zero, one, or two points. A line with intersections at two points is called a ''secant line'', at one point a ''tangent line'' and at no points an ''exterior line''. A ''chord'' is the line segment that joins two distinct points of a circle. A chord is therefore contained in a unique secant line and each secant line determines a unique chord. In rigorous modern treatments of plane geometry, results that seem obvious and were assumed (without statement) by Euclid in Euclid's Elements, ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
