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Solutions to a slope field are functions drawn as solid curves. A slope field shows the slope of a differential equation at certain vertical and horizontal intervals on the x-y plane, and can be used to determine the approximate tangent slope at a point on a curve, where the curve is some solution to the differential equation.
In science and engineering, a log–log graph or log–log plot is a two-dimensional graph of numerical data that uses logarithmic scales on both the horizontal and vertical axes. Power functions – relationships of the form y = a x k {\displaystyle y=ax^{k}} – appear as straight lines in a log–log graph, with the exponent corresponding to ...
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.
For example, a slope that has a rise of 5 feet for every 1000 feet of run would have a slope ratio of 1 in 200. (The word "in" is normally used rather than the mathematical ratio notation of "1:200".) This is generally the method used to describe railway grades in Australia and the UK.
In SI, this slope or derivative is expressed in the units of meters per second per second (/, usually termed "meters per second-squared"). Since the velocity of the object is the derivative of the position graph, the area under the line in the velocity vs. time graph is the displacement of the object. (Velocity is on the y-axis and time on the ...
The slope number of a graph of maximum degree d is clearly at least ⌈ / ⌉, because at most two of the incident edges at a degree-d vertex can share a slope. More precisely, the slope number is at least equal to the linear arboricity of the graph, since the edges of a single slope must form a linear forest, and the linear arboricity in turn is at least ⌈ / ⌉.
When plotted in the manner described above, the value of the y-intercept (at = / =) will correspond to (), and the slope of the line will be equal to /. The values of y-intercept and slope can be determined from the experimental points using simple linear regression with a spreadsheet .
As another example, to find the area of the region bounded by the graph of the function f(x) = between x = 0 and x = 1, one can divide the interval into five pieces (0, 1/5, 2/5, ..., 1), then construct rectangles using the right end height of each piece (thus √ 0, √ 1/5, √ 2/5, ..., √ 1) and sum their areas to get the approximation