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24-728: In winter 1954, the first slope , now called "Eturinne" ("Front Slope"), was opened. The current number of slopes is 41, operated with 21 ski lifts (three high-speed detachable chair with bubble and one gondola). There's effective snowmaking on all runs by almost 90 snowguns and 5 groomers. Ruka has turned into a diverse tourist centre whose year-round usage is growing. In winter there are for example Nordic skiing World Cup competitions. FIS Nordic Combined season starts yearly in Ruka (Nordic Opening). There are also annual national competitions in alpine and freestyle skiing, and there have been international competitions in these sports, too. Ruka has one of

48-430: A diagram of a road or roof, or abstract . An application of the mathematical concept is found in the 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: Special directions are: If two points of a road have altitudes y 1 and y 2 ,

72-408: A 45° rising line has slope m = +1, and a 45° falling line has slope m = −1. Generalizing this, differential calculus defines the slope of a plane curve at a point as the slope of its tangent line at that point. When the curve is approximated by a series of points, the slope of the curve may be approximated by the slope of the secant line between two nearby points. When the curve is given as

96-724: A border cross park and street as well as other activities such as snowmobiling, snowshoeing, reindeer sleigh rides, husky sled rides, visits to Santa's home, ice carting, ice fishing, ice climbing and Finnish sauna . The winter season usually runs from the beginning of December until May. In the summer the snow melts and Ruka becomes a hiking and mountain biking centre with the Oulanka National Park and world-famous Karhunkierros, 80 km hiking trail, on its doorstep. Other activities on offer are canoeing, white water rafting, fishing, birdwatching, hiking with huskies, lake cruises, bear watching, water skiing and ATV safaris. In

120-404: A curve, then the slope given by the above definition, is the slope of a secant line to the curve. For a line, the secant between any two points is the line itself, but this is not the case for any other type of curve. For example, the slope of the secant intersecting y = x at (0,0) and (3,9) is 3. (The slope of the tangent at x = 3 ⁄ 2 is also 3 −  a consequence of

144-625: A line in the plane containing the x and y axes is generally represented by the letter m , 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. This is described by the following equation: (The Greek letter delta , Δ, is commonly used in mathematics to mean "difference" or "change".) Given two points ( x 1 , y 1 ) {\displaystyle (x_{1},y_{1})} and ( x 2 , y 2 ) {\displaystyle (x_{2},y_{2})} ,

168-562: A slope given as a percentage into an angle in degrees and vice versa are: and where angle is in degrees and the trigonometric functions operate in degrees. For example, a slope of 100 % or 1000 ‰ is an angle of 45°. A third way is to give one unit of rise in say 10, 20, 50 or 100 horizontal units, e.g. 1:10. 1:20, 1:50 or 1:100 (or "1 in 10", "1 in 20", etc.) 1:10 is steeper than 1:20. For example, steepness of 20% means 1:5 or an incline with angle 11.3°. Roads and railways have both longitudinal slopes and cross slopes. The concept of

192-399: A slope is central to differential calculus . For non-linear functions, the rate of change varies along the curve. The derivative of the function at a point is the slope of the line tangent to the curve at the point and is thus equal to the rate of change of the function at that point. If we let Δ x and Δ y be the distances (along the x and y axes, respectively) between two points on

216-410: Is (−2,4). The derivative of this function is d y ⁄ d x = 2 x . So the slope of the line tangent to y at (−2,4) is 2 ⋅ (−2) = −4 . The equation of this tangent line is: y − 4 = (−4)( x − (−2)) or y = −4 x − 4 . An extension of the idea of angle follows from the difference of slopes. Consider the shear mapping Then ( 1 , 0 ) {\displaystyle (1,0)}

240-654: Is mapped to ( 1 , v ) {\displaystyle (1,v)} . The slope of ( 1 , 0 ) {\displaystyle (1,0)} is zero and the slope of ( 1 , v ) {\displaystyle (1,v)} is v {\displaystyle v} . The shear mapping added a slope of v {\displaystyle v} . For two points on { ( 1 , y ) : y ∈ R } {\displaystyle \{(1,y):y\in \mathbb {R} \}} with slopes m {\displaystyle m} and n {\displaystyle n} ,

264-472: Is the standard deviation of the y-values and s x {\displaystyle s_{x}} is the standard deviation of the x-values. This may also be written as a ratio of covariances : There are two common ways to describe the steepness of a road or railroad . One is by the angle between 0° and 90° (in degrees), and the other is by the slope in a percentage. See also steep grade railway and rack railway . The formulae for converting

SECTION 10

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288-461: Is the exact slope of the tangent. If y is dependent on x , then it is sufficient to take the limit where only Δ x approaches zero. Therefore, the slope of the tangent is the limit of Δ y /Δ x as Δ x approaches zero, or d y /d x . We call this limit the derivative . The value of the derivative at a specific point on the function provides us with the slope of the tangent at that precise location. For example, let y = x . A point on this function

312-480: Is −1. So these two lines are perpendicular. In statistics , the gradient of the least-squares regression best-fitting line for a given sample of data may be written as: This quantity m is called as the regression slope for the line y = m x + c {\displaystyle y=mx+c} . The quantity r {\displaystyle r} is Pearson's correlation coefficient , s y {\displaystyle s_{y}}

336-421: The x -axis is Consider the two lines: y = −3 x + 1 and y = −3 x − 2 . Both lines have slope m = −3 . They are not the same line. So they are parallel lines. Consider the two lines y = −3 x + 1 and y = ⁠ x / 3 ⁠ − 2 . The slope of the first line is m 1 = −3 . The slope of the second line is m 2 = ⁠ 1 / 3 ⁠ . The product of these two slopes

360-406: The mean value theorem .) By moving the two points closer together so that Δ y and Δ x decrease, the secant line more closely approximates a tangent line to the curve, and as such the slope of the secant approaches that of the tangent. Using differential calculus , we can determine the limit , or the value that Δ y /Δ x approaches as Δ y and Δ x get closer to zero ; it follows that this limit

384-406: The slope or gradient of a line is a number that describes the direction of the line on a 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 road surveyor , pictorial as in

408-471: The above equation generates the formula: The formula fails for a vertical line, parallel to the y {\displaystyle y} axis (see Division by zero ), where the slope can be taken as infinite , so the slope of a vertical line is considered undefined. Suppose a line runs through two points: P  = (1, 2) and Q  = (13, 8). By dividing the difference in y {\displaystyle y} -coordinates by

432-631: The autumn the forests of Ruka and Kuusamo change colours, with not just the trees but the forest floor, which is carpeted in plants, turning reds, oranges and yellows. In November 2010 the Ruka Pedestrian Village became ready. Ruka now has an underground car park for 320 vehicles. The car park is heated in winters. Now it is possible to walk from the hotels to the slopes without crossing any roads. 66°11′46″N 029°06′45″E  /  66.19611°N 29.11250°E  / 66.19611; 29.11250 Slope In mathematics ,

456-406: The change in x {\displaystyle x} from one to the other is x 2 − x 1 {\displaystyle x_{2}-x_{1}} ( run ), while the change in y {\displaystyle y} is y 2 − y 1 {\displaystyle y_{2}-y_{1}} ( rise ). Substituting both quantities into

480-482: The difference in x {\displaystyle x} -coordinates, one can obtain the slope of the line: As another example, consider a line which runs through the points (4, 15) and (3, 21). Then, the slope of the line is For example, consider a line running through points (2,8) and (3,20). This line has a slope, m , of One can then write the line's equation, in point-slope form: or: The angle θ between −90° and 90° that this line makes with

504-578: The graph of an algebraic expression , calculus gives formulas for the slope at each point. Slope is thus one of the central ideas of calculus and its applications to design. There seems to be no clear answer as to why the letter m is used for slope, but it first appears in English in O'Brien (1844) who introduced the equation of a line as " y = mx + b " , and it can also be found in Todhunter (1888) who wrote " y = mx + c ". The slope of

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528-439: The image has slope increased by v {\displaystyle v} , but the difference n − m {\displaystyle n-m} of slopes is the same before and after the shear. This invariance of slope differences makes slope an angular invariant measure , on a par with circular angle (invariant under rotation) and hyperbolic angle, with invariance group of squeeze mappings . The concept of

552-568: The longest ski seasons in Europe, starting usually in October and ending in June. During the main season ski lifts are open daily from 9.30 am until 7 pm. There's also night skiing on Fridays 7–11 pm (usually from December until mid-April). In the winter months Ruka is a ski resort and activity centre with 29 ski slopes, 23 flood lit slopes for night skiing, over 500 km of cross country ski tracks,

576-458: The rise is the difference ( y 2 − y 1 ) = Δ y . Neglecting the Earth's curvature , if the two points have horizontal distance x 1 and x 2 from a fixed point, the run is ( x 2 − x 1 ) = Δ x . The slope between the two points is the difference ratio : Through trigonometry , the slope m of a line is related to its angle of inclination θ by the tangent function Thus,

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