Slope Of The Tangent Line Slope Of The Tangent Line In today session we discuss the tangent line approximation. Before going on the main topic we discuss about slope of line or tangent slope. The trigonometrically tangent of the angle that a line makes with the positive direction of x axis in anticlockwise sense is called the slope or gradient of the time. The slope of a line is generally denoted by m. Thus, m = tan ?, where ? is the angle which a line makes with the positive direction of x axis in anticlockwise sense. Since, a line parallel to x axis makes an angle of 00 with x axis. Therefore, its slope is tan 00 = 0. A line perpendicular to x axis so its slope is tan π/2 = ∞. Also the slope of a line equally inclined with x axis is +1 or -1 as it makes 450 or 1350 angle with x axis. Slope of a line in terms of coordinates of any two points on it that is, let p(x1 , y1) and q(x2, y2) be two points on a line. Then its slope m is given by m = (y2 – y1)/ (x2 – x1), m = difference of ordinates / difference of abscissa,
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The slope of a line when its equation is given, The slope of a line whose equation is ax + by + c = 0 is given by, m = - a/b = - Coeff. of x / Coeff. of y. The angle ? between two lines having slope m1 and m2 is given by, tan ? = +-(m1 - m2)/(1 + m1m2), If two lines of slopes m1 and m2 are perpendicular, then m1m2 = -1. The functions which are used in tangent line approximation are related to the above functions. The tangent line approximation is the a approximation values of a function which is not easy to find. We find the value of a function by using the closest value of another function. For understand this concept let’s take an example; if we have a function g(x), then square root function is g(a) = √a. If the value of a is 16 then the square root of 16 is 4 and this can be easily found but if the value is 15.9 then it is difficult to found. We can find this value as g(15.9) = √ 16– 0.1, g (16) = √16 = 4 (as 0.1 is small) . Mathematically the tangent line approximation is expressed as, if we have function f(x + z) then it is difficult to find on other hand f(x) is easily to determined and |z| is small as compare to x. This may give the approximate of f(x + z) by using function f(x).We can use the approximate value of the function f at a point x + z. This can be done by using the tangent line graph at point x if |z| is small or x + z is close to x. The mathematical expression become,
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f( x + z ) = f(x) + z f’(x) , And the other expression is, C = f(x) + f’(x) (h - x), Where (h – x ) is any interval. Now suppose f(x) = |x|. The slope at origin gives two slopes at a particular point. Moving from right of the origin we find the slope is 1 while moving from left of the origin gives us -1 as slope. Hence there is no unique tangent at this point and slope is not defined. Differentiability suggests continuity, similarly discontinuity suggests non-differentiability. Hence we can derive that the slope of the tangent at a point, where the function is discontinuous can’t exist.
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