Direct y = f(x) will be a tangent to the schedule represented in the drawing in a point h0 under that condition if it passes through this point with coordinates (h0; f(x0)) also has slope of f' (x0). To find this coefficient, considering features of a tangent, simply.
It is required to you
- - mathematical reference book;
- - notebook;
- - simple pencil;
- - handle;
- - protractor;
- - compasses.
Instruction
1. Take into consideration that the schedule of the differentiable f (x) function in a point h0 has no distinctions with a tangent piece. Therefore it is rather close to l piece, to passing through points (h0; f(х0)) and (х0+Δx; f (x0 + Δx)). To set the straight line passing through a point And with coefficients (h0; f(х0)), specify its slope. At the same time it is equal Δy/Δx to a secant of a tangent (Δх→0) and also aspires to number f‘ (x0).
2. If values f‘ (x0) do not exist, then, perhaps, there is no tangent, or it passes vertically. Proceeding from it, presence of derivative function at a point h0 is explained by existence of not vertical tangent which adjoins to a function graph in a point (h0, f(х0)). In this case the slope of a tangent equals f' (h0). The geometrical meaning of a derivative, that is calculation of slope of a tangent becomes clear.
3. That is to find slope of a tangent, it is necessary to find value of derivative function in a contact point. Example: to find slope of a tangent to a function graph at = x³ in a point with X0 abscissa = 1. Decision: Find a derivative of this function у΄ (x) = the 3rd²; find value of a derivative in X0 point = 1. у΄ (1) = 3 × 1² = 3. The slope of a tangent = 1 is equal in X0 point to 3.
4. Draw in the drawing additional tangents so that they adjoined to a function graph in the following points: x1, h2 and h3. Note corners which are formed by these tangents with abscissa axis (the corner is counted in the positive direction - from an axis to a tangent straight line). For example, the first corner α1 will be sharp, the second (α2) – stupid, and the third (α3) will equal to zero as the drawn tangent straight line is a parallel axis OH. In this case the tangent of an obtuse angle is negative value, and a tangent of an acute angle – positive, at tg0 and the result is equal to zero.