How to find the slope of the tangent to the graph of the function

Y = f(x) will be tangent to the graph shown in the figure at the point x0 under the condition, if it passes through the point with coordinates (x0; f(x0)) and has slope f'(x0). To find this ratio, given the characteristics of a tangent, it's easy.

You will need

• mathematical Handbook;
• - notebook;
• - pencil;
• - handle;
• - protractor;
• a pair of compasses.

Instruction

1. Please note that the graph of a differentiable function f(x) at point x0 has no differences to the segment of the tangent. So it is rather close to the segment l passing through the points (x0, f(x0)) and (x0+Δx f(x0 + Δx)). To specify the straight line passing through the point A with the coefficients (x0, f(x0)), specify its angular coefficient. In this case it is equal to Δy/Δx secant tangent (DF→0) and tends to f‘(x0).
2. If the values of f‘(x0) does not exist, it is possible that a tangent there, or it is held vertically. On this basis, the presence of the derivative of the function at the point x0 due to the existence applied on other than vertical tangent, which touches the graph of the function at the point (x0, f(x0)). In this case, the slope of the tangent is f'(x0). Becomes clear the geometric meaning of derivative, that is, the calculation of the angular coefficient of the tangent.
3. That is, in order to find the slope of the tangent, you need to find the value of the derivative of the function at the point of tangency. Example: to find the slope of the tangent to the graph of the function y = X3 at the point with abscissa X0 = 1. Solution: Find the derivative of this function y(x) = 3x2; find the value of the derivative at the point X0 = 1. have(1) = 3 × 12 = 3. The slope of the tangent at the point X0 = 1 is equal to 3.
4. Draw the picture for more tangents so that they adjoined to the graph of the function at the following points: x1, x2 and X3. Mark the angles that are formed by tangent data with the abscissa axis (angle measured in the positive direction from the axis to the tangent line). For example, the first angle α1 is an acute, the second (α2) is stupid, but the third (α3) will be zero, as performed the straight line is parallel to the OX axis. In this case, the tangent of the obtuse angle has a negative value, and tangent of an acute angle is positive, when tg0 and the result is zero.