How to find the tangent of an angle tangent

The geometric meaning of the first order derivative of the function F(x) is a tangent line to its graphics, which passes through a given point of the curve and matching it at this point. Moreover, the value of the derivative at a given point x0 is the slope or tangent of the angle of inclination of the tangent line k = tg a = F`(x0). The calculation of this ratio is one of the most common tasks of the theory of functions.

Instruction

1. Write the given function F(x), e.g. F(x) = (x3 + 15x +26). If the task explicitly specify the point through which the tangent is held, for example, its coordinate x0 = -2, we can dispense with plotting functions and additional straight lines on the Cartesian system OXY. Find the derivative of the first order from the given function F`(x). In this example, F`(x) = (3x2 + 15). Substitute the given value of the argument x0 in the derivative function and calculate its value: F`(-2) = (3(-2)2 + 15) = 27. So you found tg a = 27.
2. When considering tasks where it is required to determine the tangent of the angle of inclination of the tangent to the graph of the function at the point of intersection of this graph with the abscissa axis, you need first to find the numerical value of the coordinates of the point of intersection of the function with OH. For clarity, it is best to build a graph of a function on the two-dimensional plane OXY.
3. Specify the coordinate range for x, for example, from -5 to 5 with step 1. Substituting in the function values x, calculate the corresponding ordinates y and put on the coordinate plane plotting points (x, y). Connect the dots smooth line. You will see the completed chart, the intersection of function and the x-axis. The ordinate of the function at a given point is equal to zero. Find the numerical value of the corresponding argument. For this specified function, for example F(x) = (4x2 - 16), Paranaita to zero. Solve the resulting equation with one variable and calculate x: 4x2 - 16 = 0, x2 = 4, x = 2. Thus, according to the condition of the problem, the tangent of the angle of inclination of the tangent to the graph of the function is to find the point with coordinates x0 = 2.
4. Similarly to the previously described method determine the derivative function: F`(x) = 8*x. Then calculate its value at the point with x0 = 2, which corresponds to the intersection point of the original function with OH. Substitute the obtained value into the derivative function and calculate the tangent of an angle tangent: tg a = F`(2) = 16.
5. When finding the slope at the point of intersection of the function with the ordinate axis (Oh) do the same. Only the coordinate of the initial point x0 should be set to zero.