The geometrical meaning of a derivative of the first order of the F (x) function represents the tangent straight line to its schedule passing through the set point of a curve and coinciding with it in this point. And the value of a derivative in this point h0 is slope or otherwise – a tangent of angle of an inclination of a tangent straight line of k = tg a = F' (h0). Calculation of this coefficient – one of the most widespread tasks of the theory of functions.
Instruction
1. Write down the set F (x) function, for example F(x) = (x³ + the 15th +26). If in a task the point through which the tangent, for example, it coordinate h0 =-2 is carried out is obviously specified, it is possible to do without creation of a function graph and additional straight lines on the Cartesian OHY system. Find a derivative of the first order from the set F'(x) function. In the reviewed example of F'(x) = (3x² + 15). Substitute a preset value of an argument h0 in a derivative of function and calculate its value: F' (-2) = (3(-2)² + 15) = 27. Thus, you found tg a = 27.
2. By consideration of a task where it is required to define a tangent inclination tangent of angle to a function graph in a point of intersection of this schedule with abscissa axis, you need to find at first numerical value of coordinates of a point of intersection of function with OH. It is the for descriptive reasons best of all to execute creation of a function graph on the two-dimensional OHY plane.
3. Set a coordinate row for abscissae, for example, from-5 to 5 with a step 1. Substituting in function of value x, calculate the ordinates corresponding to them at and postpone on the coordinate plane the received points (x, s). Connect points the smooth line. You will see on the executed schedule the place of crossing by function of abscissa axis. The function ordinate is equal in this point to zero. Find numerical value of the argument corresponding to it. For this purpose the set function, for example F(x) = (4x² - 16), equate to zero. Solve the received equation from one variable and calculate x: 4x² - 16 = 0, x² = 4, x = 2. Thus, according to a statement of the problem, the tangent inclination tangent of angle to a function graph needs to be found in a point with coordinate h0 = 2.
4. To the way which is similarly described earlier define a function derivative: F' (x) = 8*x. Then calculate its value in a point with h0 = 2 that corresponds to a point of intersection of initial function with OH. Substitute the received value in a derivative of function and calculate a tangent inclination tangent of angle: tg a = F'(2) = 16.
5. When finding slope in a function graph point of intersection with ordinate axis (OY) perform similar operations. Only the coordinate of a required point h0 should be accepted equal to zero at once.