If the schedule of a derivative has obviously expressed signs, it is possible to speculate about behavior of an antiderivative. At creation of a function graph check the drawn conclusions on characteristic points.
Instruction
1. If the schedule of a derivative — direct, parallel axes OH, then its equation Y' = k, then required function Y = k*x. If the schedule of a derivative — the straight line passing under some corner to numerical axes, then a function graph — a parabola. If the schedule of a derivative is similar to a hyperbole, then prior to its research it is possible to assume that the antiderivative is function of a natural logarithm. If the schedule of a derivative — a sinusoid, then function is an argument cosine.
2. If the schedule of a derivative — a straight line, then its equation in a general view it is possible to write down Y '=k*x+b. For determination of coefficient of k at a variable x draw a straight line parallel to the set schedule through the beginning of coordinates. Remove from this auxiliary schedule of coordinate x and at any point and calculate k = y/x. Establish sign k in the direction of the schedule of a derivative — if with increase in value of an argument the schedule rises, therefore k> 0. The value of the free member of b is equal to value Y' at x =0.
3. Determine a function formula by the worked-out derivative equation: Y=k/2 * x²+bx+ with the Free member with cannot be found according to the schedule of a derivative. The provision of a function graph along axis Y is not fixed. On points construct the schedule of the received function — a parabola. Branches of a parabola are directed up at k> 0 and down at k
The schedule of a derivative exponential function coincides with the schedule of the function as at differentiation the exponential function does not change. The control point of the schedule has coordinates (0, 1) since any number is equal in zero degree to unit.
If the schedule of a derivative - a hyperbole with branches in the first and third quarters of a coordinate axis, then Y derivative equation' = 1/x. Therefore the antiderivative will be function of a natural logarithm. Control points at creation of a function graph (1.0) and (e, 1).
4. The schedule of a derivative exponential function coincides with the schedule of the function as at differentiation the exponential function does not change. The control point of the schedule has coordinates (0, 1) since any number is equal in zero degree to unit.
5. If the schedule of a derivative - a hyperbole with branches in the first and third quarters of a coordinate axis, then Y derivative equation' = 1/x. Therefore the antiderivative will be function of a natural logarithm. Control points at creation of a function graph (1.0) and (e, 1).