Let the parametric equations x (t) and y (t) be the second derivatives:

The first derivative is the rate of change of independent variables, and the second derivative is the rate of change of the first derivative, that is, the rate of change of the first derivative.

The first derivative of a continuous function is the corresponding tangent slope. If the first derivative is greater than 0, it will increase; If the first reciprocal is less than 0, decrease; If the first derivative is equal to 0, it will not increase or decrease.

The second derivative can reflect the inhomogeneity of the image. When the second derivative is greater than 0, the image is concave; The second derivative is less than 0, like convex; The second derivative is equal to 0, neither concave nor convex.

The extreme value of a function can be obtained by combining the first derivative and the second derivative. When the first derivative is equal to zero and the second derivative is greater than zero, it is a minimum point; When the first derivative is equal to zero and the second derivative is less than zero, it is the maximum point; When the first derivative and the second derivative are equal to zero, it is the stagnation point.

Extended data:

If the acceleration is not constant, the expression of the acceleration at a certain point is: a = lim Δ t→ 0 Δ v/Δ t = dv/dt (that is, the first derivative of speed with respect to time).

And because v=dx/dt, there is: a=dv/dt=d? x/dt？ That is, the second derivative of element displacement with respect to time.

Applying this idea to functions is the so-called second derivative f' (x) = dy/dx (the first derivative of f (x)). f''(x)=d？ y/dx？ = d (dy/dx)/dx (the second derivative of f (x)).

If a function f(x) has f''(x) (that is, the second derivative) in an interval I >: 0, then the line segment connected by any two points on the image of f(x) in the interval I, the function image between these two points is below the line segment, and vice versa.

When describing the law of motion with parametric equations, it is often more direct and simple than using ordinary equations. It is very suitable for solving a series of problems such as maximum voyage, maximum altitude, flight time or trajectory. For some important but complex curves (such as the involute of a circle), it is difficult or even impossible to establish their ordinary equations, and the listed equations are complex and difficult to understand.

Drawing curves according to equations is very time-consuming; However, it is often easy to indirectly relate two variables X and Y by using parametric equation, and the equation is simple and clear, and drawing is not too difficult.

Baidu Encyclopedia-Secondary Derivation