Basic elementary functions: inverse function (y=arctanx), logarithmic function (y=lnx), power function (y=x), exponential function (), trigonometric function (y=sinx) and constant function (y=c).
Piecewise function is not an elementary function.
Infinitesimal: high order+low order = low order for example:
Two important limitations:
Empirical formula: when,
For example:
What is derivable must be continuous, and continuous is not necessarily derivable. For example: continuous but non-conductive.
Definition of derivative products:
Derivation of composite function;
For example:
Derivation of implicit function: (1) direct derivation method; (2) Derive both sides of the equation at the same time, and get dy/dx.
For example:
Derivative of a function determined by a parametric equation: If, then its second derivative:
Approximate calculation of differential: for example, calculation
Types of function discontinuities: (1) The first category: removable discontinuities and jumping discontinuities; For example: (x=0 is the broken point of the function), (x=0 is the jump discontinuity of the function) (2) The second category: oscillation discontinuous point and infinite discontinuous point; For example: (x=0 is the oscillation discontinuity point of the function), (x=0 is the infinite discontinuity point of the function)
Asymptote:
Horizontal asymptote:
Vertical asymptote:
Oblique asymptote:
For example, find the asymptote of a function.
Stationary point: let the function y=f(x), and if f'(x0)=0, x0 is called stationary point.
Extreme point: let the function y=f(x), give a small neighborhood u(x0, δ) of x0, for any x∈u(x0, δ), there is f(x)≥f(x0), x0 is called the minimum point of f(x); Otherwise, x0 is called the maximum point of f(x). The minimum point and the maximum point are collectively called extreme points.
Inflection point: the boundary point between the upper concave arc and the lower concave arc on a continuous curve arc, which is called the inflection point of the curve arc.
Judgement theorem of inflection point: Let function y=f(x), if f "(x0) = 0, and x
Necessary condition of extreme point: Let the function y=f(x) be derivable at point x0, and x0 is an extreme point, then f'(x0)=0.
A point that changes monotonicity: a non-existent and discontinuous point (in other words, the extreme point may be a stagnation point or a non-derivative point).
The point that changes the concavity and convexity: it does not exist (in other words, the inflection point may be the point where the second derivative is equal to zero or the second derivative does not exist).
The extreme point of the derivable function f(x) must be the stagnation point, but the stagnation point of the function is not necessarily the extreme point.
Mean value theorem:
(1) Rolle theorem: continuous on [a, b] and derivable on [a, b], then there is at least one point, so,
(2) Lagrange mean value theorem: If it is continuous on [a, b] and (a, b) is internally derivable, then there is at least one point, so
(3) integral mean value theorem: integrable on the interval [a, b], with at least one point, so that