1.
Definition: a function with the shape of y = x a (a is constant), that is, a function with the base as the independent variable and the exponent as the dependent variable is called a power function.
Domain and Value Domain:
When a is a different numerical value, the different situations of the domain of the power function are as follows: if a is any real number, the domain of the function is all real numbers greater than 0; If a is a negative number, then X must not be 0, but the definition domain of the function must also be determined according to the parity of Q, that is, if Q is even at the same time, then X cannot be less than 0, then the definition domain of the function is all real numbers greater than 0; If q is an odd number at the same time, the domain of the function is all real numbers that are not equal to 0. When x is different, the range of power function is different as follows: when x is greater than 0, the range of function is always a real number greater than 0. When x is less than 0, only when q is odd and the range of the function is non-zero real number. Only when a is a positive number will 0 enter the value range of the function.
Nature:
For the value of a nonzero rational number, it is necessary to discuss their respective characteristics in several cases:
First of all, we know that if a=p/q, q and p are integers, then x (p/q) = the root of q (p power of x), if q is odd, the domain of the function is r, if q is even, the domain of the function is [0, +∞). When the exponent n is a negative integer, let a=-k, then x = 1/(x k), obviously x≠0, and the domain of the function is (-∞, 0)∩(0, +∞). So it can be seen that the limitation of X comes from two points. First, it can be used as a denominator, but it cannot be used as a denominator.
Rule out two possibilities: 0 and negative number, that is, for x>0, then A can be any real number;
Rule out the possibility of 0, that is, for X.
The possibility of being negative is ruled out, that is, for all real numbers with x greater than or equal to 0, a cannot be negative.
2. Senior three compulsory three mathematics knowledge points induction
The scope of the 1. function depends on the defined scope and the corresponding rules. No matter what method is used to find the range of a function, we should first consider defining the range. The common methods to find the range of a function are: (1) direct method, also known as observation method. For the function with simple structure, the range of the function can be directly observed by applying the properties of inequality to the analytical expression of the function.
(2) Substitution method: A given complex variable function is transformed into another simple function re-evaluation domain by algebraic or trigonometric substitution. If the resolution function contains a radical, algebraic substitution is used when the radical is linear and trigonometric substitution is used when the radical is quadratic.
(3) Inverse function method: By using the relationship between the definition domain and the value domain of the function f(x) and its inverse function f- 1(x), the value domain of the original function can be obtained by solving the definition domain of the inverse function, and the function value domain with the shape of (a≠0) can be obtained by this method.
(4) Matching method: For the range problem of quadratic function or function related to quadratic function, the matching method can be considered.
(5) Evaluation range of inequality method: Using the basic inequality a+b≥[a, b∈(0, +∞)], we can find the range of some functions, but we should pay attention to the condition of "one positive, two definite, three phases, etc." Sometimes you need skills such as Fang.
(6) Discriminant method: y=f(x) is transformed into a quadratic equation about x, and the definition domain is evaluated by "△≥0". The characteristic of the question type is that the analytical formula contains roots or fractions.
(7) Finding the domain by using the monotonicity of the function: When the monotonicity of the function on its domain (or a subset of the domain) can be determined, the range of the function can be found by using the monotonicity method.
(8) Number-shape combination method to find the range of function: using the geometric meaning expressed by the function, with the help of geometric methods or images, to find the range of function, that is, finding the range of function through the combination of numbers and shapes.
2. Find the difference and connection between the maximum value of the function and the range.
The common method of finding the maximum value of a function is basically the same as the method of finding the function value domain. In fact, if there is a minimum (maximum) number in the range of a function, this number is the minimum (maximum) value of the function. Therefore, the essence of finding the maximum value of a function is the same as that of the evaluation domain, but the angle of asking questions is different, so the way of answering questions is different.
For example, the value range of the function is (0, 16), and the value is 16, and there is no minimum value. For example, the range of the function is (-∞, -2]∩[2,+∞), but this function has no value and minimum value, only after changing the definition of the function, such as X >；; 0, and the minimum value of the function is 2. The influence of definition domain on the range or maximum value of a function can be seen.
3. The application of maximum function in practical problems.
The application of function maximum is mainly reflected in solving practical problems with function knowledge, which is often expressed in words as "the lowest project cost", "profit" or "area (volume) (minimum)" and many other practical problems. When solving, we should pay special attention to the restriction of practical significance on independent variables, so as to get the maximum value correctly.
3. Senior three compulsory three mathematics knowledge points induction
The first definition of (1) derivative lets the function y=f(x) be defined in a certain area of point x0. When the independent variable x has increment △ x at x0 (x0+△ x is also in the neighborhood), the corresponding function gets increment △ y = f (x0+△ x)-f (x0); If the ratio of △y to △x has a limit when △x→0, the function y=f(x) can be derived at point x0, and this limit value is called the derivative of function y=f(x) at point x0, which is also called f'(x0), which is the first definition of derivative.
(2) The second definition of derivative
Let the function y=f(x) be defined in a domain of point x0. When the independent variable x changes △ x at x0 (x-x0 is also in the neighborhood), the function changes △y=f(x)-f(x0) accordingly. If the ratio of △y to △x is limited when △x→0, then the function y=f(x) is derivable at point x0. This limit value is called that the derivative of function y=f(x) at point x0 is f'(x0), which is the second definition of derivative.
(3) Derivative function and derivative
If the function y=f(x) is differentiable at every point in the open interval I, it is said that the function f(x) is differentiable in the interval I. At this time, the function y=f(x) corresponds to a certain derivative of each certain value of x in the interval I, and forms a new function, which is called the derivative function of the original function y=f(x), and is denoted as y' and f'. Derivative function is called derivative for short.
Monotonicity and its application
1. General steps to study monotonicity of polynomial functions with derivatives
(1) Find f \u( x)
(2) Make sure that f￠(x) is in (a, b). Symbol (3) If F ￠ (x) >: 0 is a constant on (a, b), then f(x) is a increasing function on (a, b); If the intersection of the solution set of f￠(x)0 and the domain corresponds to an increasing interval; F￠(x)0, translate upward; b