① Empty set: Because an empty set is a subset of any set and an empty set is a proper subset of any non-empty set, we must consider the case that B is an empty set when we encounter or! This situation often occurs when solving parameter set problems.

② Three characteristics of a set: the elements in the set are deterministic, disordered and different from each other. Among the three properties, the main point to be investigated is the mutual dissimilarity, especially in the set containing letters, which will automatically hide some requirements for letters-no element in the set is equal to each other.

Second, simple logic.

(1) Whether a proposition is negative or not: they are two different concepts, and the negation of a proposition usually has multiple description forms (negative forms of "yes" and "no"). Negative proposition is one of the four major propositions, and we need both negative conditions and negative conclusions when dealing with it, which is very common in compound propositions.

② Necessary and sufficient conditions: for two conditions, if, it is a sufficient condition and a necessary condition; If yes, it is a sufficient condition and a necessary condition. When solving the problem, we must be clear about who pushed who out.

③ NOR: To solve this kind of problem with parameter range, we can combine NOR with the intersection and complement of sets and solve it through the operation of sets.

Third, function

① Monotonous interval: When studying the function problem, we should think of the image of the function, learn to use the idea of combining numbers and shapes, analyze the problem from the image of the function, and find a solution. For several different monotonous intervals of the function, remember not to use union, but only to explain the monotonicity of the function in these intervals.

② Parity: To judge the parity of a function, we should first consider the definition domain of the function. The necessary condition for a function to have parity is that the domain of the function is symmetric about the origin. If not, it must be odd and even.

③ Zero Theorem: A function has signed zero and signed zero. There is no way to solve the zero theorem of sign-changing zero function. We must pay attention to solving this problem, but many times we can convert the sign-changing zero into the sign-changing zero.

Fourth, derivative products

Geometric meaning: the derivative value of a function at a certain point is the tangent slope of the function image at the change point, but in many problems, it is often necessary to solve the problem that a point outside the function image leads to the tangent of the function image. The basic idea to solve this kind of problem is to set the tangent coordinates, write the tangent equation according to the geometric meaning of the derivative, and then solve it according to other conditional equations or equations given in the topic, so it is necessary to make clear whether you are doing "tangency at a certain point" or "tangency at a certain point"

② Derivative and extreme value:' () = 0 is only a necessary condition for the derivative function to get the extreme value at, that is, this condition is necessary, but only this condition is not enough. It is also necessary to consider whether the derivative values on the left and right sides are different in sign. In addition, the extreme value of a function at a certain point is not necessarily derivable at that point.

Verb (abbreviation of verb) trigonometric function

① Monotonicity of trigonometric functions: For monotonicity of functions, both positive and negative factors should be considered when judging. Combined with monotonicity, the monotonicity judgment principle of compound functions-the same increase but different decrease (odd-numbered layers decrease, even-numbered layers decrease) can be used, and the image of trigonometric functions can be processed when judging, and the idea of combining numbers and shapes can be applied.

② Image transformation: There are many different ways to define sine function transformation, including first translation and then expansion, first expansion and then translation. The translation amount of the two ways is different. It must be pointed out that we will only convert a single OR.

Six, vector

① Zero vector: A zero vector is a very special vector with a length of 0 and any direction. Therefore, when dealing with parallel lines or * * * line vectors, zero vectors must be considered.

② Vector included angle: The problem should be considered comprehensively when solving the problem. There are often some factors that are easily overlooked by candidates in mathematics test questions. Whether these factors can be taken into account when solving problems is the key to solving problems successfully. For example, the angle between and is not necessarily obtuse, and it should be noted that θ = π.