1. Given that a function f(x) is derivable at or above the second order, we should immediately think of expanding f(x) into Taylor formula at a specified point.
2. If there is a definite integral expression in the problem setting condition or the conclusion to be proved, the integral expression should be treated with the integral mean value theorem first.
3. If the function f(x) is continuous in [a, b] and derivable in (a, b), and f(a)=0 or f(b)=0 or f(a)=f(b)=0, it should be treated by Lagrange mean value theorem first.
4. For definite integral or variable limit integral, if the integrand function or its main part is a composite function, first make variable substitution to make it a simple form f(u).
linear algebra
1. If the condition is related to the algebraic cofactor Aij or A*, it is immediately related to the row (column) expansion theorem with determinant and a * = a * a = | a | e.
2. If it comes to whether A and B are interchangeable, that is, AB=BA, it is immediately associated with the definition of inverse matrix for analysis.
3. If the n-order square matrix A satisfies f(A)=0, to prove that aA+bE is reversible, first decompose the factor aA+bE.
4. Prove a set of vectors a 1, a2,? Because it has nothing to do with linearity, consider using definitions first.
5. If AB=0 is known, take each column of B as the solution of Ax=0.
6. If the parameters are determined according to the requirements of the topic, then think about whether there is a determinant that is zero.
7. If the eigenvector ζ0 of A is known, firstly, it is treated by defining ζ 0 = λ 0 ζ 0.
8. If we want to prove that the abstract N-order real symmetric matrix A is a positive definite matrix, we should use the definition to deal with it.
probability theory
1. If you need the probability of "at least" one of several events, immediately connect it with the probability addition formula; When the event groups are independent of each other, the probability formula of opposing events is used.
2. If the occurrence of an event is accompanied by the occurrence of a complete event group, then the occurrence probability of this event is calculated by the total probability formula. Key: find a complete event group.
3. If the random variable X ~ N is given in the problem, it is immediately associated with the standardized ~ N(0, 1) to deal with the related problems.
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