Substitution method is also called auxiliary element method and variable substitution method. By introducing new variables, scattered conditions can be linked, implicit conditions can be revealed, or conditions can be linked with conclusions. Or turn it into a familiar form to simplify complicated calculation and derivation.
It can transform high order into low order, fraction into algebraic expression, irrational expression into rational expression, transcendental expression into algebraic expression, and has a wide range of applications in the study of equations, inequalities, functions, sequences, triangles and other issues.
The substitution methods include: local substitution, triangle substitution, mean substitution and so on.
The types of substitution are: isoparametric substitution and unequal generation.
Local substitution, also known as global substitution, means that an algebraic expression appears many times in the known or unknown, and it is replaced by a letter to simplify the problem. Of course, sometimes it is discovered through deformation.
Example: Find the range of y=x+√( 1-x),
If you start directly, it is difficult, but we can assume that:
T=√( 1-x), the inverse solution is: x = 1-t 2, (note: t≥0, "√" stands for the root sign).
So the original formula is equivalent to: y = 1-t 2+t =-t 2+t+ 1 (quadratic function is familiar to us), and its value range is:
(-∞,5/4]。
When trigonometric substitution is applied to the root-removed sign, or it is easy to find the triangular form, it is mainly to use a certain point in the known algebra for substitution.
If you find the range of function y = √1-x 2, if x∈[- 1, 1], let x=sin α, sinα∈[- 1]. Why did you think of this setting? Mainly to find the relationship between the range of values and the need to remove the root sign.
Another example is that variables X and Y meet the condition X2+Y2 = R2 (r >); 0), x=rcosθ and y=rsinθ can be transformed into trigonometric problems by trigonometric substitution.
When the mean value is substituted in the form of x+y=2S, let x= S+t, y = s-t and so on.
For example, given that A and B are nonnegative real numbers, m = A 4+B 4, a+b= 1, find the maximum value of m.
Let a = 1/2-t, b = 1/2+t (0 ≤ t ≤ 1/2) be brought into m, and m = 2× (t 2+3/4) 2- 1, from.
Equivalent substitution let x+y=3 let x=t+2 and y=v-3 be used for double integration.
Unequal substitution u=(x+y)+3(x+y) and x+y=S, also called global substitution method.
Application skills When we use the substitution method, we should follow the principles that are conducive to operation and standardization. Pay attention to the selection of the new variable range after substitution, and make sure that the new variable range corresponds to the value range of the original variable, which cannot be reduced or expanded.
You can observe the formula first, and you can find that there is always the same formula in this formula for substitution, and then replace it with a letter to calculate the answer. Then, if there is this letter in the answer, bring the formula in and the calculation will come out.
Sometimes in factorization, you can choose the same part of the polynomial, replace it with another unknown, then factorize it and finally convert it back. This method is called substitution method.
Note: don't forget to return the RMB after exchange.
Example 1, when decomposing (x 2+x+1)-12, you can make y = x 2+x, then the original formula = (y+1) (y
Example 2, (x+5)+(y-4)=8.
(x+5)-(y-4)=4
Let x+5 = m and y-4 = n.
The original equation can be written as
m+n=8
m-n=4
The solution is m = 6 and n = 2.
So x+5 = 6 and y-4 = 2.
So x= 1, y=6.
Features: The two equations contain the same algebraic expressions, such as x+5 and y-4 in the title, and the simplification of the equations after substitution is also the main reason.