How to solve 40 and how to solve the substitution method

The methods of commuting are: local conversion, triangular conversion, mean conversion, etc.

Local commutation: also known as global commutation, is in the known or unknown, a certain algebraic formula appears several times, and a letter is used to replace it to simplify the problem, of course, sometimes through deformation to discover. For example, solving inequalities:

4 +2 2 0, first deformed to let 2 =t(t>0), trigonometric commutation: applied to the root number, or transformed into a trigonometric form when easy to find, mainly using the known algebraic formula and a certain point in the trigonometric knowledge of the connection for commutation. For example, when finding the range of the function y= 1-x 2, if x [-1,1], let x=sin sin [1,1 ], the problem becomes the familiar domain of finding trigonometric functions.

The main reason why I thought of such a setting should be to discover the connection of the value range, and there is a need to remove the root number. If the variables x and y fit the conditions x 2 + y 2 =r 2(r>0), then the trigonometric substitution x=rcos and y=rsin can be used to transform the trigonometric problem.

Mean commutation: If you encounter the form of x+y=2s, set x= s+t, y= s t, and so on.

For example, in the Tsinghua University independent admissions examination questions, it is known that a, b are non-negative real numbers, m=a 4+b 4, a+b=1, and find the maximum value of m.

can make a=1 2-t,b=1 2+t(0 t 1 2), bring in m,m=2 (t 2+3 4) 2-1, and know from the quadratic function properties m(min)=1 8,m(max)=1

Equivalence transformation unknowns. For example, x 2 + y 2 = r 2(r>0), which can be replaced by x=rcost, y=rsint, and many other variations, this is just the most common one.

The methods of commuting are: local conversion, triangular conversion, mean conversion, etc.

Local commutation: also known as global commutation, is in the known or unknown, a certain algebraic formula appears several times, and a letter is used to replace it to simplify the problem, of course, sometimes through deformation to discover. For example, solve the inequality: 4

2 0, first deformed to set 2

t(t>0), trigonometric commutation: It is applied to the root number, or when it is transformed into a trigonometric form that is easy to find, it mainly uses a certain point in the known algebraic formula to have a connection with trigonometric knowledge for commutation. For example, when finding the range of the function y= 1-x 2, if x [-1,1], let x=sin

sin [1,1, the problem becomes the familiar domain of trigonometric functions. Why did you think of such a set, the main thing should be to discover the connection of the value range, and there is a need to go to the root number. For example, variables x and y fit condition x

2+y^2r

2(r>0), then the trigonometric substitution x=rcos and y=rsin can be used to transform the trigonometric problem.

Mean commutation: If x+y=2s is encountered, set x=

s+t，y=

s t and so on this hu.

For example, in the Tsinghua University independent admissions examination questions, it is known that a, b are non-negative real numbers, m=a 4+b 4, a+b=1, and find the maximum value of m.

can make a=1 2-t,b=1 2+t(0 t 1 2), bring in m,m=2 (t 2+3 4) 2-1, and know from the quadratic function properties m(min)=1 8,m(max)=1

Let x-1 t under the root number x-1 t x t +1 so 1 t +1 2 then f(t) 2(t +1)+t, i.e., f(t) 2t +t+2 find the range of t and bring in the available value range.

Let y=(x+1) x, then the original equation becomes y-1 y=2 3, solve this unary quadratic equation, and get y=2, y=-1 2 return to the original equation:

2=1+1/x，-1/2=1+1/x---x=1，x=-2/3