After learning irrational numbers, a mathematical interest group launched an inquiry activity: estimating the approximate value of 13.

Xiao Ming's method:

∵9< 13< 16,

Let 13 = 3+k (0 < k < 1).

∴( 13)2=(3+k)2.

∴ 13=9+6k+k2.

∴ 13≈9+6k.

The solution is k ≈ 46.

∴ 13≈3+46≈3.67.

Question:

(1) Please estimate the approximate value of 4 1 according to Xiao Ming's method;

(2) Please use the above concrete examples to summarize the formula for estimating m: non-negative integers A, B and M are known; If A < M < A+ 1 and m=a2+b, then m≈a+b2a (represented by an algebraic expression containing a and b);

(3) Please use the conclusion in (2) to estimate the approximate value of 37.

Test center: estimate the size of irrational numbers.

Special topic: reading type.

Analysis: (1) According to the topic information, find two square numbers before and after 4 1 to determine 4 1 = 6+k (0 < k < 1), and then approximately solve it according to the topic information;

(2) According to the solution provided by the topic, first find the value of k, and then add a;

(3) Replace A with 6 and B with 1, and substitute them into the formula to get the solution.

Solution: (1) ∵ 36 < 4 1 < 49,

Let 4 1 = 6+k (0 < k < 1),

∴(4 1)2=(6+k)2，

∴4 1=36+ 12k+k2，

∴4 1≈36+ 12k.

The solution is k≈5 12,

∴4 1≈6+5 12≈6+0.42=6.42;

(2) let m = a+k (0 < k < 1),

∴m=a2+2ak+k2≈a2+2ak，

∫m = a2+b，

∴a2+2ak=a2+b，

The solution is k=b2a,

∴m≈a+b2a；

(3)37≈6+ 1 12≈6.08.

Comments: This question examines the estimation of irrational numbers, provides information by reading the stem, and then changes the data according to the methods in the information. It's not difficult or interesting.

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