[Jia Liwei university mathematics magazine] No. 245th Xiamen University 2011 mathematical analysis postgraduate exam reference

1. multiple choice questions (This question includes five full scores of 30 points and six points for each question)

(1) function $ Y = f (x) $ has a continuous second derivative in a neighborhood of $ x_0 $, satisfying $ f' (x_0) = 0 $, and $ f'' (x_0) <0 $, then ()

A. $ f (x) $ obtain the maximum value at the Point $ x_0 $;

B. $ f (x) $ obtain the minimum value at $ x_0 $;

C. $ (x_0, F (x_0) $ is the inflection point of the curve $ Y = f (x) $;

D. $ f (x) $ monotonically decreases in a neighborhood of $ x_0 $.

Proof: Actually, $ \ Bex f (x) = f (x_0) + f' (x_0) (x-x_0) + \ frac {f'' (\ xi_x )} {2} (x-x_0) ^ 2, \ quad x \ In U (x_0 ). \ EEx $

 

(2) function $ f (x) = \ ln x-x $ the number of zeros in the range $ (0, \ infty) $ is ().

A. 0;

B. 1;

C. 2;

D. Not sure.

Proof: by $ \ Bex f' (x) = \ frac {1} {x}-1 \ sedd {\ BA {ll}> 0, & 0 <x <1, \ <0, & x> 1 \ EA} \ EEx $ Zhi $ \ DPS {\ Max _ {x \ In (0, \ infty)} f (x) =-1} $.

 

(3) When $ x \ to 0 $ is known, the functions $ e ^ {\ SiN x}-e ^ x $ and $ x ^ N $ are of the same order, then $ n = (c) $.

A. 1;

B. 2;

C. 3;

D. 4.

Proof: by $ \ Bex e ^ {\ SiN x}-e ^ x = e ^ \ XI (\ SiN x-x) \ sim-\ frac {x ^ 3} {6} \ Quad (x \ to 0) \ EEx $.

 

(4) which of the following statements is true? (B ).

A. if $ f (x) $ on $ [a, B] $, and there is an original function $ f (x) $, then $ \ Bex \ SEZ {\ int_0 ^ x F (t) \ RD t} '= f (x); \ EEx $

B. $ f (x) $ Riann product on $ [a, B] $ | f (x) | $ product on $ [a, B] $;

C. if $ f ^ 2 (x) $ can accumulate in Riann on $ [a, B] $, then $ | f (x) | $ must be in $ [, b] $ on Riann product;

D. if $ | f (x) | $ Riann product on $ [a, B] $, then $ f (x) $ in $ [, b] $ ON THE Riann product.

Answer: Use the Equivalent Definition of the Riann integral directly (the one with an amplitude ).

 

(5) set $ f (x) $ to meet $ f'' (x)> 0 $ in $ (-) $, and $ | f (x) | \ Leq x ^ 4 $, $ \ DPS {I = \ int _ {-1} ^ 1 F (x) \ RD x} $, then (B) is required ).

A. $ I = 0 $;

B. $ I> 0 $;

C. $ I <0 $;

D. Not sure.

Proof: by $ \ beex \ Bea f (x) & = \ int_0 ^ x f' (t) \ RD t \\\<\int_0 ^ x \ SEZ {f' (0) + \ int_0 ^ t f' (s) \ RD s} \ RD t \ & = f' (0) x + \ int_0 ^ x \ int_0 ^ t f' (s) \ RD s \ RD t \ & = f' (0) x + \ int_0 ^ x (x-S) f' (s) \ RD s \ EEA \ eeex $ Zhi $ \ beex \ Bea \ int _ {-1} ^ 1 F (x) \ RD x <=\ int _ {-1} ^ 1 (X-S) f'' (s) \ RD s \ RD x \\\\\int _ {-1} ^ 0 \ int_x ^ 0 (S-x) f'' (s) \ RD s \ RD x + \ int_0 ^ 1 \ int_0 ^ x (x-S) f'' (s) \ RD s \ RD x \ &> 0. \ EEA \ eeex $

 

2 ($10 '$) $ f (x) $ on $ [0, 1] $ top second-order bootable with $ F (0) = F (1) = 0 $, $ \ DPS {\ min _ {x \ in [0, 1]} f (x) =-1} $. proof: $ \ Xi \ In (0, 1) $ exists, making $ f'' (\ xi) \ geq 8 $.

Proof: Set $ x_0 \ in [0, 1] $ to make $ \ DPS {f (x_0) = \ min _ {x \ in [0, 1]} f (x) =-1} $, then $ f' (x_0) = 0 $, instead of Taylor, $ \ Bex 0 = f (0) =-1 + \ frac {f'' (\ xi_1)} {2} x_0 ^ 2, \ quad 0 = F (1) =-1 + \ frac {f'' (\ xi_2)} {2} () ^ 2. \ EEx $ therefore, if $ \ DPS {0 \ Leq x_0 \ Leq \ frac {1} {2 }}$, then $ \ xi = \ xi_1 $; if $ \ DPS {\ frac {1} {2} \ Leq x_0 \ Leq 1 }1, $ \ xi = \ xi_2 $.

 

3 ($ 10' $) set the function $ f \ in C [0, 1] $, which proves: $ \ Bex \ lim _ {\ Lambda \ to + \ infty} \ int_0 ^ \ Lambda f \ sex {\ frac {x }{\ Lambda }}\ frac {1} {1 + x ^ 2} \ RD x = \ frac {\ PI} {2} f (0 ). \ EEx $

Proof: $ \ beex \ Bea \ lim _ {\ Lambda \ to + \ infty} \ int_0 ^ \ Lambda f \ sex {\ frac {x} {\ Lambda} \ frac {1} {1 + x ^ 2} \ rd x & =\ LiM _ {\ Lambda \ to + \ infty} \ int_0 ^ 1 \ frac {\ Lambda f (x )} {1 + \ Lambda ^ 2t ^ 2} \ RD t \\\&=\ LiM _ {\ Lambda \ to + \ infty} \ int_0 ^ 1 \ frac {\ Lambda [F (t) -F (0)]} {1 + \ Lambda ^ 2t ^ 2} \ rd t + \ frac {\ PI} {2} f (0) \ & =\ LiM _ {\ Lambda \ to + \ infty} \ int_0 ^ \ Delta + \ int _ \ Delta ^ 1 \ frac {\ Lambda [F (t) -F (0)]} {1 + \ Lambda ^ 2t ^ 2} \ rd t + \ frac {\ PI} {2} f (0) \ & =\ frac {\ PI} {2} f (0 ). \ EEA \ eeex $ the last step is because the optional $ \ Delta $ makes $ \ int_0 ^ \ Delta \ cdots $ sufficiently small; then the $ \ Delta $, when $ \ Lambda $ is sufficiently large, $ \ int _ \ Delta ^ 1 \ cdots \ Leq \ frac {\ Lambda} {1 + \ Lambda ^ 2 \ Delta ^ 2} \ cdots $ small enough.

 

4 ($ 15' $) set $ f (x) $ to be continuous on $ [] $. proof of $ \ forall \ t> 0 $, $ \ Bex \ SEZ {\ int_0 ^ 1 \ frac {f (x )} {t ^ 2 + x ^ 2} \ RD x} ^ 2 \ Leq \ frac {\ PI} {2 t} \ int_0 ^ 1 \ frac {f ^ 2 (x)} {t ^ 2 + x ^ 2} \ rd x. \ EEx $

Proof: It is derived from the Gini-Schwarz inequality, $ \ beex \ Bea \ EEA \ eeex $ \ beex \ Bea \ SEZ {\ int_0 ^ 1 \ frac {f (x )} {t ^ 2 + x ^ 2} \ RD x} ^ 2 & =\ SEZ {\ int_0 ^ 1 \ frac {1} {\ SQRT {t ^ 2 + x ^ 2 }}\ cdot \ frac {f (x )} {\ SQRT {t ^ 2 + x ^ 2 }}\ RD x} ^ 2 \ & \ Leq \ int_0 ^ 1 \ frac {1} {t ^ 2 + x ^ 2} \ RD x \ cdot \ int_0 ^ 1 \ frac {f ^ 2 (x )} {t ^ 2 + x ^ 2} \ RD x \\\\=\ frac {1 }{t }\ arctan \ frac {1 }{ t} \ cdot \ int_0 ^ 1 \ frac {f ^ 2 (x )} {t ^ 2 + x ^ 2} \ RD x \ & \ Leq \ frac {\ PI} {2 t} \ int_0 ^ 1 \ frac {f ^ 2 (x)} {t ^ 2 + x ^ 2} \ rd x. \ EEA \ eeex $

 

5 ($ 15' $) set $0 <\ Lambda <1 $, $ \ DPS {\ lim _ {n \ To \ infty} a_n = A }$, verify: $ \ Bex \ lim _ {n \ To \ infty} \ sex {\ Lambda ^ na_0 + \ Lambda ^ {n-1} a_1 + \ cdots + \ Lambda A _ {n-1} + a_n }=\ frac {A} {1-\ Lambda }. \ EEx $

Proof: write $ \ lim \ frac {A} {1-\ Lambda }=\ LiM _ {n \ To \ infty} (1 + \ Lambda + \ cdots + \ Lambda ^ {n-1} + \ Lambda ^ N) A \ EEx $.

 

6 ($ 20' $) set $ f_n (x) = E ^ {-NX ^ 2} \ Cos x $, $ x \ in [-] $, $ N $ is a positive integer. proof:

(1) $ f_n $ inconsistent convergence on $ [-1, 1] $;

(2) $ \ DPS {\ lim _ {n \ To \ infty} \ int _ {-1} ^ 1 f_n (X) \ RD x = \ int _ {-1} ^ 1 \ lim _ {n \ To \ infty} f_n (x) \ RD x} $.

Proof:

(1) by $ \ Bex e ^ {-n \ cdot \ sex {\ frac {1} {\ SQRT {n }}^ 2} \ cos \ frac {1 }{\ SQRT {n }}>\ frac {1} {2E} \ Quad (n \ Gg 1) \ EEx $ conclusion.

(2) by $ \ Bex \ sev {\ int_0 ^ 1 f_n (X) \ RD x} \ Leq \ int_0 ^ 1 e ^ {-NX ^ 2} \ RD x \ Leq \ int_0 ^ \ Delta \ RD x + \ int _ \ Delta ^ 1 E ^ {-n \ Delta ^ 2} \ RD x \ to 0 \ quad \ sex (n \ To \ infty) \ EEx $ conclusion.

 

Note: This indicates that consistent convergence is only a sufficient, not a necessary condition for the order of the limit and the point exchange.

 

7 ($ 10' $) proves that continuous functions in the finite closed interval can obtain the minimum value.

Proof: we must first prove bounded, and then obtain the minimum value. The Weierstrass clustering theorem is used for both cases.

 

8 ($ 15' $) set $ f (x) $ to be micro at $ A $, and $ F (a) \ NEQ 0 $, limit $ \ DPS {\ lim _ {n \ To \ infty} \ SEZ {\ frac {f \ sex {A + \ frac {1} {n }}{ F ()}} ^ n} $.

A: Set $ F (a) to 0 $, and $ \ beex \ Bea \ lim _ {n \ To \ infty} \ SEZ {\ frac {f \ sex {A + \ frac {1} {n }}}{ f ()}} ^ N & =\ exp \ SEZ {\ lim _ {n \ To \ infty} \ frac {\ ln f \ sex {A + \ frac {1} {n }}- \ ln f ()} {1/N }}\\\&=\ exp \ SEZ {\ lim _ {x \ to 0 }\frac {\ ln f (a + x) -\ ln f ()} {x }}\\\=\ exp \ SEZ {\ lim _ {x \ to 0} \ frac {f' (a + x )} {f (a + x) }\\\& = e ^ \ frac {f' (a)} {f ()}. \ EEA \ eeex $

 

9 ($ 15' $) calculate $ \ DPS {\ iint _ {[0, \ Pi] \ Times [0, 1]} Y \ sin (xy) \ RD x \ RD y} $.

Answer: You can get the answer $1 $ directly as the accumulated credits.

 

10 ($ 10' $) calculate $ \ DPS {\ lim _ {n \ To \ infty} \ sum _ {k = n ^ 2} ^ {(n + 1) ^ 2} \ frac {1} {\ SQRT {k }}$.

Answer: by $ \ Bex \ frac {1} {n + 1} [(n + 1) ^ 2-N ^ 2 + 1] \ Leq \ sum _ {k = n ^ 2} ^ {(n + 1) ^ 2} \ frac {1} {\ SQRT {k }}\ Leq \ frac {1} {n} [(n + 1) ^ 2-N ^ 2 + 1] \ EEx $ knows that the original limit is equal to $2 $.