[Jia Liwei university mathematics magazine] 248th Northeast Normal University 2013 mathematical analysis postgraduate exam

1 calculate $ \ Bex \ lim _ {x \ To \ infty} \ sex {\ frac {4x + 3} {4x-1 }}^ {2x-1}. \ EEx $

2 calculate $ \ Bex \ lim _ {x \ To \ infty} \ frac {1} {n} \ sum _ {I = 1} ^ n \ ln \ frac {I \ PI} {n }. \ EEx $

3. Evaluate the derivative of the implicit function $ x ^ 2 + y ^ 2 = \ cos (xy) $.

4 calculate $ \ Bex \ lim _ {x \ to 0} \ frac {x \ int_0 ^ x e ^ {t ^ 2} \ RD t} {\ int_0 ^ x Te ^ {t ^ 2} \ RD t }. \ EEx $

5 Calculate $ \ Bex \ int_0 ^ 1 \ rd y \ int_0 ^ y e ^ {-x ^ 2} \ rd x. \ EEx $

6 calculate $ \ Bex \ iiint _ {x ^ 2 + y ^ 2 + Z ^ 2 \ Leq 1} (x ^ 2 + y ^ 2 + Z ^ 2) \ RD x \ rd y \ rd z. \ EEx $

7. Determine the monotonicity, extreme value, inflection point, and concave-convex of the function $ f (x) = 2x ^ 3-5x ^ 2 + 4x + 2 $.

8. Set the function $ f (x) $ to be continuous on $ [a, B] $, $ f (x)> 0 $. set $ \ Bex f (x) = \ int_a ^ x F (t) \ rd t + \ int_ B ^ x \ frac {1} {f (t )} \ rd t. \ EEx $ authentication:

(1) $ f' (x) \ geq 2 $;

(2) equation $ f (x) = 0 $ has a unique solution in $ (a, B) $.

9 known functions $ f (x) $ perform micro-consecutively on $ [0, 1] $, perform second-order export, and $ f'' (x) \ geq 1 $. verification: $ | f' (0)-f' (1) | \ geq 1 $.

10 known $ f (x) $ continuous export. Verification:

(1) $ \ Bex \ int_0 ^ 1 x ^ n f (x) \ RD x = \ frac {F (1 )} {n + 1}-\ frac {1} {n + 1} \ int_0 ^ 1 x ^ {n + 1} f (x) \ rd x; \ EEx $

(2) $ \ Bex \ lim _ {n \ To \ infty} \ int_0 ^ 1 x ^ n f (x) \ rd x = F (1 ). \ EEx $

11 known $ f (x) = \ Pi (E ^ x + e ^ {-x}) e ^ \ PI + e ^ {-\ PI} $, evaluate the Fourier series of $ f (x) $ on $ [-\ Pi, \ Pi] $.

12 calculate $ \ DPS {\ sum _ {n = 1} ^ \ infty \ frac {(-1) ^ n} {(2n) ^ n }}$.

13 set $ f (x) $ to continuous on $ [0, \ infty) $, $ \ DPS {\ int_0 ^ \ infty f (x) \ rd x, \ int_0 ^ \ infty f' (x) \ RD x} $ all converge. proof: $ \ DPS {\ lim _ {x \ To \ infty} f (x) = 0} $.