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Final exam 28.01.10 Name:

Exercise 1. Compute the limit of the sequence:

an= sin(√ n + 2) n2+ 1 .

Solution:

1

(2)

Exercise 2. Check whether the following series converges:

X∞

n=1

2n n√

2n+ 3n.

Solution:

2

(3)

Exercise 3. For which values of parameters a, b the function f is continuous:

f (x) =





x : x < 0 x2 + ax + b : 0 ≤ x < 1

x + 3 : 1 ≤ x.

Solution:

3

(4)

Exercise 4. Compute the derivative of the function:

f (x) = e√

log(x). Solution:

4

(5)

Exercise 5. Compute the limit:

x→0lim

2 cos(x) − x2+ 2 x sin(x) − x2 . Solution:

5

(6)

Exercise 6. Compute the indenite integral:

Z dx

1 +√3 x + 1. Solution:

6

Cytaty

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