SOLUTIONS PRE-INTERMEDIATE

Вопрос № 202

$$1 Test the function of two variables on extreme points:

(0;-2) min

 

 

Вопрос № 203

$$1 Test the function of two variables on extreme points:

(2;-3)local min

Вопрос № 204

$$1 Test the function of two variables on extreme points:

(-4;-2)local max

Вопрос № 205

$$1 Test the function of two variables on extreme points:

(2;3) saddle point

 

1.

$$1 Determine the domain of the function:

 

$$2

 

$$3

 

$$4

 

$$5

 

$$6

 

2.

$$1 Find the limit of the function:

 

$$2 4

 

$$3 0

 

$$4

 

$$5 -4

 

$$6 2

 

3.

 

$$1 Take derivative of the function:

 

$$2 2 x lnx + x

 

$$3 2x

 

$$4 2 x lnx

 

$$5 2 x lnx - x

 

$$6 x (lnx + 1)

 

4.

$$1 Find the interval where the function is increasing:

 

$$2

 

$$3

 

$$4

 

$$5

$$6

 

5.

$$1 Find the extreme points of the function:

 

$$2 y(1)-max and y(3) -min

 

$$3 y(1)-min and y(3) -max

 

$$4 y(-1)-min and y(-3) -max

 

$$5 y(-1)-min and y(-3) -max

 

$$6 y(-4)-max and y(3) –min

 

6.

 

$$1 Find the interval where the function is concave:

 

$$2

 

$$3

 

$$4

 

$$5

 

$$6

 

7.

 

$$1 Find the points of inflection of the function:

 

$$2 y(2)

 

$$3 y(-2)

 

$$4 y(1) and y(3)

 

$$5 y(-3)

 

$$6 y(-3) and y(-1)

 

8.

 

 

$$1 Find the equations of the vertical asymptotes of the function:

 

$$2 x = -3 and x=3

 

$$3 x= 2

 

$$4 x= 2/9

 

$$5 y= -3 and y=3

$$6 y= 2x-3

 

9.

$$1 Find the equations of the horizontal asymptotes of the function:

 

$$2 y=2

 

$$3 x= 2

 

$$4 y= 2/9

 

$$5 y= -3 and y=3

 

$$6 y= 2x-3

 

10.

$$1 Find the partial derivative of the function of two variables:

 

$$2

 

$$3

 

$$4

 

$$5

 

$$6

 

11.

 

 

$$1 Evaluate the indefinite integral:

 

$$2

 

$$3

 

$$4

 

$$5

 

$$6

 

12.

 

 

$$1 Formula of Integration by parts is:

 

$$2

 

$$3

 

$$4

 

$$5

 

$$6

 

13.

 

 

$$1 What property of indefinite integrals is wrong:

 

$$2

 

$$3

 

$$4

 

$$5

 

$$6 if

 

14.

 

$$1 Find the area bounded between two curves:

 

$$2 32/3

 

$$3 2/9

 

$$4 4/3

 

$$5 1

 

$$6 -1/2

 

15.

 

 

$$1 Average value over interval (a; b) of continuous function is defined by integral: .

Find the average demand of the function: D(x) = if x -output is changed: .

 

 

$$2 6

 

$$3 1

 

$$4 8

 

$$5 12/7

 

$$6 25/3

 

16.

 

 

$$1 Find the general solution of differential equation:

 

$$2

 

$$3

 

$$4

 

$$5

 

$$6

 

17.

 

 

$$1 Find general solution of differential equation:

 

$$2

 

$$3

 

$$4

 

$$5

 

$$6

 

18.

 

$$1 Evaluate the iterated integral:

 

$$2 23/3

 

$$3 2/5

 

$$4 15/4

 

$$5 1

 

$$6 ½

 

19.

$$1 Reduce the double integral to iterated integral: .

D -region is depicted

 

$$2

 

$$3

 

$$4

 

$$5

 

$$6

 

20.

$$1 Take the limit using L^,Hopital^,s rule:

 

$$2

 

$$3 1/3

 

$$4 5

 

$$5 7

$$6 5/3

 

 

21.

 

$$1 Take the limit using L^,Hopital^,s rule:

 

$$2

 

$$3 12

 

$$4 ln5

 

$$5 4/5

 

$$6 12/5

 

22.

 

 

$$1 Take the limit using L^,Hopital^,s rule:

 

$$2

 

$$3 2

 

$$4 3

 

$$5 6

 

$$6 3/2

 

23.

 

 

$$1 Take the indefinite integral:

 

$$2

 

$$3

 

$$4

 

$$5

 

$$6

 

24.

 

$$1 Take the indefinite integral:

 

$$2

 

$$3

 

$$4

 

$$5

 

$$6

 

25.

 

 

$$1 Take the indefinite integral:

 

$$2

 

$$3

 

$$4

 

$$5

 

$$6

 

26.

 

$$1 Take the indefinite integral:

 

$$2

 

$$3

 

$$4

 

$$5

$$6

 

27.

$$1 Take the indefinite integral:

 

$$2

 

$$3

 

$$4

 

$$5

 

$$6

 

28.

 

 

$$1 Which of the following functions is inverse to the function:

 

$$2

 

$$3

 

$$4

 

$$5

$$6

 

29.

 

$$1 Which of the following functions are odd functions:

a) ; b) ; c) d) e)

 

$$2 b) and c)

 

$$3 a)

 

$$4 e)

 

$$5 a) and c)

 

$$6 no one

 

30.

 

 

$$1 Take differential of the function:

 

 

$$2

 

$$3

 

$$4

 

$$5

 

$$6

 

31.

 

$$1 Test the function of two variables on extreme points:

It is known that It is known that If at critical point: a) - Heissian > 0 and =>

b) > 0 and => ; c) < 0 => - saddle point;

d) = 0 => test fails

 

 

$$2

 

$$3

 

$$4

$$5

 

$$6 test fails

 

32.

 

 

$$1 Find the general term of the number series:

 

$$2

 

$$3

 

$$4

 

 

$$5

 

 

$$6

 

33.

 

 

$$1 Let given two series: and

If for any n: u_n ≤ v_n, then

a) from convergence of follows the convergence of .

b)from divergence of follows the divergence of .

The test is known as:

 

$$2 Comparison test

 

$$3 Ratio test

 

$$4 Root test

 

$$5 Integral test

 

$$6 Necessary test for convergence

 

34.

 

$$1 Let for (u_n ≥0) the limit exist: , then

a) the series converges if ;

b) the series diverges if ;

c) the test fails if .

The test is known as:

 

$$2 Ratio test

 

$$3 Comparison test

 

$$4 Root test

 

$$5 Integral test

 

$$6 Necessary test for convergence

 

35.

 

 

$$1 Let for (u_n ≥0) the limit exist: , then

a) the series converges if ;

b) the series diverges if ;

c) the test fails if .

The test is known as:

 

$$2 Root test

 

$$3 Comparison test

 

$$4 Ratio test

 

$$5 Integral test

 

$$6 Necessary test for convergence

 

36.

 

$$1 Let u_n=f(n) (f(x) is continuous and positive). Then the and the integral both are convergent or divergent.

The test is known as:

 

$$2 Integral test

 

$$3 Comparison test

 

$$4 Ratio test

 

$$5 Root test

 

$$6 Necessary test for convergence

 

37.

 

$$1 What series among harmonic series is/are convergent:

a) ; b) ; c) ; d) ; e)

 

$$2 d)

 

$$3 a)

 

$$4 b)

 

$$5 c)

 

$$6 e)

 

38.

 

 

$$1 What series among harmonic series is/are divergent:

a) ; b) ; c) ; d) ; e)

 

$$2 a)

 

$$3 d)

 

$$4 b)

 

$$5 c)

 

$$6 e)

 

39.

 

$$1 What series among following series is/are convergent:

a) ; b) ; c) ; d) ; e)

 

$$2 b) and e)

 

$$3 a) and c)

 

$$4 b) and d)

 

$$5 c) and a) and d)

 

$$6 c) and d)

 

40.

 

$$1 What series among following series is/are divergent:

a) ; b) ; c) ; d) ; e)

 

$$2 a) and c) and d)

 

$$3 b) and e)

 

$$4 b) and d)8

 

$$5 c) and e) and d)

 

$$6 c) and d)

 

41.

 

 

$$1 Determine the interval of convergence with radius of the following power series:

 

$$2

 

$$3

 

$$4

 

$$5 everywhere convergent

 

$$6 everywhere divergent

 

 

42.

 

$$1 Determine the interval of convergence with radius of the following power series:

 

$$2

 

$$3

 

$$4

 

$$5 everywhere convergent

 

$$6 everywhere divergent

 

43.

 

 

$$1 What series among following alternating series is/are divergent:

a) ; b) ; c) ; d) ; e)

 

$$2 a) and c) and d)

 

$$3 b) and e)

 

$$4 b) and d)

 

$$5 c) and e) and d)

 

$$6 c) and d)

 

44.

 

$$1 What series among following alternating series is/are convergent absolutely:

a) ; b) ; c) ; d) ; e)

 

$$2 b) and e)

 

$$3 a) and c) and d)

 

$$4 b) and c)and e)

 

$$5 c) and e) and d)

 

$$6 c) and d)

 

45.

 

 

$$1 What series among following alternating series is/are convergent conditionally:

a) ; b) ; c) ; d) ; e)

 

$$2 a) and c)

 

$$3 a) and b) and c) and d)

 

$$4 b) and c)and e)

 

$$5 c) and e) and d)

 

$$6 e)

 

 

SOLUTIONS PRE-INTERMEDIATE