84268 Sol 06 T2 C

SOLUTIONS to Test 2 c 11.06.07

Tusk 1 (lOp.) Lind the limils:n) lim

SOLUTIONS to Test 2 c 11.06.07

(2x +1)

M II jOPifS _A,    *— Cfl

b) lim, —    - p    ( *^^{0 Ł}^v hi

^K**° UJ X-*0*

*■=>0

^ jUm    =

/-^cr

= e ■= i

r fe, t 5- . iiś2s_ _i>

---- r> CCS fi *i«r

H-S) O*"

x-=»o»-

o

Task 2 (5p.) Find all the local extrema of the function intervals of concavity and the inflection points of f(x) = x³ - 6x² +9x .

¹ X^Ł = 2

P"in)*-o &> x n) ■    ^

X-K--łl - C/^lęcKoh potnfł

Task 3 (5p.) Compute the the 2-nd degree Taylor polynomial and the remainder term for f(x) = ln(l + 2x) at point a = 0 and estimate the value of ln(l .2).

e,/*V f[ H-t£cr^s*^ł

? S.    f (ó| '^^=0 J \lo)'- 0+ fi*x - %/■

r (*) r-a(^ 2*)^ A p(x>4 i^^f-2

H1 MM

1 §§§¹

Task 4 (5p.) Calculate the area of the region bounded by the curves: y = 5x - x* and y = x.

w_ ■■ Si isBS

^ |    X ( n-y)-O

Task 5 (5p.) Find the volume of the solid of revolution obtained by revolving the region

bounded by y = e^x , x = 0, x = 1 and y = 0 around the x-axis.

UL* I - fili Si# {«:“!• fil

11 p fi

*0

i*-i

Task 6 (5p.) Write out the form of the partial fractions expansion of the rational function R(x). Do not determine the numerical values of the coefficients.

no

R(x) =

B K +

(x-l)(x'^t+x + l) X- 4 Task 7 (15p.) Find the following antiderivatives

f0<) *x    1

a) |x sin3xdx = ( ,    ..

^J ’ s('‘>t^u,i1x śfii

r. - cosSa + j|    dx = - i

-    I0s5>< +

i- Z-

o a

b), |

2x

dx =

fc = x-k _9*

(Mr> Zx cUr-

1

_dfc_

rr

0: *c(i: ~ v £= ~2. * fx²^ * £

■>J

2x +4

- dx

ifc

oU

jlo|t| *C -- IrHifjl

d.x

9^ - dx

JL