Sol 06 T2 B

B

SOLUTIONS to Test 2

11.06.07

Task 1 (lOp.) Find the limits: a) lim-ⁿ^^{x x}    £lo£&zA

^x-*° x Vpl _x-*o d

rvX ^    .    »— /r>\    4r

b) lim, x ^l~^x =    £

^x-+1    x ■* AT

^ ^ it

x-^-    V®/    *-=>r -d

- - a ; =

Task 2 (5p.) Find the smallest and lareest value of f(xl = 3x² -12x + 5 on the interyal [0.3] f^ł(y)^r(ox-da-o    x j aso v=a -v^_n

f (o)^5-| ?(&>Vq -AQ*2+5-=-T- J £(s) = 2f-36

so s.raoJJLev-4 i*3 ‘-“ł-¹ i •^o.np-e^rt- jw| ‘S^ ask 3 (5p.) Compute the the 2-nd degree Taylor polynomial and the remainder term f<

Ti f(x) =

for

1

(1 t 3rf.

at point a = 0 and estimate the value of

wĘm _r(c)zrS,._iZ[M^

[iyifj----*u —

T,|0)=A'4y+^X

e,(o^

_ -S^ •12-

ai

(At3c)

-3“

X

e. >x

(A^3)^ł "    ^

Task 4 (5p.) Calculate the area of the region bounded above by y = e^x , bounded below by v = x, and bounded on the sides by x = 0. x = 1.

JU" ¹

⁼tF-—^>

flV(¹|e%Ux -    W1-1=

Task 5 (5p.) Find the volume of the solid of revolution obtained by revolving the region bounded by y = — , x — 1, x = 2 and y = 0 around the x-axis.

j.    .. TT- f* A. ------

Tf> ^Vcrrl    ^l*-f^łir=£

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

R(x)J^i-    ^

X³-X²    x^z(x-1)    ^ A^a

X'f

Task 7 (15p.) Find the following antiderivatives

fffrO-tnb-l f’W'- >T (    ,

;) ^^{ln|x|dx F}    g,_w„nr( "    =|^UW-

ł\a-

b), J-~— dx 3

^Jx^J+8

"kI

=)J

x +2x + 20

		fTl4_|	^r
" J (xm)^ł-*->H	' 1<1		' dtr \ W
ifUl fcł C *	Pt? . Ktl arc Km t== 4. q ^T		c k