SOLUTIONS lo Tcsl 2 X

11.06.07

Tusk 1 (1 Op.) lind ihc llmils: a) lim--jt /^£-11 JUm — - a O

>-*“ x + e ^ J »^>«o Iłor

b) lim,(sin */'. g* (?-%2. -jr¹ &) '    ‘ f =

f - Vip* ■=¥%r^* -%0-^f^/-⁰/

2. Find the intervals of concavity and the inflection points of f(x) = — x⁴ --x³ 4-x² +1.

Alwo.i^ COHU^A, O0    pOĆrUi

'    <*s *f- 4* o

Task 3 (5p.) Compute the the 2-nd degree Taylor polynomial and the remainder term for f(x) = x e* at point a = 0 and estimate the value of 'tfe, j -j-    - 0    ^ 'X f ^ ^

o

e - i

fis-*'-¹ r ‘•w--w

0 e'i «-^/-t '

wV3 ,Task 4 (5p.) Calculate the area of the region bounded belowby y = 4x², bounded aboveby 'it    y = x² + 3, and bounded on the sides by x = -1, x = 1.

ixf_ t-/4.0 b-V --* jp>-^)dx -- J*-*³ | -^-±.(0-0) =2

Pff C^#*l

PY0*3*^ł»«'®^c I sfrfe^--    1 (*]"• ii

fs^X9 i.

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

bounded by y = Vx , x = 1, x = 2 and y = 0 around the x-axis,

'    V ’Ttj>Vv * ir p «, - TT ii |\ f[. % . Tl %, % JT

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 coeffidents.

R(x) = -

3x + 4

(x — l)²(x — 2) j H (X-D* x'j2 Task 7 (15p.) Find the following antiderivatives

a) H ln|x² ₊ l|dx J-V " l„ j 11 U,)l\ M - ł-laW- ^{fc tC 1} I    ( eli 1 Z.* ł'*>

b), :|xe« dx p I    *'^W=3 i - 3-x/^A - (

■i fflb s'o)=^( ^J

r» i a r/ t, 5~/

Mir Ili 'ś?^{ł +}c

PO

d/

. ( t^sV" I- 4-f 'im / ) ‘•J

!_{(x + 3)!+4}^dx'- ^ jUu.J

9 dJb

1 I    - i (xrt hbn &• \d ~

= 1 o-rdTayi-^r + C