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[.
22th of January,20t4.
You needreachat least20 points to pass.
1. (6 p.) Find the derivativosof the functions:
z .\Íw)
) J \ n )_- x2. E+-cost4x
,.
_I
b.)gír)
, - r * l : -]l n T t * u \r , ,
c.) h(,) : (x4+ sinx)a'.
2. (4+3p.) a.) Show thatthe lines
x:-2+t,
l:3+2t,
.z-4-t
x:3-t,
!:4-2t,
z:t
and
are parallel and find an equationof the plane they determine.
b.) Solve tfreeOuationand give the resultin algebricform:
- Dra - 2i : 0.
3. (6 p.) Sketch the graphof
*'*4
r/\I.v- )/ - 2
X'
4. (6 p.) Find the following limits, if they exists:
a.)fr_gxta"-'?,
b.)
5. (7 p.) Let
l:: "
* gnlzx)- ln(x+ I )),
..) J!*(tanx)#.
Í(i: tr'_sin,"í*).
a.) Show thatthe inverseof/exists and find it.
b.) Find the rangeoff(x),and thedomainandtherangeof f-l(x).
c.) Sketch the graphof the functionsÍ(*) andf_1(x).
d.).Givethe equationof the tangentline of Í( atxo : 3n.
6. (6 p.) Evaluatethe indefinitintegrals:
b.)Ixln(I+xz)dx.
fr-f
") I ffi*,
7 . (6 p.) Evaluatethe definiteintegrals:
b,s
a.) I t-
,
+
dl I arcsin(2x)dx
.
8. (6 p.) {ind the areabetweenthe curves
y:
-)(v-1\*-1r)
e -r-,,,!:
e^ ano l:
e-.
, Name:
MathematicsI. (BSc)- Exam Test VI.
2 8 t ho f A u g . 2 0 1 5 .
You needreachat least20 pointsto pass.
1. (6 p.)Let
.:._l / l -x \ .
-ÍU)_ | -- r , f , \ T ;
)
a.)Find thedomain,therangeandthederivativeof ./(x).
b.) Show thatthe inverseof/exists and find it.
c.) Find the domain,the rangeandthe derivativeof the inverseof f(x).
2. (3+4p.)a.) Find an equationfor theplanepassingthroughthegivenpointP(0, l, -l and
)
parallelto the lines
c1 :2x: l:32
a n d c z : 5 ( x+ 7 ) : - 2 ( t t* 3 ) : l 0 ( z - a ) .
b.) Solvetheequationandgivetheresultin algebricform:
d1 -Dro-2i:0.
3. (7 p.) Sketchthe graphof
.Í(i: ,, .!i ! ,
4. (6 p.) Let given
and s(x): tan
+.
.Í(i:3 cos-lG, - t)
a.)Í"8(x) : ? a n d D f g : ?
b.)(/..9)'(x) : ?
c . ) l i m . / ' "g ( x ) : l
5 . (6 p.)a.)Determinethe20l5th derivarives
1|{zott)(")?) of the f"ll;g
Í(*):(,,_|)g_,,
b.)|im -ÍU): ?
.t-
functions:
c.)lim 'ÍU): ?
X+-t1)
l-
6. (7 p.) Evaluatethe integrals:
..)íiffia,,
7. (6 p.)Evaluatethedefiniteintegrals:
2
u.lI
' J La*.
t+e^
0
2
b)I ffidx
0
8. ( 5 p.)Draw theareabetweenthegivencurvesandc a l c u l a t et h e v a l u e o f t h e a r e a :
!:3+2x-X2,
!:3-x.
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i
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Mathematics I. (BSc)- Exam Test I.
23th of December,2014.
You needreachat least20 pointsto pass.
1 . (6 p.) Let
- í ( x -) 1 + s i n' \-2l í+]xl ;)r' ) .
a.)Find the domain,the rangeandthe derivativeof /(x).
b.) Show thatthe inverseof/exists and Íindit.
c.) Find the domain,the rangeand the derivativeof the inverseof f(x).
2 . (3+4p.) a.)Find an equationfor the planepassingthroughthe givenpointP(3,1,-1) and
parallelto the lines
ct :2x: -!:32
a n d c 2 : 5 ( x- 7 ) :
- 2 ( y+ 3 ) - 1 , 0 ( z - 4 ) .
b.) Solvethe equationand give theresultin algebricform:
+ )za +2i = O,
(í
3. (7 p,) Sketchthe graphof
r/\í' -r /) :
x2+x- l
x+4
4. (6 p.)Let given
and s(x): tan-l+.
ÍU): 3cos(* - t)
a . ) , f , s ( x )- ? a n d D p r = )
b.)(/"g)'(r): ?
c.)li*/"g(x):
?
(r) : ?) of the following functions:
5. (6 p.) a.)Determinethe2Al4th derivativesU<2014)
,ÍU)=@2_|)e,.
b.)limÍU) : ?
c.)lim .ÍU): z
x-l-
6. (6 p.) Evaluatethe integrals:
o) í ^[*zJax.
a.)r| ${a*,
xr(x- l)
7. (6 p.) E,valuate
integrals:
thedeflrnite
2
. f .Ji
--l
I
a.)
. J| ftdx,
11
b .' J) |
IX
-3
=2x+4 4r.
xz+6x+10
8. ( 6 p.)Drawtheareabetween
thevalueof the area
thegivencurvesandcalculate
s*pposingx>o:
y:;+
r,
I +x'
!:x2.
Narne:
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I. (BSc)- Exam Test
Mathernatics
Sth of Okt..2008.
1 . ( 5 p . ) F i n d a n e q u l a t i oonf t h e p l a n ec o n t a i n i n tgh e l i n e . r: - 2 + 3 t . ! : 4 + 2 t .
z : 3 - / a n d p e r p e n d i c u l at ro t l t ep l a r r ex - 2 1 t * z : 5 .
2 . ( 7 p . ) a . ) G i v e t h e c o m p l e x n u m b e r sz w , # i n a l g e b r a i cf o r m w h e r e
z : 2 + 3i and v) : -l - 2i and give the value of arg(z+ w).
c . ) S o l v e t h e e q u a t i o nz 6 + 8 l z 2 : 0 .
3 . ( 6 p . ) S l < e t c tl rh e g r a p ho [ . ( x ) :
#F.
t a n g e n tl i n e o f / ( x ) a t x s : - 2 .
a n d g i v e t h e e c i u a t i o no f t l r e
4 . ( 6 p . t l ' i r r dt i r et b l l o w i n g l i m i t s .i t ' t h e ye x i s t s :
a.)111;(t
*2x)z,*. b)lTl+t,
c.)ll'n(* - *)
5 . ( 8 p . )L e t/ ( x ) : f + c o s - r ( t- + x ; .
a) Findthedomainandtherangeof.f'and.f't.
b) Give the inversefunctionof it.
c) Findthederivative
of thefunctions
.f andJ't.
6. (9 p.)Evaluatetheintegrals:
a.)lfidx,
b) Í
kdx,
c.)Jx5|_2x4,,
7 . ( a p . ) F i n d t h e l i m i t o f e a c hc o n v e r g e nste q u e n c e :
a.)l,'_l,J(#)
"', b ) l,'-l.l
+*-
8 . ( j p . , lF i n d t i r ea r e ab e t w e e nt l r ec u r v e SJ , , : | ' Y +l , ! :
_ x : + 2 x + 3 a r r cJj, , : Ü .