Mathematics Department
King Abdul Aziz University Faculty of Sciences
Time 120 m
Final Test Spring 2011 (40 Marks)
Math 110
Student Number:
Student Name:
~
1) The domain of log, (x + 3) is
~ (-00,00)
~ (0,00)
I£]
[4] (-3,00)
(3,00)
limx-1_
2) x--*l lnx ~oo
3)
f
If
(x) = coC\x ) and g(x) = cot(x) then
[J
~1
{x ElR:X
cotx cot" x
f
4)The function
~
(x) =
:;t±7}
2 -
if
0
g )(x) =
[4]
I£] x
cotx
x +1
is continuous on
X2 -49
~[-7,7]
I4J {±7}
I£] (-00,-7)U(7,00)
5) If x
[4]0
1£]1
1B2
4 = 3xy - y
~ 3y -2x
2y -3x
2x
I£]
3-2y
IB~
[4]
2,
then y , =
y
2x + Y
3x -2y
6) If Y =3x tanx , then y' =
~
IB 3x ln3tanx +3x sec" x
3x ln3tanx _3x sec+x
I£] 3x tanx - 3x sec" x
I4J 3x tanx + 3x sec" x
lnx
7) If f (x ) = -2 ' then f '(1) =
x
[4]1
~O
I£] 2
1B4
8) If Y =(2x
~ 7( 2x 2 + cscx
I£] 7 (2x
2
r
2
+ cscx
r,
then y' =
(4x - cscx cotx )
+ cscx )\ 4x + cscx cotx )
[J
7(2x
2
+cscx
I4J 28x ( 2x
2
r
+ cscx
r
9) The absolute minimum point of f (x) = 3x 2 -12x + 2 in [0,3] is
~
(3,-7)
IB (0,2)
I£] (2,-10)
[4]
(2,-12)
9
If In(3 - x ) = 5, then x
10)
~3e-5
=
14]
3 _e5
~-2
l£] 3-e
11) The slope of the perpendicular line to the line 2y + 3x - 6 = 0 is
2
[qJ2
~-l£]-~
3
3
2
2
x-I
lim
=
12)
x--+2X 2(X +2)
14]~
~!
~ does not exist
14]~16
8
csc-1(2) =
13)
~7r
[qJ7r
2
l£]7r
4
6
lim tan5x
x--+o 3x
14)
1
~-3
~
l£]!
4
I4J7r
3
=
3
14]53
[£]-5
~5
15)
If f (x ) = 2x + 3, then
f
x +3
2
~~-3
2
I4J x -3
2
(x) = cosx , then
(x ) =
l£] ~+3
2
16) If
f
~sinx
f
(47)
-1
(x ) =
~-sinx
.J
17) lim ( x 2 + X
x--+ 00
1
~-2
- X )
14] -cosx
l£] cos x
=
14] -!2
[£]0
~1
18) If Y = sin 4(3x ), then y =
I
[qJ 12sin 3 (3x ) cos (3x )
~
l£] 3cos" (3x )
14] 3sin4(3x)
19)
~120o
4sin3 (3x )cos(3x )
+ 12sin3 x cosx
27r
-rad=
3
~1500
[£] 2700
14] 2100
10
-
--
--~-----
If e -x = 5 , then x =
20)
~ 5-1
~-ln5
~ln5
~-5
21) The tangent line equation to the curve y = ~
x-I
~
y =-2x-1
~y
=2x +1
~y
at the point (0,0) is
=2x
~y
=-2x
22) If the graph of the function f (x) =ex is shifted a distance 2 units to the right,
then the new graph represented the graph of the function
x
x 2
[] eX +2
[£] ex-2
IEJ e +
l4Je -2
23) The distance between the points (-1,2) and (2,-1) is
~ 213
1EJ3
~3fi
~9
24) The horizontal asymptote of f (x) = I-x
is
3x +1
IEJ x
~y=-
~ y=~
=!
3
3
25)If Ix + 41= 3 , then x =
1EJ7
[] 1or
~
3
kJ -1 or -7
7
~x
=--
1
3
[4]1
26) Ify =sin-1(eX),theny'=
x
~_
1
.J1-e
eX
.J1-e
~-
27)
~
2x
~
2x
29)
1EJ00
1
.J1-e2x
~ [0,(0)
~ (0,00)
~ (-00,0)
limx+1=
x~3- x -3
~-3
~oo
~3
2x
The range of e" is
~ (-00,00)
28)
e
.J1-e
~-oo
limx-1=
x~1 lnx
[£]0
~2
[4]1
30) If Y =xx, then y' =
~
l s- lnx
~
XX
~ X X(1 + lnx )
~
XX lnx
11
31)
Thevalues in (0,2) whichmakesf(x)=x3-3x2+2x
Rolle's Theorem on [0,2] are
413
13
~ 1+-E[0,2]
6
32)
~
~ (-00,-2)
~
~
~
IB -2,1
~
~
(-oo,-X)
39)
~
(-oo,-X)
40)
~
3
(-1,2)
(1,-){)
(X, -){2)
[ill (-00, -2] U [2,00)
I£](-2,2)
f (x ) = l.x 3 +.!.x 2 - 2x + 1 are
~
2
I£] (_2,I~O
(_1,1%)
~
~
3
2
2
-2x +1
I£] (- X,oo)
f (x ) = -x1
3
3
1
+ -x
2
2 -
I£] (- X,oo)
f (x) = l.x 3 +.!.x
3
IB (X,~)
I4J (2,%)
(_2,1%)
f (x )=_x13 +-x1
(-oo,X)
The function
(2,%)
2
~
(-oo,X)
The graph of
~
f (x) = l.x 3 +.!.x 2 - 2x + 1 has a relative minimum point at
3
The graph of
I4J (-00, -1) U (2, 00)
I£](-2,1)
f (x) = l.x 3 +.!.x 2 - 2x + 1 has a relative maximum point at
(-1,l%)
~
(-oo,-1)U(2,00)
2
3
The function
~
(-2,1)
f (x ) = l.x 3 +.!.x 2 - 2x + 1 is decreasing on
(-oo,-2)U(1,00)
[]
~ -1,-2
2
3
The function
2
.
f (x) = l.x 3 +.!.x 2 - 2x + 1 IS.. mcreasmg
on
The function
38)
~
The function
35)
37)
U (2,00)
I£]-1,2
IIJ (-oo,-2)U(1,00)
(1,-}{)
.
3
(-1,2)
36)
(0,2)~ 1+-E(0,2)
6
IS
The critical numbers of the function
~ 1,2
34)
13
(0,2)1£]-1+-E
3
Th e d omamo
. f ~ x +3
4_x2
[-2,2]
33)
13
~1 +-E
3
+5 satisfied
2 -
2
I£](- X ,2J(2)
~
concave upward on
(X,oo)
2x + 1
concave downward on
[4] (X, 00)
2x + 1
has an inflection point at
I4J (- X, 4%4)
12