UNIVERSITY OF NAIROBI
UNIVERSITY EXAMINATIONS 2014/2015
FIRST YEAR CAT EXAMINATION FOR THE DEGREE OF BACHELOR OF SCIENCE
CSC 114 DIFFERENTIAL AND INTEGRAL CALCULUS
DATE: _______________TIME: _________________
Answer Question one and any other two Questions
Question I (30 marks)
[5]
(a) Find dy/dt of y = e3t sin 4t ; y’ =
( c) Find dy/dx of ye x  3 y x 2  4 ;
[5]
(b) Find dy/dx of
1
( d ) Evaluate
y=
2 xe 4 x
;
sin x
[5]
 3xe dx [5]
 3. [5]
(2 x2 1)
0
(e ) Find the area of the region bounded by x  y  3, y  x
( f ) Sketch the region R bounded by the graphs of the given equations and find the volume of the solid
2
generated by revolving R about the y axis: x  y 2 ; y  x  2  0 about the y axis. [5]
Question II (10 marks)
5sec x
; [3] ( b ) Find dy/dx of y  l n(4 x  5) tan3 x [3]
23 x
(c ) Given f(x) = x3  6 x 2  9 x  1; find x values corresponding to all local extrema, points of
(a) Find dy/dx of y 
inflections; and find regions of concavity; and where f increases/decreases; Sketch the graph of f.[4]
Question III (10 marks)
(a) Find dy/dx of y  52 x  2 x  1 .[3];
3
2/3
(b) Evaluate
2 x 2  9 x  35
 ( x  1)( x  2)( x  3) dx;
(c) Find the area of the region bounded by y 2  4  x, y  x  2.
Question IV (10 marks)
lim 3x 2  13x  10
2
( a ) Find x  5 2 x  7 x  15 ;
[3]
(b) Evaluate
. (c) Find the area of the region bounded by y  4  x 2 , y  5.
[3];
[4]
 3x ln 5xdx . [3]
[4]
Question V (10 marks).
(a) A window has the shape of a rectangle surmounted by an equilateral triangle. If the
perimeter of the window is 15 ft, find the dimensions that will allow the maximum
amount of light to enter.
[5]
( b ) Sketch the region R bounded by the graphs of the given equations and find the volume of the solid
generated by revolving R about the x axis: y  x 2 , y  4  x 2 about the x axis. [5]
-1-
1.  e 2 x sin 3xdx  =
1 2x
2
  cos 3x 
  cos 3x 
2x
2 x   cos 3 x 
2x
e ( sin 3x3dx)  e 2 x 
   e 2dx
e 
  2  e cos 3x3dx 

3
3
3
3





 3
e
2x
e
2x


2 2x
4
  cos 3x  2 2 x
2x
2 x   cos 3 x 
2x
sin 3xdx  e 2 x 
  2 e sin 3x  2  e sin 3xdx  e 
  2 e sin 3x  2  e sin 3xdx
3
3
3
3
3




sin 3xdx 

9  2 x   cos 3x  2 2 x
e 
  2 e sin 3x  Ans

13  
3
 3

2.
sin 2 x 5 ln 3 5 x
sin 2 x 5 ln 3 5 x

3 sin 2 xdx  35 x

3  sin 2 x 2dx  

2
2
2
4 
sin 2 x 5 ln 3  5 x
5 ln 3 5 x

35 x

3 cos 2 x 
3 cos 2 xdx


2
4 
1

3
cos 2 xdx  35 x
3
4
 sin 2 x 5 ln 3 5 x

cos 2 xdx  35 x

3 cos 2 x 
Ans
2
2
4

 4  5 ln 3
5x
5x
3.
2
 x sin 4 xdx 
(b)
 x2
cos 4 x
 x 2 cos 4 x
sin 4 x 1
cos 4 x   2 x
dx 
x
 cos 4 x Ans
4
4
4
8
32
lim 1  cos x
x  0 3 x2
=
lim 1  cos x
lim 1/ 3 lim 1  cos x

x
 o [5]
2/3
x0 x
x0
x0
x
1
1
(c ) Given y  52 x ln 3x ; dy / dx  52 x (2 ln 5 ln 3x  x )  52 x (ln 52 ln 3x  x ) [5].
1
(d )  t sin( 3t )dt =
0
t
cos(3t )
3
1
0
 3  sin( 3t ) 10 
(e) The area is bounded by x  2 y 2 ;

y 1 / 2
y  1
1  y  2 y dy  32 y
(f) Given 
2
3

y2
y
2
2
1
=0.106103[5]
3
x  y  1Sol: 2 y 2  y  1  0  2 y  1 y  1
1/ 2
1

1 1 1 2 1
7 9
     1  2   = 1.125[5].
12 8 2 3 2
8 8
2 xdx
; 2x  3A  Bx   A  3B;  9B  B  2; B  1 / 4; A  3 / 4 ;
x  33x  1
-2-
 x  33x  1   4x  3   123x  1  ln x  3
2 xdx
3dx
3dx
3/ 4
 ln 3x  1
1 / 12
+C[5]
Question II (20 marks)
 t e dt  t e
3 t
(a) E
3 t
 3[t 2 et  2[tet  et ]]  t 3et  3t 2 et  6et [t  1] +C[5]
(b)
lim x  4x  4
lim
lim x 2  16
=

 2  x x  4   32 [5]
x4
x 4 2 x x 4 2 x
(c)


 /3
/4

tan xdx = ln | cos x |  // 34  ln( 1 / 2)  ln 2 / 2  ln 2 / 2  0.3466 [5]
Question III (20 marks)
(a)
d csc 2 x
 32 x  2 csc 2 x cot 2 x   32 x 2 ln 3 csc 2 x [4].
2x
dx 3
(b)  sec 3 x tan xdx   sec 2 d sec x  (sec 3 x) / 3  c [4].
12]
Question IV (20 marks)
ln 3 x  2 
3 cot 3 x
(a) y 
; dy / dx 
 3 cot 4 x csc 2 x ln 3x  2 [6]
3
cot x
3x  2




(b)  cos 5 x sin 4 xdx   1  sin 2 sin 4 x cos xdx   sin 4 x  2 sin 5 x  sin 8 cos xdx  .
sin 5 2 sin 7 x sin 9 x


 C [6]
5
7
9
(c) The areaof the region bounded by y  x; y  5x  x 2

 4 x  x dx   2 x
4
2

0
=
4
2

x3 
 32  64 / 3  32 / 3  10.6667 [8]
3  0
Question V (20 marks) .
( c ) Sketch the region R bounded by the graphs of the given equations and find the volume of the
solid generated by revolving R about the indicated axis: x  y 2 ; x  1; about x  1.
1
1
16
V  2  2 1  x x1/ 2 dx  4  x1/ 2  x 3 / 2 dx  8 1 / 3  1 / 5 
 3.35  1.069
0
0
15

1

V    1 y
1

2 2

1

2 y3 y5 
16
dy   1  2 y  y dy   y 
  
 1.069  3.35
1
3
5  1 15

1

2
4

[7]
-3-
lim 4 x  tan 2 x
;[6]
x  0 sin 5 x
(a) Determine

(c) Evaluate
3
0


2 x 3x 2  1
1/2
dx;
[6]
(b) Find dy/dxof y  e5 x  tan 3x  ln 2 x  54 x [6].
(e) Find the area of the region bounded by
x  2 y  y 2 , x  3 y  2. [6]
Question II (20 marks)
lim x 2  16
(b) Determine
; [5] (c ) Evaluate :
x 4 2 x
 5x  4 x
Question III (20 marks)
5sec x
(a) Find dy/dx of 3 x ; [8Question IV (20 marks)
2
ln(4 x  5)
(a) Find dy/dxof y 
[6] (b) Evaluate
tan 3 x
2

3
 3 dx . [5
 3x ln 5xdx . [6] (c)
Find the area of the
region bounded by y  4  x 2 , y  4. [8]
Question V (20 marks) .
3x 2
3
(a) Given f(x) = x 
 18 x  5; find x values corresponding to all local extrema, points of
2
inflections; and find regions of concavity; and where f increases/decreases; then sketch the graph of f.
[10]
(b) Sketch the region R bounded by the graphs of the given equations and find the volume of the solid
generated by revolving R about the indicated axis: y  x 2 , y  4  x 2 about the x axis. [10]
lim 4 x  tan 2 x
(a) Determine
;[6](b) Find dy/dxof y  e5 x  tan 3x  ln 2 x  54 x [6].
x  0 sin 5 x
(c) Evaluate

3
0


2 x 3x 2  1
1/2
dx;
[6
(e) Find the area of the region bounded by x  2 y  y 2 , x  3 y  2. [6]
Question II (20 marks)
lim x 2  16
(b) Determine
; [5](c ) Evaluate :
x 4 2 x


3
2
 5 x 4 x  3 dx . [5
Question III (20 marks)
5sec x
ln(4 x  5)
(a)Find dy/dx of 3 x ; [(a) Find dy/dxof y 
[6] (b) Evaluate
2
tan 3 x
Find the area of the region bounded by y  4  x 2 , y  4.
 3x ln 5xdx . [6](c)
[8]
Question V (20 marks).
-4-
3x 2
 18 x  5; find x values corresponding to all local extrema, points of
2
inflections; and find regions of concavity; and where f increases/decreases; then sketch the graph of f.
(a) Given f(x) = x3 
[10]
(b) Sketch the region R bounded by the graphs of the given equations and find the volume of the solid
generated by revolving R about the indicated axis: y  x 2 , y  4  x 2 about the x axis. [10]
lim
2 x2  5x  3
(a) Determine
;[5](b) Find dy/dtof 52t ln 3t;
2
x  1/ 2 6 x  7 x  2
1
(c) Evaluate  3xe(2 x
2
1)
0
dx;
[5]
[5](d)
(e) Find the area of the region bounded by y  x 2  1, x  1; x  2; y  0 ; [5
Question II (20 marks)
lim 1  cos x
b) Determine
[5](c ) Evaluate
x  0 3 x2
5 x 2  2 x  19
  x  3 x  1
2
dx. [5
Question III (20 marks)
(a)
[8]
Find dy/dx of f(x) = e3 x tan x  ln csc2 3x ;
Question IV (20 marks)
2 xe 4 x
(a) Find dy/dx of f(x) =
; [6](b) Evaluate  x 2 sin xdx [6]
sin x
(c) Sketch the region R bounded by the graphs of the given equations and find the volume of the solid
generated by revolving R about the indicated axis: x  y 2 ; y  x  2  0 about the y axis. [8]
Question V (20 marks)
(a) Given f(x) = 2 x3  x 2  4 x  2; find x values corresponding to all local extrema, points of
inflections; and find regions of concavity; and where f increases/decreases; Sketch the graph of
f.[10]
(b) Find the area of the region bounded by x  y  3, y  x 2  3. [10]
lim 3x 2  13x  10
(a) Find
;
x  5 2 x 2  7 x  15
(c) Evaluate:
1
 2xe
0
6 x 1
dx;
[5](b
5
) Evaluate   3 x  1 dx;
1
0
[5]
[5 (e
Question II (20 marks)
lim 2 x3  6 x 2  x  3
; [4]Question III (20 marks)
(a) Find
x 3
x 3
(a) Find dy/dtof y  52 x  2 x  1 .[6](b) Evaluate
3
2/3
 /3
 3sin
2
3xdx;
[6](c) Find the area of the
0
region bounded by y 2  4  x, y 2  x  2. [8]
-5-
Question IV (20 marks)
1
(a) Find dy/dxof ye  3 x  4 ;[6](b) Evaluate
x
y
2
 5xe
4x
dx;
[6]
0
(c) Sketch the region R bounded by the graphs of the given equations and find the volume of the solid
generated by revolving R about the indicated axis: y  x 2  4 x, y  0. [8]
Question V (20 marks) .
2 x 2  9 x  35
(a)Find dy/dtof e sin 4t; [6] (b) Evaluate 
dx; [8]
( x  1)( x  2)( x  3)
(c ) Given f(x) = x3  6 x 2  9 x  1; find x values corresponding to all local extrema, points of
3t
inflections; and find regions of concavity; and where f increases/decreases; Sketch the graph of f.[6]
Vertical, Horizontal and ObliqueAssymptotes:
x2  9
5
 x / 2 1
with oblique assymptotes y  x / 2  1; VA : x  2;
2  x  2
2x  4
2 x2
2 x2

; VA : x  3; x  3; H . A : y  2
9  x 2  3  x  3  x 
Sketch the graphs of f and determine the asymptotes for
3
;
2x 1
x2  4 x2  3
;
x 1 x  2
lim x  8
lim

x 2/3  2 x1/3  4  12 Ans ; [4]
3
x 8 x 2
x 8
2
lim
2 x  5x  3
lim
(2 x  1)( x  3)
3.5
(b)


 7 Ans ;[4]
2
x  1 / 2 6 x  7 x  2 x  1 / 2 (2 x  1)(3 x  2) 0.5
lim 1  cos x
lim 1/3 1  cos x

x
 0.0  0 Ans ; [4]
(c)
3
2
x0
x0
x
x
1. (a)


2. (a) Given f(x) = e3 x tan x  ln csc2 3x ;
1
6csc2 3x cot 3x
f’(x) = 3e3 x tan x  e3 x
sec2 x 
csc2 3x
2 x
1
 3e3 x tan x  e3 x
sec2 x  6cot 3x Ans[5]
2 x
3
sec x
cot (3x  1)3sec3 x tan x  3csc2 (3x  1)sec3 x
(b) y 
; y’ =
Ans[5]
cot(3x  1)
cot 2 (3x  1)
3
3. (a) (52t ln 3t )'  (52t 2ln 5ln 3t )  (52t )
Ans[5]
3t
2sin xe 4 x  4 x  1  2 xe 4 x cos x sin x  4 x  1 2e 4 x  2e 4 x x cos x
2 xe 4 x
(b) (
)' 

Ans[5]
2
2
sin x
 sin x 
 sin x 
f’(x) = 3x 2  4 x  1  3  1  2. x 
4  42  12 2  2 7
22 7
; hence

or
6
3
3
-6-
x3  2 x 2  x  3 on [1,1] satisfies the Hypotheses of Mean Value theorem. Ans[10]
5. Given f(x) = (2 x3  x 2  4 x  2)''  2(3x 2  x  2)'  2[(3x  2)( x  1)]'  2(6 x  1)
At x = -2/3 f is max; at x = 1; f is min.
Pi is at x = 1/6; hence incr. on (  , 2 / 3
1,  ; fdecr. on  2 / 3,1.
f is CD on  ,1/ 6 . and is CU on  1/ 6, . Ans[8]Sketch the graph of f.
 /3
6.  (2sec2   3cos2 )d   2tan  / 3  (3 / 2)sin 2 / 3   2tan  / 6  (3 / 2)sin  / 3
 /6

3 3 2 3 3 3
2 3 4 3
 2 3 


Q E D [10]

2
4   3
4 
3
3

5 x 2  2 x  19
A
Bx  B
C
Ax 2  2 Ax  A  Bx 2  x(C  B)  3Bx  3(C  B)
7.




2
2
 x  3 x  1 x  3 ( x  1)2 ( x  1)2
 x  3 x  1
5 = A + B; -2 = -2A + C – B +3B = -2(5-B) +C+2B = -10 +4B + C
8 = 4B + C; -19 = 5 – B -3B + 3C ; -24 = -4B +3C = -4B +24 – 12B
-48 = -16B. B = 3; A = 2; C = -4
5 x 2  2 x  19
2dx
3dx
4dx
.
dx.  


 2ln( x  3)  3ln  x  1  4( x  1) 1  C
2
2
x

3
x

1
(
x

1)
 x  3 x  1
2ln( x  3)  3ln  x  1  4( x  1)1  ln( x  3)2  x  1  4( x  1)1 Ans[10]
3
=
Ans
[10]
9.
Evaluate
1
0
10.
x
2
x
2
1


2
2
 2 tan 2tdt    2 1  sec 2t dt   2t  tan 2t 0  tan 2  2 Ans[10]
1
0
sin xdx   x 2co s x  2  xco s xdx;  xco s xdx  x sin x   sin xdx  x sin x  cos x


sin xdx   x 2co s x  2  x sin x  cos x   cos x 2  x 2  2 x sin x Ans[10]
-7-