GTU BE 1st and 2nd Semester 110014 Calculus Summer 2018 Question Paper

1
Seat No.: ________ Enrolment No.___________
GUJARAT TECHNOLOGICAL UNIVERSITY
BE- SEMESTER? 1
st
/ 2
nd
? EXAMINATION ? SUMMER 2018
Subject Code:110014 Date: 21-05-2018
Subject Name: Calculus
Time: 02:30 pm to 05:30 pm Total Marks: 70
Instructions:
1. Attempt any five questions.
2. Make suitable assumptions wherever necessary.
3. Figures to the right indicate full marks.
Q.1 (a) (1)Evaluate
?
?
?
?
?
?
?
?
?
?
?
1 log
1
lim
1
x
x
x
x
03
(2)Show that the sequence
?
?
?
?
?
?
? 1
2
n
n
is monotonic decreasing and bounded. Is it
convergent?
04
(b)Find expansion of ?
?
?
?
?
?
?
4
tan
?
x is ascending powers of x upto terms in
4
x and
find approximately the value of ) 43 tan(
?
07

Q.2 (a)(1) Find expansion of ). 1 log( x ? 03
(2)Test the convergence of the series …
! 4
4
! 3
3
! 2
2
1
2 2 2
? ? ? ?
04
(b)Determine absolute or conditional convergence of the series.
?
?
?
?
?
1
3
2
1
) 1 (
n
n
n
n
07

Q.3 (a) (1)Evaluate
?
?
1
2
1
dx
x
03
(2) Find the linearization of xz yz xy z y x f ? ? ? ) , , ( at the point (1,0,0) 04
(b) Trace the curve 0 ), cos 1 ( ? ? ? a a r ? 07

Q.4 (a)(1)Show that y x y x f 2 ) , (
2
? ? is continuous at (1,2).
03
(2)If
2 2 2 1
tan a y x where
y
x
u ? ?
?
?
?
?
?
?
?
?
?
?
find
dx
du
.
04
(b)State Euler?s theorem. If ?? = ?????? -1 (
?? 2
+?? 2
?? +?? ),prove that
?? 2
?? 2
?? ?? ?? 2
+ 2????
?? 2
?? ???????? + ?? 2
?? 2
?? ?? ?? 2
= -2?????? 3
??????????
07

Q.5 (a) (1)If
t
e y t x y x u ? ? ? , log ,
3 2
find
dt
du
03
(2) Find the equation of the tangent plane and normal line to the surface (1,1,1). point at the 3
2 2 2
? ? ? z y x
04
(b)Change the order of integration and Evaluate for
? ?
a ax
a
x
dydx xy
4
0
2
4
2
07
1
Seat No.: ________ Enrolment No.___________
GUJARAT TECHNOLOGICAL UNIVERSITY
BE- SEMESTER? 1
st
/ 2
nd
? EXAMINATION ? SUMMER 2018
Subject Code:110014 Date: 21-05-2018
Subject Name: Calculus
Time: 02:30 pm to 05:30 pm Total Marks: 70
Instructions:
1. Attempt any five questions.
2. Make suitable assumptions wherever necessary.
3. Figures to the right indicate full marks.
Q.1 (a) (1)Evaluate
?
?
?
?
?
?
?
?
?
?
?
1 log
1
lim
1
x
x
x
x
03
(2)Show that the sequence
?
?
?
?
?
?
? 1
2
n
n
is monotonic decreasing and bounded. Is it
convergent?
04
(b)Find expansion of ?
?
?
?
?
?
?
4
tan
?
x is ascending powers of x upto terms in
4
x and
find approximately the value of ) 43 tan(
?
07

Q.2 (a)(1) Find expansion of ). 1 log( x ? 03
(2)Test the convergence of the series …
! 4
4
! 3
3
! 2
2
1
2 2 2
? ? ? ?
04
(b)Determine absolute or conditional convergence of the series.
?
?
?
?
?
1
3
2
1
) 1 (
n
n
n
n
07

Q.3 (a) (1)Evaluate
?
?
1
2
1
dx
x
03
(2) Find the linearization of xz yz xy z y x f ? ? ? ) , , ( at the point (1,0,0) 04
(b) Trace the curve 0 ), cos 1 ( ? ? ? a a r ? 07

Q.4 (a)(1)Show that y x y x f 2 ) , (
2
? ? is continuous at (1,2).
03
(2)If
2 2 2 1
tan a y x where
y
x
u ? ?
?
?
?
?
?
?
?
?
?
?
find
dx
du
.
04
(b)State Euler?s theorem. If ?? = ?????? -1 (
?? 2
+?? 2
?? +?? ),prove that
?? 2
?? 2
?? ?? ?? 2
+ 2????
?? 2
?? ???????? + ?? 2
?? 2
?? ?? ?? 2
= -2?????? 3
??????????
07

Q.5 (a) (1)If
t
e y t x y x u ? ? ? , log ,
3 2
find
dt
du
03
(2) Find the equation of the tangent plane and normal line to the surface (1,1,1). point at the 3
2 2 2
? ? ? z y x
04
(b)Change the order of integration and Evaluate for
? ?
a ax
a
x
dydx xy
4
0
2
4
2
07

2
Q.6 (a) (1)Evaluate ? ?
? ? ?
?
?
1
1
2
0
1
0
3
dzdydx y xz
03
(2) State fundamental theorem of calculus. Use first fundamental theorem of
calculus to find area under the curve f(x) given as an integrand.
?
2
1
log dx x
04
(b)Find the extreme values of the function 7 3 3 3
2 2 2 3
? ? ? ? y x xy x
07

Q.7 (a) (1)Evaluate ? ?
? ?
?
2
1
1
0
3 1 dxdy xy
03
(2)If ? ? sin , cos r y r x ? ? find
) , (
) , (
) , (
) , (
y x
r
and
r
y x
?
?
?
? ?
?
04
(b) Find the volume generated by revolving the area bounded by
0 , 4 , 2
2
? ? ? y x x y about x -axis.
07
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