No.
of Printed Pages : 8 I MTE-7 I
BACHELOR'S DEGREE PROGRAMME
Term-End Examination
•zr
June, 2010
..o
n ELECTIVE COURSE : MATHEMATICS
O
MTE-7 : ADVANCED CALCULUS
Time : 2 hours Maximum Marks : 50
Note : Question No. 1 is Compulsory. Solve any four form
Question No. 2 to 7 Calculators are not allowed.
1. State whether the following statements are true 10
or false. Give reasons :
The function f : R2 ---> R, f (x, ecos,rr is
continuous at (1, 2).
The domain of f/g, where
f (x, y) = 2sin x + sin y and g (x, y) = 2cos y
is R2 — i(o, 7/2)}.
Function F (x, y) = In x+Y is not a
Y
homogeneous function.
Every stationary point is a saddle point.
(e) The mass of a cube [0, 1] x [0, 1] x [0, 1] with
density given by 5(x, y, z) =1+ x is 3/ 2.
MTE-7 1 P.T.O.
2. (a) Evaluate the following limits : 5
. x — sinx
li m
x->0 X 3
lim (x — V X 2 + X)
x->00
(b) Show that the function f defined by 5
\_ 2y 2
f ( x y)i (x-02 + x 2 y2 ' (x, y) (0,0)
,
=0 , (x, y)= (0, 0)
is discontinuous at the origin.
3. (a) Describe and draw a rough sketch of the 2
level curves of f (x, y) = x2 + y.
du
If u = x2 + 02, x = sin2t, y = cost2, find — 3
dt
using the chain rule.
Check whether the repeated limits of the 5
function f :R2 --> R, defined by
‘ xy
f (x, y)
X2 + y2 ' (x, y) (0, 0)
= 0 , (x, y)= (0, 0)
exists or not ? Does the simultaneous limit
exist ? Justify your answer.
MTE-7 2
4. (a) Find the area of the region bounded by y = x2 3
and x = y2.
Find an approximation to the function 5
f (x, y) = sin(x + 2y) by a second degree
2 , 0).
polynomial at 1 2
Check if the function F : R 2 ----> R2 , defined 2
by F (x, y) = (exY , ex+Y ) is conservative.
5. (a) Find the points (x, y) on the unit circle at 5
which the product xy is maximum ?
(b) Draw a sketch of the region of integration 5
/1 x2
in dxdy and evaluate by
reversing the order of integration.
6. (a) Find the directional derivative of the 3
function f : R2 --> R defined by
f (x, y)= exY at (1, 0) in the direction 0 = 3 •
Calculate the Jacobian for the following 3
mapping :
w = x2 + cos y, z= yex at (L, ).
Calculate the work done by a force 4
F = (4x2y,2xy2) in moving a particle from
(0, 0) to (1, 1) along y= x2.
MTE-7 3 P.T.O.
7. (a) Using cylindrical coordinates, evaluate 4
2 J4 — x 2 1
I f f x2 dzdydx •
2 - J4 - x 2 ( x 2 + y 2 )2
State Implicit Function Theorem for two 3
variables. Use the Theorem to show that
there exists a unique solution of the equation
X eY y ex = 0 in a neighbourhood of the
point (0, 0).
If a=(1, 2) and b= (2, 0) are two points in 3
R2, then find I x — y I and I 3x —y I , where
x=a-2b and y=2a+b.
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1
f (x, y) = 2sinx + siny g (x, y) = cosy.
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--- ki li tild 1110-1 -ffe
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tI
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(e) f (x, y, z) =1+ x i i titi-fifird curl
[0, 1]x [0, 1] x [0, 1] 3/2 t I
MTE-7 5 P.T.O.
2. (a) F-H-ikifig o 4-Haff Maui Wit4R : 5
x - sinx
m
x-40 x3
lim (x — V-x 2 x)
x—>00
(b) tlITR : 5
x2y2
—
f (x ' Y) (x—y) 2 x2y2 , (x • y) (0, 0)
= 0 , (x, y) = (0, 0)
iti tiftiTTRIU Lholf 1f0f TIT 3-Tttdff t
3. (a) f (x, y)=x2 +y kciT Gish) a u k 3 2
7W TV' "i77
irc u= x 2 +0 2 , x=sin2t, y =cost2 sitg of 3
du
iti+i —
dt
ATri i 5
f (x, y)- xY (x, y) # (0,0)
x` + y `
=0 , (x , y) = (0, 0)
gRI trffiltrd f : R 2 R rutirT
Htaff 3Tft7.m t zrr 971? azrr TIrrq7ftgri
t? 3174 drIt iftz wrr--4R
MTE-7 6
4. y = x2 3 ix = y2 gm 1:111-44 5Rq1 TT tITR ;Ito 3
W-77
( 72- 0) ITT tt zT bud WETR g itT 5
f (x, = sin(x +2y) Wf I'd ii
(c) ^f . 1f77 f ITT T77 2
F (x, y) = ( e xY , e X+Y ) gl7T trft iltrd 4101
F:R 2 ->#T-111-71"Tel
5. r*--r/ TR (x,y)ic WIC- 717 5
xyZi g t I
ki4-1 1cbc1-1 f eX2 dXd Y 5
'47-4•7 3 w-li cnol TR drdi et) dlich I
itti TlfA7
6. (a) f:R 2 -> R, fT4T1 0 =
3
-4 (1, o) TR 3
f (x, y)=exli T -f-T 31-q-w-67 Ta. Tii7R
(b) (2,TR 4--4-41- 3
2
-urff :
w= x2 + cosy, z = yex.
(c) y = x2 k ar
lfqT (0, 0) tf (1, 1) cif W:IT *4' A- 4
‘111 1;17ST q ci F = (4x2y,2xy2) 'gRT T17
a.)14 4ficiirria W--A71
MTE-7 7 P.T.O.
7. (a) - . rI kli cb
40 91* I4 I .Wrk 1 4-ironsto T 4
te",-41chl :
2 V4 — x 2 1
f f fx
2 dzdydx
z — — x2 (x2 + y2 )2
re-R 3WRZ 41(1-1 31147W c1f‘IR I 3
Tr4zrgru f-4-gfrffw f-4-srr-d-4vr
ki+-1 Mx° yex = 0 t 3Trgtzr
3ifk71 t I
R2 t fF3If a= (1, 2) at b = ( 2, 0) i
t Sri 3
x—y I at 3x — y I Tff Wri-4R ‘714x=a-2b
y=2a+b.
MTE-7 8
