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MTH301 Calculus
Final Term Examination – Spring 2005
Time Allowed: 150 Minutes

Please read the following instructions carefully before

attempting any of the questions:

1-Duration of the paper is 150 minutes and use of calculator is

allowed
2. Attempt all questions. Marks are written adjacent to each
question.
3. Do not ask any questions about the contents of this
examination from anyone.
a. If you think that there is something wrong with any of
the questions, attempt it to the best of your understanding.
b. If you believe that some essential piece of information is
missing, make an appropriate assumption and use it to
solve the problem.
c. Write all steps, missing steps may lead to deduction of
marks.
4- Pasting the equations of math type from word file into software may
cause some visibility problem, so please note that do not copy equations
of math type into software from word file. Paste the equations from math
type directly into software.

Total Marks: 65 Total Questions: 12

Question No. 1 Marks : 08

Evaluate the iterated integral

∫∫ x
R
(1− x2 ) dA R ={(x, y) :0 ≤ x ≤1,2 ≤ y ≤ 3}
 Question No. 2 Marks : 06

D r f (−2, 0) 2
f ( x, y) = x − 3xy + 4 in the direction
Find the directional derivative u of
r 3
u = $i + 2 j y
of at the point (-2, 0).

Question No. 3 Marks : 05

2
Find the centre and radius of the sphere x y2 +z2 −8x+6y +12z = 3.
+

Question No. 4 Marks : 02

r 4
a = i$ + $j r
Let 3 and
b = 4 $i 3 ,then
+ $
j
r r
o a and b
are orthogonal
π
o The angle between a and b is 4 .
o a and b are parallel.

r r π
o The angle between a and b is 6 .

Question No. 5 Marks : 02

The Laplace transform of

e2t is

1
2 if s > 2
o s +4
1
if s > 2
o s −4
1
if s > 2
o s +2
1
if s>2
o s −2

Question No. 6 Marks : 02

2
 π
If (4, 3 ,- 4) is a point in the Cylindrical coordinates then the same point in Rectangular
coordinates is given by

o (2,2 3 ,4)

o (2,2 3 ,4)
o (2,2 3 ,-4)
o (- 2,2 3 ,4)
o (2 3 ,2,-4)

Question No. 7 Marks : 02

The differential
dz = Mdx + Ndy is an exact differential if we have

∂M ∂N
=
∂y ∂y
o
∂M ∂N
=
∂y ∂x
o
∂M ∂N
=
∂x ∂y
o
∂M ∂N
=
∂x ∂x
o

Question No. 8 Marks : 10

ur 2$ 3 $ 2 $
Compute Curl and Divergence for the given vector field F = x i + 4 xy j + y xk .

Question No. 9 Marks : 02

The function
y = x2/ 3 + 6 is

o Even function
o Odd function
o Neither even nor odd
o Constant
 Question No. 10 Marks : 06

Determine whether the differential dz = ( xlny + xy) dx + ( ylnx + xy) is exact

or not? dy

Question No. 11 Marks : 10

s − 19
L { s 2 + 3s − 10}
-1
Determine by using the concept of partial fraction.

Question No. 12 Marks : 10

Locate all the relative maxima, relative minima and saddle points for function
f(x, y) = x 2 +y2 - x 2 y
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FINALTERM EXAMINATION
SPRING 2006 Marks: 55
MTH301 - CALCULUS II (Session - 2 ) Time: 120min

StudentID/LoginID:

Student Name:

Center Name/Code:

Exam Date: Thursday, August 17, 2006

Please read the following instructions carefully before attempting any

of the questions:
1. Attempt all questions. Marks are written adjacent to each question.
2. Do not ask any questions about the contents of this examination
from anyone.
a. If you think that there is something wrong with any of the
questions, attempt it to the best of your understanding.
b. If you believe that some essential piece of information is
missing, make an appropriate assumption and use it to solve the
problem.
c. Write all steps, missing steps may lead to deduction of marks.
3. Calculator is allowed.

**WARNING: Please note that Virtual University takes serious note of unfair
means. Anyone found involved in cheating will get an `F` grade in this
course.

For Teacher's use only

Question 1 2 3 4 5 6 7 8 9 10 Total
Marks
Question 11 12 13
Marks

Question No: 1 ( Marks: 2 ) - Please choose one

Laplace transform of ‘t ‘ is
 ► 1
s

► 1
2
s
► e− s
► s

Question No: 2 ( Marks: 2 ) - Please choose one

Symmetric equation for the line through (1,3,5) and (2,-2,3) is

► y+2 z−3
x−2= − =−
3 5

► x+2= − y+3= − z+5

5 2

► y−3 z−5
x − 1= − =−
5 2

► x + 1= y + 3 = z−5
5 5

Question No: 3 ( Marks: 1 ) - Please choose one

The level curves of f(x, y) = y Cscx are parabolas.

► True.

► False.
Question No: 4 ( Marks: 1 ) - Please choose one

z= r
The equation is written in

► Rectangular coordinates

► Cylindrical coordinates

► Spherical coordinates

► None of the above

Question No: 5 ( Marks: 5 )
 1 3
∫ ds, 2

C
1+x x=t , y= t ,0≤t≤3
3

Question No: 6 ( Marks: 5 ) 2

∫∫ (3x + y 2 − z)dV over the box 0 ≤ x ≤ 2 , − 1 ≤ y ≤ 1 , − 3 ≤ z ≤ 0

Q

Evaluate

Question No: 7 ( Marks: 10 )

Us e St okes theorem to evaluate the integral

J JG G
v∫
C
F.d r ,
JG ∧ ∧ ∧
where F = 2z i + 3x j + 5 y k , C is the circle x 2 + y 2 = 1
In the xy-plane with counter clockwise orientation looking down the positive z-axis.

Question No: 8 ( Marks: 3 )

f wwx for function 2

f (w, x, y, z) = w xyz − 5e
wz
2

Find

Question No: 9 ( Marks: 4 )

Locate critical points for the function

a 3 b3
f ( x, y) = xy + + (a ≠ 0 , b ≠ 0)
x y

Question No: 10 ( Marks: 4 )

z= x +y , y= 4− x ,
2 2 2

Set up a double integral for the volume bounded by surface and first
octant.
(Do not evaluate.)

Question No: 11 ( Marks: 4 )

G G
u and v
Determine whether make an acute angle ,an obtuse angle , or are orthogonal.
G ∧ ∧ ∧ G ∧ ∧ ∧

u = 7 i +3 j +5k , v = − 8 i + 4 j +2k
Question No: 12 ( Marks: 4 )

JG ∧ ∧ ∧
JG
F = x 2 i − 2 j + yz k F
Given .Find div(Curl ).

Question No: 13 ( Marks: 10 )

a0 , an a n d bn
Determine the Fourier coefficients for a periodic function of period 2 π defined
by

⎧−x −π≤ x<0

f(x) = ⎨
⎩0 0≤ x<π
 www. vujannat.ning.com

MTH301 Calculus II
Mid Term Examination – Spring 2006
Time Allowed: 90 Minutes

Please read the following instructions carefully before

attempting any of the questions:
1. Attempt all questions. Marks are written adjacent to each
question.
2. Do not ask any questions about the contents of this
examination from anyone.
a. If you think that there is something wrong with any of the
questions, attempt it to the best of your understanding.
b. If you believe that some essential piece of information is
missing, make an appropriate assumption and use it to solve
the problem.
c. Write all steps, missing steps may lead to deduction of
marks.
3. Calculator is allowed.

**WARNING: Please note that Virtual University takes serious note of

unfair means. Anyone found involved in cheating will get an `F` grade in
this course.

Question No. 1 Marks : 5

Find the directional derivative D uG f (1 , − 1,1) f ( x, y, z) = x 2 z + y 3 z 2 − in the

G of xyz
direction of u = (−1, 0 , at the point (1 , − 1,1) .
3)

Question No. 2 Marks : 1

1
∫ 4x 2
+
2
dx = ln 4 x + x + c

x
 True.

ﾐ False

Question No. 3 Marks : 10

Find
∂z ∂z
and , where z(s, t) = f (r(s, t), θ (s, t)) ,
∂s ∂t
2r
f (r, θ ) = e sin(3θ ) , r(s, t) = st − , θ (s, t) = s2 + t 2
2
t

Question No. 4 Marks : 10

Compute the volume of the solid bounded by the given surface

z =1 − y , z = 0 , y = 0 , x = 1, x = 2

Question No. 5 Marks : 1

Not all critical points need to be local extrema.

ﾐ True.
ﾐ False

Question No. 6 Marks : 1

f (x, y) = 3 − x 2 − is continuous at
2
y

 x2 + y 2 = 3
2 2
ﾐ x +y ≥3
2 2
 x +y ≤3
 x2 + y 2 < 3

Question No. 7 Marks : 5

Find an equation of the plane through the origin that is parallel to the plane:
4 x − 2 y + 7 z + 12 = 0
Question No. 8 Marks : 10

Use double integrals to compute the area of the region bounded by the curves:
2
y = x and y = 8 − x 2

Question No. 9 Marks : 1

If f ′(a ) is +ve then the graph of function is concave up.

 True.

 False

Question No. 10 Marks : 1

Direction cosines of straight line through A(1,1,1) and B(0,0,-1) are

2 2 1
 , ,
3 3 3
2 −1 1
 , ,
3 3 3
1 1 2
 , ,
6 6 6
 None of these.
 MTh301-Calculus II
Midterm Special 2006
www.
vujannat.ning.com
¾ At critical points function can assume
o maximum value.
o minimum value.
o both maximum and minimum value.
o zero value

¾ A plane can be perfectly determined by

o one point.
o one point and normal vector.
o one point and parallel vector.
o one normal vector.

¾ The length and width of a rectangle are measured with the errors of at most 4 %
and 3%, respectively. Use differentials to approximate the maximum percentage
error in the calculated area.
Solution:
Let
x = length

y = width

A = area

Then
A = xy So
∂A
=

y
∂x

∂A
=x
∂y

Therefore,

∂A ∂A
∂A ≈ + ∂y
∂x ∂y
= y∂x + x∂y
We desire percentage change in A, which is relative change multiplied by 100 so
let’s work out relative change first. This is given by
∂A y∂x x∂y
≈ +
A A A
∂x
= ∂y +x y

Since

A = xy

∂x ∂y
−0.04 ≤ ≤ 0.04 and −0.03 ≤ ≤ 0.04
x y

∂A
The maximum possible value for given the above constraints. This happens
A
∂x ∂y ∂A
when = 0.04 and = 0.03 , giving = 0.07 . This is relative error, so the
x y A
percentage error is 7%.

Convert the Cartesian coordinates (1, 2 , −2) into cylindrical coordinates.

Solution:

We are to convert (x,y,z) int (r, θ ,z)

Given:

x=1

y= 2

z = −2

We know that:

2 2
r= x +y

Put value of x and y in the above equation:

 2 2
r = (1) + 2 )
(

r= 1+ 2

r= 3

y
tan θ =
x

−1 y
θ = tan ( )
x

−1 2
θ = tan ( )
1

−1
θ = tan 2

θ = 54.73

z = −2

So the cylindrical coordinates are ( 3,54.73,−2)

======================================================
G ^ ^ ^
a = i+ j − k
¾ A unit vector in the direction of is

1 1 −1
, ,
o 3 3 3
1 1 −1
, ,
o 3 3 3
1−1 −1
, ,
o 2 2 2
1 1 −1
, ,
o 2 2 2

Find equation of the tangent plane to the surface z = x 3 + at the point (2,1, 3) .
Solution:
y3
f x = 3x − y
fy =3y−x

f z = −1

fx ( P) =6−1=5

fy ( P) =3−2=1

fz ( P) = −1

Equation of tangent line to the surface through P is:

5 ( x − 2 ) + 1 ( y − 1) − 1 ( z − 3 ) = 0

5x −10 + y −1 − z + 3 = 0

Hence:

5x + y − z − 8 = 0
======================================================
G ^ ^ ^ G ^ ^ ^
a = i + 2 j + k and b = 2 i + 3 j + 2 k
Dot product of is
o 8
o 5
o 10
o 13
∫∫R x 2 dA ;
¾ Evaluate the double integral R is the region bounded
16
y= , y = x , and x = 8
by x .
Solution:
∫∫
2
Area of R = x dA
R
8 x
=∫ ∫ 16 x2 dydx
2
x

x
8 x
=∫ ∫ 1 6
2
16 dx
2
x x
x
8
= ∫ x 3 − 16 xdx
2
 4
8 x
16x
2
=∫ −
2 4 2
4
8 x
=∫ − 8x
2
2 4

⎛ 84 ⎞ ⎛ 24 ⎞
= ⎜ − 8 × 8 ⎟− ⎜ − 8× 2⎟
⎝ 4 ⎠ ⎝ 4 ⎠
= 972

x+ y
¾ Domain of is
x+ y> 0
o
≥ ≥
o x 0,y 0
o x > 0, y > 0
o x+ y≥ 0

¾ Use double integrals to compute the volume of the solid bounded by the
cylinder x 2 + y 2 = 9 and the planes y + z = 9 and z = 0
Solution:
∫∫R
(9 − y)da

R as a type 1 region we find that:

2 9− x 2
V= ∫∫
−2 − 9− x 2
(9 − y ) dydx

2
2 ⎡ 1 9−x
= 9y− y
2⎤
∫ −2
2 2
dx
y= − 9−x

2
2

=
∫ 18 9 − x dx
−2

⎛1 2 ⎞
= 18⎜ π 9 ⎟ = 729π
2
⎝ ⎠