Download PTU B.Sc (Non-Medical) 2020 March 1st Sem 75746 Differential Calculas Question Paper

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1 | M-75746 (S105)-1773
Roll No. Total No. of Pages : 02
Total No. of Questions : 09
B.Sc. (Non Medical) (2018 & Onwards) (Sem.?1)
DIFFERENTIAL CALCULAS
Subject Code : BSNM-105-18
M.Code : 75746
Time : 3 Hrs. Max. Marks : 50
INSTRUCTIONS TO CANDIDATES :
1. SECTION-A is COMPULSORY consisting of TEN questions carrying ONE mark
each.
2. SECTION-B contains FIVE questions carrying FIVE marks each and students
have to attempt any FOUR questions.
3. SECTION-C contains THREE questions carrying TEN marks each and students
have to attempt any TWO questions.

SECTION-A
1. Answer briefly :
a) Define sequence.
b) Define limit inferior with example.
c) Define Left Hand Limit.
d) Define uniform continuity.
e) Define Right hand derivatives.
f) Find
( , )
( , )
f g
x y
?
?
if f = x
2
? x sin y and g = x
2
y
2
+ x + y
g) Show that the function f (x, y) = | x | + | y | is continuous at the origin.
h) State Euler?s Theorem on homogeneous function.
i) Prove that a real polynomial function is continuous everywhere.
j) Give an example of a decreasing sequence which diverges to ? ?.

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1 | M-75746 (S105)-1773
Roll No. Total No. of Pages : 02
Total No. of Questions : 09
B.Sc. (Non Medical) (2018 & Onwards) (Sem.?1)
DIFFERENTIAL CALCULAS
Subject Code : BSNM-105-18
M.Code : 75746
Time : 3 Hrs. Max. Marks : 50
INSTRUCTIONS TO CANDIDATES :
1. SECTION-A is COMPULSORY consisting of TEN questions carrying ONE mark
each.
2. SECTION-B contains FIVE questions carrying FIVE marks each and students
have to attempt any FOUR questions.
3. SECTION-C contains THREE questions carrying TEN marks each and students
have to attempt any TWO questions.

SECTION-A
1. Answer briefly :
a) Define sequence.
b) Define limit inferior with example.
c) Define Left Hand Limit.
d) Define uniform continuity.
e) Define Right hand derivatives.
f) Find
( , )
( , )
f g
x y
?
?
if f = x
2
? x sin y and g = x
2
y
2
+ x + y
g) Show that the function f (x, y) = | x | + | y | is continuous at the origin.
h) State Euler?s Theorem on homogeneous function.
i) Prove that a real polynomial function is continuous everywhere.
j) Give an example of a decreasing sequence which diverges to ? ?.

2 | M-75746 (S105)-1773
SECTION-B
2. State and prove Cauchy?s first theorem on limits.
3. State and prove Bolzano?s Intermediate Value Theorem.
4. Prove that the function f (x, y) = | | xy is not differentiable at the origin but it is
continuous at the origin, both fx, fy exist at the origin & have the value 0.
5. Apply Taylor?s Theorem with Lagrange?s form of remainder to the function f (x) = log x
in [1, x].
6. If v
1
= x
1
+ x
2
+ x
3
+ x
4

v
1
v
2
= x
2
+ x
3
+ x
4

v
1
v
2
v
3
= x
3
+ x
4

v
1
v
2
v
3
v
4
= x
4
, show that
3 2 1 2 3 4
1 2 3
1 2 3 4
( , , , )
( , , , )
x x x x
v v v
v v v v
?
?
?


SECTION-C
7. a) Show that the alternating series
1
1
1 2 3 4 5
( 1) .........
1 2 3 4
n
n
n
n
?
?
?
? ? ?
? ? ? ? ? ?
? ?
? ?
?
oscillates
finitely.
b) Use definition of limit to prove that lim
x ?3
(1 ? 3x) = ? 8.
8. a) Show that the function f (x) =
1
x
is differentiable at x > 0.
b) If z = x
2

1 2 1
tan tan
y x
y
x y
? ?
? , prove that
2 2 2
2 2
z x y
x y x y
? ?
?
? ? ?
, x ?0, y ? 0.
9. a) Prove that sequence
1
n
n
? ?
? ?
?
? ?
is Cauchy sequence.
b) Prove that if f is continuous at x = a, the | f | is also continuous at x = a.

NOTE : Disclosure of Identity by writing Mobile No. or Making of passing request on any
page of Answer Sheet will lead to UMC against the Student.
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