Paper II ( CALCULUS ) Prof. R. B. Patel Art, Science & Comm. College, Shahada Dr. B. R. Ahirrao Jaihind College, Dhule Prof. S. M. Patil Art, Science & Comm. College, Muktainagar Prof. A. S. Patil Art, Science & Comm. College, Navapur Prof. G. S. Patil Art, Science & Comm. College, Shahada Prof. A. D. Borse Jijamata College, Nandurbar
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Paper II
( CALCULUS )
Prof. R. B. Patel Art, Science & Comm. College, Shahada
Dr. B. R. Ahirrao Jaihind College, Dhule Prof. S. M. Patil Art, Science & Comm.
College, Muktainagar Prof. A. S. Patil Art, Science & Comm.
College, Navapur Prof. G. S. Patil Art, Science & Comm.
College, Shahada Prof. A. D. Borse Jijamata College,
Nandurbar
Unit I
Limit, Continuity, Differentiability and Mean Value Theorem
Q.1 Objective Questions Marks – 02
1. 2
25
4 5lim 2 35x
x x is equal tox x→
− −+ −
a) 1 b) 12
c) 12− d) none of these
2. 1
coslim 1x
x is equal tox→ −
a) 0 b) 1 c) -1 d) none of these
3. Evaluate 30
tanlimx
x xx→
−
a) 13− b) 1
3 c) 0 d) 1
4. The value of the 0
log(sin 2 )limlog(sin )x
xx→
is
a) 2 b) 0 c)1 d) -1
5. limn
xx
xe→∞
is equal to
a) 1 b) -1 c) 2 d) 0
6. 0
log(sin )lim , ( , 0)log(sin )x
ax a bbx→
> is equal to
a) -1 b) 1 c) 0 d) none of these
7. 0
lim logxx x
→ is equal to
a) 0 b) 1 c) 2 d) -1
8. 0
1 1limsinx x x→
⎡ ⎤−⎢ ⎥⎣ ⎦ is equal to
a) 0 b)1 c) -1 d) none of these
9. 0
1 1lim1xx x e→
⎡ ⎤−⎢ ⎥−⎣ ⎦ is equal to
a) 1 b) 12− c) 1
2 d) 0
10. 0
lim x
xx
→ is equal to
a) 1 b) -1 c) 2 d) none of these
11. ( )tan 2
4
lim tan x
xx
π→ is
a) e b) 1e
c) 1e− d) – e
12. The function 1( ) sin , 0
(0) 0 , 0
f x x for x andx
f for x
= ≠
= =
a) Continuous and derivable
b) Not continuous but derivable
c) Continuous but not derivable
d) Neither continuous nor derivable at the point x = 0
13. The function 2 1( ) sin , 0
(0) 0 , 0
f x x for x andx
f for x
= ≠
= = is
a) Continuous and derivable
b) Not continuous but derivable
c) Continuous but not derivable
d) Neither continuous nor derivable
14. For which value of ( , )c a b∈ , the Roll’s theorem is verified for the function
[ ]2
( ) log ,( )x abf x defined on a bx a b
⎡ ⎤+= ⎢ ⎥+⎣ ⎦
a) Arithmetic mean of a & b b) Geometric mean of a & b
c) Harmonic mean of a & b d) None of these .
15. For which value of ( , ) (0, 2 )c a b π∈ = , the Rolle’s theorem is applicable for
the function [ ]( ) sin , 0,2f x x in π=
a) 0 b) 4π c)
2π d)
3π
16. For which value of 0,2
c π⎛ ⎞∈⎜ ⎟⎝ ⎠
, the Rolle’s theorem is applicable for the
function ( ) sin cos 0,2
f x x x in π⎡ ⎤= + ⎢ ⎥⎣ ⎦
a) 0 b) 4π c)
3π d)
6π
17. For which value of (1,5)c∈ , the Rolle’s theorem is verified for the function
[ ]2( ) 6 5 1,5f x x x in= − +
a) 1 b) 2 c) 3 d) 4
18. for which value of c∈ (-2, 3) . the L.M.V.T. is verified for the function
[ ]2( ) 3 2 2,3f x x x in= − + −
a) 1 b) 12
c) 12− d) 0
19. L.M.V.T is verified for the function [ ]2( ) 2 7 10 2,5f x x x in= − +
a) 52
b) 12
c) 0 d) 72
20. For which value of 0,2
c π⎛ ⎞∈⎜ ⎟⎝ ⎠
C.M .V.T. is applicable for the function
f(x) = sin x , g(x) = cos x in [0, π/2]
a) 0 b) 3π c)
6π d)
4π
21. If the C.M.V.T. is applicable for the function
f(x) = ex , g(x) = e-x , in [a, b] find the value of ( , )c a b∈
a) 2a b+ b) ab c) a + b d) none of these
22. If the C.M.V.T. is applicable for the function
f(x) =1/x2 , g(x) = 1/x , in [a, b] find the value of C.
a) 2a b+ b) ab c) 2ab
a b+ d) none of these
23. If log log 5( ) , 55
xf x xx−
= ≠−
is continuous at x = 5 then find f(5)
a) 5 b) -5 c) 15
d) 15−
24. If 2
1 sin( ) ,( 2 ) 2
xf x xx
ππ−
= ≠−
is continuous at 2
x π= then f(π/2) is
a) 18
b) 23
c) 1 d) -1
25. If 1 cos( ) , 0sin
xf x xx
−= ≠ is continuous at x = 0 then value of f(0) is
a) 0 b) 1 c) -1 d) none of these
26. I f ( ) ,x aa af x x aa x−
= ≠−
is continuous at x = a , then find f(a)
a) logaa a b) logaa a− c) log a d) none of these
27. Evaluate 0
lim sin logx
x x→
a) 0 b) 1 c) -1 d) 2π
28. Evaluate 0
lim tan logx
x x→
.
a) 0 b) 1 c)-1 d) none of these
29. 1
1 1limlog 1x x x→
⎡ ⎤−⎢ ⎥−⎣ ⎦
is equal to
a) 12− b) 1
2 c) 2 d)-2
30. 1 cos
0
1limx
x x
−
→
⎛ ⎞⎜ ⎟⎝ ⎠
is equal to
a) -1 b) 1 c) 2 d) 12
31. Evaluate lim
a) 12− b) 1
2 c) 1 d) -1
32. Evaluate lim
a) 12− b) 1
2 c) 1 d) -1
33. Evaluate 1
lim(1 ) xx
x→∞
+
a) -1 b) 2 c) -2 d) 1
34. If 2
sin 4( 3)( ) , 32xf x x
x−
= ≠−
is continuous at point x = 3 find
f(3)
a) 23
b) 13
c) 32
d) none of these
35. Evaluate sin
0lim
sin
x x
x
e ex x→
−−
a) 1 b) -1 c) 2 d) -2
Q.2 Examples Marks – 04
1. Evaluate 0
tanlimsinx
x xx x→
−−
2. Evaluate 0
1limlog(1 )
x
x
e xx x→
− −+ −
3. Evaluate 0
log(tan 2 )limlog(tan )x
xx→
4. Evaluate 220
1lim cotx
xx→
⎛ ⎞−⎜ ⎟⎝ ⎠
5. Evaluate ( )tan 2
lim 2xa
x a
xa
π
→
⎛ ⎞−⎜ ⎟⎝ ⎠
6. Evaluate 0
lim(cot ) , 0x
xx x
→>
7. Evaluate 1
log
0lim(cot ) x
xx
→
8. Evaluate 1
1lim tan2
x
xxπ −
→∞
⎡ ⎤−⎢ ⎥⎣ ⎦
9. Examine for continuity, the function
2
3
2
( ) , 0
0 , 0
,
xf x a for x aa
forxaa forx ax
= − < <
= =
= − >
10. Using δ∈− definition , prove that
2 1( ) cos , 0
0 , 0
f x x if xx
if x
= ≠
= = is continuous at x = 0
11. Examine the continuity of the function
1
11( ) , 01
0 , 0
x
x
ef x if xe
if x
−= ≠
+= =
at the point x = 0.
12. Examine the continuity of the function
2
2
9( ) , 0 33
6 , 318 8 , 3
xf x for xx
for x
for xx
−= ≤ <
−= =
= − >
at the point x = 3.
13. Examine the continuity of the function
2
2
( ) 4, 0 44
2 , 464 4 , 4
xf x for x
for x
for xx
= − < <
= =
= − >
at the point x = 4..
14. If the function
sin 4( ) , 05
4 , 0 1 , 0
xf x a for xx
x b for xfor x
= + >
= + − <= =
is continuous at x = 0 , then find the values of a & b.
15. If f(x) is continuous on [ ],π π−
( ) 2sin , -2
- sin , 2 2
cos , 2
& .
f x x for x
x for x
x for x
Find
ππ
π πα β
π π
α β
−= − ≤ ≤
= + < <
= ≤ ≤
16. Define differentiability of a function at a point and show that ( )f x x= is
continuous, but not derivable at the point x = 0.
17. Discuss the applicability of Rolle’s Theorem for the function
[ ]( ) ( ) ( ) ,m nf x x a x b defind in a b= − − where m, n are positive integers.
18. Discuss the applicability o Rolle’s Theorem for the function
5( ) (sin cos ) ,4 4
xf x e x x in π π⎡ ⎤= − ⎢ ⎥⎣ ⎦ .
19. Verify Langrange’s Mean Value theorem for the function
( ) ( 1)( 2)( 3)f x x x x= − − − defined in the interval[ ]0,4 .
20. Find θ that appears in the conclusion of Langrange’s Mean Value theorem
for the function 3 1( ) , 1,3
f x x a h= = = .
21. Show that 1 12 2tan tan , 0
1 1b a b ab a if a bb a
− −− −< − < < <
+ +.
And hence deduce that 13 4 1tan4 25 3 4 6π π− ⎛ ⎞+ < < +⎜ ⎟
⎝ ⎠
22. For 0 < a < b , Prove that 1 log 1a b bb a a
− < < − and hence show that
1 6 1log6 5 5< <
23. If < a < b <1 , then prove that 1 1
2 2sin sin
1 1b a b ab aa b
− −− −< − <
− −
Hence show that 11 1 1sin6 4 62 3 15π π−− < < −
24. Show that 12 tan , 0
1x x x xx
−< < >+
25. For x > 0 , prove that 2 2
log(1 )2 2(1 )x xx x x
x− < + < −
+
26. Separate the interval in which 3 2( ) 8 5 2f x x x x= + + − is increasing or
decreasing.
27. Show that log(1 ) , 01x x x xx< + < ∀ >
+
28. Show that 1
2
1 tan 1, 01
x xx x
−
< < ∀ >+
29. With the help of Langrange’s formula Prove that
2 2tan tan , 0cos cos 2
whereα β α β πα β β αβ α
− −< − < ≤ ≤ ≤
30. Verify Cauchy’s Mean Value theorem for the function
f (x) = sinx, g (x) = cosx in 02
x π≤ ≤
31. Show that sin sin cot , 0cos cos 2
whereα β πθ α θ ββ α−
= < < < <−
32. If 2
1 1( ) ( )f x and g xx x
= = in Cauchy’s Mean Value Theorem, Show that
C is the harmonic mean between a & b.
33. Discuss applicability of Cauchy’s Mean Value Theorem for the function
f (x) = sinx and g (x) = cosx in [ ],a b .
34. Verify Cauchy’s Mean value theorem [ ]1( ) , ( ) ,f x x g x in a bx
= =
35. Find (0,9)c∈ such that
(9) (0) '( )(9) (0) '( )f f f Cg g g c
−=
− where 3( ) ( ) 2f x x and g x x= = −
36. Discuss the applicability o Rolle’s Theorem for the function
( )2 12( ) log 3,4xf x inx
⎡ ⎤+= ⎢ ⎥
⎣ ⎦ .
37. Verify Langrange’s Mean Value theorem for the function
1( ) ( 1)( 2) 0,2
f x x x x in ⎡ ⎤= − − ⎢ ⎥⎣ ⎦
38. Discuss the applicability o Rolle’s Theorem for the function
-( ) cos ,2 2
xf x e x in π π⎡ ⎤= ⎢ ⎥⎣ ⎦ .
39. Verify Langrange’s Mean Value theorem for the function
[ ]2( ) 2 10 29 2,7f x x x in= − + .
Q.3 Theory Questions Marks – 04 / 06
1. If a function f is continuous on a closed and bound interval [ a, b] ,then show
that f is bounded on [a, b].
2. Show that every continuous function on closed and bounded interval attains its
bounds.
3. Let [ ]: ,f a b R→ be a continuous on [a ,b] and if ( ) ( ),f a k f b< < then
show that there exists a point ( , )c a b∈ such that f (x) = k.
4. If f (x) is continuous in [a, b] and ( ) ( )f a f b≠ , then show that f assume
every value between f (a) and f (b).
5. If a function is differentiable at a point then show that it is continuous at that
point. Is converse true? Justify your answer.
6. State and Prove Rolle’s theorem OR
If a function f(x) defined on [a,b] is
i)continuous on [a,b] ii) Differentiable in (a, b) iii) f (a ) = f( b)
then show that there exists at least one real number ( , )c a b∈ such that f’(c)=0.
7. State and Prove Langrange’s Mean Value Theorem. OR
If a function f(x) defined on [a,b] is i) continuous on [a,b]
ii) differentiable in (a, b)
then show that there exixt at least one real number ( , )c a b∈ such that
( ) ( )'( ) f b f af cb a−
=−
8. State and Prove Cauchy’s Mean Value Theorem. OR
If f(x) and g(x) are two function defined on [a,b] such that
i) f(x) and g(x) are continuous on [ a, b]
ii) f(x) and g(x) are differentiable in (a,b)
iii) '( ) 0, ( , )g x x a b≠ ∀ ∈
then show that there exist at least one real number ( , )c a b∈ such that
'( ) ( ) ( )'( ) ( ) ( )f c f b f ag c g b g a
−=
−
9. State Rolle’s Theorem and write its geometrical interpretation.
10. State Langrange’s Mean Value Theorem and write its geometrical
interpretation.
11. If f(x) is continuous in [a,b] with M and m as its bounds then show that f(x)
assumes every value between M and m.
12. Using Langrange’s Mean Value Theorem show that
cos cos , 0a b b a ifθ θ θθ−
≤ − ≠
13. If f(x) be a function uch that '( ) 0, ( , )f x x a b= ∀ ∈ then show that
f(x) is a constant in this interval.
14. If f(x) is continuous in the interval [a,b] and '( ) 0, ( , )f x x a b> ∀ ∈ then show
that f(x) is monotonic increasing function of x in the interval [a,b].
15. If a function f(x) is such that i) it is continuous in [a, a+h]
ii) it is derivable in (a, a+h)
iii) f(a) = f(a+h)
then show that there exist at least one real numberθ such that '( ) 0,f a hθ+ =
where 0<θ <1.
16. If the function f(x) is such that i) it is continuous in [a, a+h]
ii) it is derivable in (a, a+h)
then show that there exists at least one real number θ such that
( ) ( ) '( ), 0 1f a h f a hf a h whereθ θ+ = + + < <
17. If f(x) is continuous in the interval [a,b] and '( ) 0, ( , )f x x a b< ∀ ∈ then show
that F(x) is monotonic decreasing function of x in the interval [a, b].
Unit II
Successive Diff. And Taylor’s Theorem,
Asymptotes, Curvature and Tracing of Curves
Q-1.Question (2-marks each)
1. State Leibnitz theorem for the thn derivative of product of two functions.
2. Write thn derivative of axe .
3. Write thn derivative of sin( ).ax b+
4. Write thn derivative of cos( ).ax b+
5. State Taylor’s theorem with Langrange’s form of reminder after thn term.
6. State Maclaurin’s infinite series for the expansion of f(x) as power series in
[0,x].
7. Define Asymptote of the curve.
8. Define intrinsic equation of a curve.
9. Define point of inflexion.
10. Define multiple point of the curve.
11. Define Double point of the curve.
12. Define Conjugate point of the curve.
13. Define Curvature point of the curve at the point.
Q-2 Examples ( 4- marks each)
1. If 2
3 2
4 1 , .2 2 nx xy find y
x x x+ +
=+ − −
2. If 2cos sin , .axny e x x find y=
3. If 28sin(3 7), .y x x find y= +
4. If 1 2
2 22 1
(sin ) P(1 ) (2 1) 0n n n
y x rovethatx y n y n y
−
+ +
=
− − + − =
5. 1
2 2 22 1
cos( sin ) P (1 ) (2 1) ( ) 0n n n
If y m x rove thatx y n xy m n y
−
+ +
=
− − + + − =
6. 21 1
tan(log ) P (1 ) (2 1) ( 1) 0n n n
If y y rove thatx y nx y n n y+ −
=
+ + − + − =
7. 11
m
2 2 22 1
y +y =2 P ( 1) (2 1) ( ) 0
m
n n n
If x rove thatx y n xy n m y
−
+ +− + + + − =
8. ( ) ( )n1
2 22 1
cos log P
(2 1) 2 0n n n
y xIf rove thatb n
x y n xy n y
−
+ +
=
+ + + =
9. Find 2
( 2)(2 3)n
xy if yx x
=+ +
10. Find 4 cosny if y x=
11. 2 22 1
cos(log ) sin(log ) P (2 1) ( 1) 0n n n
If y a x b x rove thatx y n xy n y+ +
= +
+ + + + =
12. 1
22 1
tan P (1 ) 2( 1) ( 1) 0n n n
If y x rove thatx y n xy n n y
−
+ +
=
+ + + + + =
13. Find logxny if y e x=
14. Find cos cos 2 cos3ny if y x x x=
15. ( )2 2 y sin cos P
4 .cos 4 28
n
n
If x x rove that
ny x π
=
−= +
16. 2 n
22 1
y=(x -1) P ( 1) 2 ( 1) 0n n n
If rove thatx y xy n n y+ +− + − + =
17. 1cos
2 2 22 1
y P ( 1) (2 1) ( ) 0
m x
n n n
If e rove thatx y n xy n m y
−
+ +
=
− − + − + =
18. ( )m2 2
2 2 2 22 1
y P
( ) (2 1) ( ) 0n n n
If x x a rove that
x a y n xy n m y+ +
= + −
− + + + − =
19. 1
2 2 22 1
sin( sin ) P (1 ) (2 1) ( ) 0n n n
If y m x rove thatx y n xy n m y
−
+ +
=
− − + − − =
20. 2 22 1
cos(log ) P (2 1) ( 1) 0n n n
If y x rove thatx y n xy n y+ +
=
+ + + + =
21. Use Taylor’s theorem to express the polynomial 3 22 7 6x x x+ + − in powers
of ( x-2 ) .
22. Expand sinx in ascending powers of ( )2x π−
23. Assuming the validity of expansion , prove that 3 4 5
cos 13 6 30
x x x xe x x= + − − − + − − − −
24. Assuming the validity of expansion , prove that 2 4 52 16sec
2! 4! 6!x x xx = + + + − − − −
25. Expand log(sinx) in ascending powers of ( x- 3).
26. Expand tanx in ascending powers of ( )4x π−
27. Prove that 1 3 51 1tan3 5
x x x x− = − + -------- and hence find the value of π