Continuity & Differentiabilityt

8/2/2019 Continuity & Differentiabilityt

1/4

S. K. Jain - 9868445172 Page 1

CONTINUITY & DIFFERENTIABILITY

Q.1) Discuss the continuity of the function, if 0

( )

0, if 0

xx

f x x

x

(A) x = 0 (B) x > 0 (C) x < 0 (D) 0x

(Q.2) If f is derivable at x = 1,then

( ) ( )

x a

xf a af x

x aLt

is equal to

(A)'

f (a) (B)'

af (a)'

f (a) (C)f(a)a'

f (a) (D) none of these

(Q.3) If f be a function such that f (9) = 9,'

f (9) = 3, then9

( ) 3

3x

f x

xLt

is equal to

(A) 9 (B) 3 (C) 1 (D) none of these.

(Q.4) Let f (x) = x |x| then'

f (0) is equal to

(A) 1 (B)1(C) 0 (D) none of these.

(Q.5) If f(x) = ex sin x in 0, , then c in Rolles theorem is

(A)6

(B)

4

(C)

2

(D)

3

4

(Q.6) Let2

, if 1( )

1, if 1

x a xf x

ax x

and f is continuous at 1, then the value a so that f is derivable

at 1, is

(A)1

2(B) -

1

2(C) 1(D) can not be determined.

(Q.7) If a > 0, a 1, then log na

x =

(A) n loga x (B) n + loga x (C)

1

loga xn (D) none of these

(Q.9) If , if 0

( ), if 0

x x xf x

x

( )f x = then f is continuous at x = 0 if

(A) =1 (B) = 0 (C) = 1 (D) none of these

(Q.10) Let

1sin , if 0

( )

, if 0

x xf x x

k x

f (x) = then f is continuous at x = 0 if

(A) k = 1 (B) k = 0 (C) k = 2 (D) none of these.

(Q.11)

1

2

2

sin 1

d x

dx x

=

(A)2

2

1 x(B)

2

2

1 x

(C)2

2 2

2(1 ), 1

1 (1 )

xx

x x

(D) none of these

(Q.12) If 1 11 1

sin sec1 1

x xy

x x

then

dy

dxis,


2/4

S. K. Jain - 9868445172 Page 2

(A) 0 (B) 1 (C)1 (D) none of these

(Q.13) If the functionsin( 1)

( )1 cos

xx e

f xx

is continuous at 0, then f(0) =

(A) 1 (B) 0 (C) 2 (D)1

2

(Q.14) Rolles theorem is not applicable to f(x) = |x| in [2, 2] because(A)fis not continuous in [2, 2] (B)fis not derivable in (2, 2) (C) (2) ( 2)f f in (2, 2)

(D) none of these

(Q.15) Lagranges mean value theorem is not applicable to f(x) in [1, 4] where f(x)=

(A) x22x (B) | x2 | (C) x | x | (D) x

2

(Q.16) 2log 1d x xdx

=

(A) 21 x (B)21

x

x(C) 21x x (D)

2

1

1 x

(Q.17) If sin cos y a mx b mx , then

2

2

d y

dx is equal to

(A)2m y (B)

2m y (C) my (D)my

(Q.18) If

1, 3

( ) , 3 5

7, 5

ifx

f x ax b if x

if x

. Determine the values of a and b so that f(x) is

continuous.

(A) a = 3, b =8 (B) a =3, b = 8 (C) a = 2, b =8 (D) None of these

(Q.19) If , then f(5) is(A) 0 (B) 1 (C) 6 (D) 2

(Q.20) The function2

3

4( )

4

xf x

x x

(A) discontinuous at only one point (B) discontinuous exactly at two points

(C) discontinuous exactly at three points (D) none of these(Q.21) The function f(x) = sin

1 (cos x) is

(A) discontinuous at x = 0 (B) continuous at x = 0

(C) differentiable at x = 0 (D) none of these.(Q.22) If f(x) = x2 is differentiable at x = 1, then f(1) equal to

(A) 0 (B) 1 (C) 2 (D) 4

(Q.23) If f(x)= x2+ 2x + 7, f (3) equals

(A) 4 (B) 6 (C) 7 (D) 8(Q.24) The value of c in Rolles theorem for the function f(x) = x

33x in the interval 0, 3 is

(A) 1 (B)1 (C) 3/2 (D) 1/3

(Q.25) Differentiate the1

xy

x

with respect to x.

(Q.26) Find the points where the constant function ( ) f x k is continuous.

(Q.27) Examine the continuity of function f given by ( ) 2 5 f x x at 1x .


3/4

S. K. Jain - 9868445172 Page 3

(Q.28) Prove that the identity function on real numbers given by ( ) f x x is continuous at every

real number.

(Q.29) If f and g be two real functions continuous at a real number c then f g is continuous

at x c .

(Q.30) Find the point of discontinuity of f where f is defined by

2 3, if 2

( ) 2 3, if 2

x x

f x x x

(Q.31) If a function f is real and c is its domain then the derivative of f at c is defined by.

(Q.32) Differentiate the cos(sin )x with respect to x.

(Q.33) Differentiate the2xy e with respect to x

(Q.34) Differentiate the sin( ) y xy with respect to x

(Q.35) If 5cos 3sin y x x , prove that2

20

d yy

dx

(Q.36) If1cosy x Find

2

2

d y

dxin terms ofy alone.

(Q.37) Find2

2

d y

dxif

1siny x

(Q.38) Finddy

dxif

1sin(tan )xy e

(Q.39) Finddy

dxif

31

2

3tan

1 3

x xy

x

(Q.40) If

cos, if

2 2( )

3, if2

k xx

xf x

x

find the value of k if f is continuous at2

x

(Q.41) Examine the continuity of ( ) 5 f x x

(Q.42) Differentiate: log log log ... y x x x

(Q.43) If ( sin ), (1 cos ) x a y a find2

2

d y

dxat

2

(Q.44) Differentiate 11 sin 1 sin

tan1 sin 1 sin

x x

x x

w.r.t. 1tan x

(Q.45) If1cos , 1 1a x y e x

, show that2

2 2

2

(1 ) 0d y dy

x x a ydx dx

(Q.46) If 2 21 1 ( ) x y a x y , then prove that2

2

1

1

ydy

dx x

(Q.47) Using mathematical induction prove that 1,n

ndxnx n I

dx


4/4

S. K. Jain - 9868445172 Page 4

(Q.48) If2 2 2( ) ( ) x a y b c , for some 0c prove that

3 22

2

2

1dy

dx

d y

dx

is a constant,

independent of a and b .

(Q.49) If 1 1 0 x y y x for 1 1x , prove that2

1

(1 )

dy

dx x

(Q.50) Verify the Mean value theorem for the function: ( ) loge f x x on 1,2

(Q.51) Verify Rolles Theorem for the following functions:2( ) ( 1)( 2) f x x x on 1,2

(Q.52) Differentiate2

2

( 3)( 4)

3 4 5

x x

x x

w.r.t. x .

(Q.53) If1 2(tan )y x , show that 2 2 22 1( 1) 2 ( 1) 2 x y x x y

(Q.54) If

1 1sin cos

,t t

x a y a

show thatdy y

dx x .

(Q.55) Finddy

dxif cos log tan , sin

2

t x a t y a t

(Q.56) Finddy

dxif 1

y xx y

(Q.57) Differentiatesin cos(sin )x x y x x w.r.t. x .

(Q.58) Determine if defined by

2 1sin , if 0

( )

0 , if 0

x xf x x

x

is a continuous function.

(Q.59) Find the value of a and b so that the function defined by5, 2

( ) , 2 10

21, 10

if x

f x ax b if x

if x

is continuous function.

(Q.60) Find the value of a and b so that the function defined by1, if 3

( )3, if 3

ax xf x

bx x

is

continuous at 3x .

Documents

Continuity & Differentiabilityt