CENGAGE/GTEWANIMATHSSOLUTIONS

CHAPTER CONTINUITY AND DIFFERENTIABILITY ||DPPDAILYPRACTICEPROBLEMS

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1

CENGAGE_MATHS_DPP DAILY PRACTICE PROBLEMS_CONTINUITY ANDDIFFERENTIABILITY_Miscellaneous

fisacontinousfunctionin ;gisacontinuousfunctionin[b,c].Afunctionh(x)isdefinedas

iff(b)=g(b)then

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2

CENGAGE_MATHS_DPP DAILY PRACTICE PROBLEMS_CONTINUITY ANDDIFFERENTIABILITY_Miscellaneous

Ifthefunction definedas definedas

iscontinuousat then b. c. d.

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3

CENGAGE_MATHS_DPP DAILY PRACTICE PROBLEMS_CONTINUITY ANDDIFFERENTIABILITY_Miscellaneous

A twice differentiable function f(x)is defined for all real numbers and satisfies thefollowingconditions

. The function is defined by , where 'a' is anyconstantIf .Findthevalue(s)of'a'

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CENGAGE_MATHS_DPP DAILY PRACTICE PROBLEMS_CONTINUITY ANDDIFFERENTIABILITY_Miscellaneous

If anditfollowstherelation ,thenfind(i) and(ii)

[a, b]

h(x) = f(x)f or x ∈ [a, b),

g(x)f or x ∈ (b, c]

f(x) f(x)f(x)

=⎧⎨⎩3, x = 0(1 + ) ,

x > 0

ax + bx3

x2

1x

x = 0, a = 0 b = e3 a = 1 b = (log)e3

f(0) = 2; f' (0) − − 5 and f(0)= 3

g(x) g(x) = eax + f(x) ∀x ∈ Rg' (0) + g(0) = 0

y = y(x) exy + y cosx = 2 y' (0)

(0)

4 .

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5

CENGAGE_MATHS_DPP DAILY PRACTICE PROBLEMS_CONTINUITY ANDDIFFERENTIABILITY_Miscellaneous

If

areconstants then (b) (d)

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6

CENGAGE_MATHS_DPP DAILY PRACTICE PROBLEMS_CONTINUITY ANDDIFFERENTIABILITY_Miscellaneous

If

b. c. d.

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7

CENGAGE_MATHS_DPP DAILY PRACTICE PROBLEMS_CONTINUITY ANDDIFFERENTIABILITY_Miscellaneous

Given

y(0)

y = e− x cos xandyn + kny = 0,

whereyn = andkndny

dxn∀n ∈ N , k4 = 4 k8 = − 16 k12 = 20 k16 = − 24

y3 − y = 2x,

then(x2 − ) + x =127

d2y

dx2

dy

d

yy

3y

9y

27

f(x)

= {3 − [cot−1( )]for x > 0{x2}cos(e )f or x

< 0

2x3 − 3

x2

1x

(where {} and [] denotes the fractional part and the integral part functionsrespectively). Then which of the following statements do/does not hold good?

b. c. if , then is continuous at d.irremovablediscontinuityof at

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8

CENGAGE_MATHS_DPP DAILY PRACTICE PROBLEMS_CONTINUITY ANDDIFFERENTIABILITY_Miscellaneous

If

. Then at is continuous and differentiable is continuous but notdifferentiable not continuous but differentiable is neither continuous nordifferentiable

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9

CENGAGE_MATHS_DPP DAILY PRACTICE PROBLEMS_CONTINUITY ANDDIFFERENTIABILITY_Miscellaneous

Let (where [.]denotes thegreatest integer function)and

.Thenfor existsbutnotcontinuousContinuous

but not differentiable at Differentiable at does not

exist iscontinuousbutnotdifferentiable

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10

CENGAGE_MATHS_DPP DAILY PRACTICE PROBLEMS_CONTINUITY ANDDIFFERENTIABILITY_Miscellaneous

Let and If iscontinuousanddifferentiableforallnumbersinitsdomainthen

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11

CENGAGE_MATHS_DPP DAILY PRACTICE PROBLEMS_CONTINUITY ANDDIFFERENTIABILITY_Miscellaneous

If isanevenfunctionsuchthat hassomefinitenon-zero

value,thenprovethat isnotdifferentiableat

f(0− ) = 0 f(0+ ) = 3 f(0) = 0 f(x) x = 0f x = 0

f(x) = {s ∈ (cos− 1 x)

+ cos(sin−1 x), x ≤ 0s

∈ (cos−1 x)

− cos(sin−1 x, x > 0)x = 0 f(x) f(x)f(x) f(x)

f(x) = {[x]x ∈ Ix − 1x ∈ I

g(x) = {sin x + cosx, x < 01, x≥ 0

f(g(x))atx = 0 ( lim )x→0f(g(x))

x = 0 x = 0 ( lim )x→0f(g(x))

f(x)

g(x) = 3x2 − 4√x + 1, x < 1 g(x) = ax + b, x ≥ 1. g(x)

f ( lim )h→0

f(h) − f(0)

hf(x) x = 0.

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12

CENGAGE_MATHS_DPP DAILY PRACTICE PROBLEMS_CONTINUITY ANDDIFFERENTIABILITY_Miscellaneous

If

, then [where [.] and {.] represent the greatest integer and fractional part functionsrespectively] is continuous at is not continuous at isdifferentiableat doesnotexist

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13

CENGAGE_MATHS_DPP DAILY PRACTICE PROBLEMS_CONTINUITY ANDDIFFERENTIABILITY_Miscellaneous

Numberofpointswhere

isnon-differentiableisa.0b.1c.2d.3

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14

CENGAGE_MATHS_DPP DAILY PRACTICE PROBLEMS_CONTINUITY ANDDIFFERENTIABILITY_Miscellaneous

Let be differentiable for real such that If then the

valueof isa.1b.2c.0d.4

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CENGAGE_MATHS_DPP DAILY PRACTICE PROBLEMS_CONTINUITY AND

f(x) = {[x] + √{x}, x

< 1 , x ≥ 11

[x] + {x}2

f(x) x = 1 f(x) x = 1 f(x)x = 1 ( lim )

x→1f(x)

f(x) = x2 − ∣∣x2 − 1∣∣ + 2||x| − 1|

+ 2|x| − 7

f(x) x f ′ (x) > 0on( − ∞, − 4),f ′ (x) < 0on( − 4, 6), f ′ (x) > 0on(6, ∞), g(x) = f(10 − 2x),

g ′ (2)

15

DIFFERENTIABILITY_Miscellaneous

The differential coefficient of with respect to

is b. c. d.

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16

CENGAGE_MATHS_DPP DAILY PRACTICE PROBLEMS_CONTINUITY ANDDIFFERENTIABILITY_Miscellaneous

Let isa twicedifferentiable functiononsuchthat Thevalueof equals__________

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17

CENGAGE_MATHS_DPP DAILY PRACTICE PROBLEMS_CONTINUITY ANDDIFFERENTIABILITY_Miscellaneous

If at isequalto (b)e(c)1(d)zero

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CENGAGE_MATHS_DPP DAILY PRACTICE PROBLEMS_CONTINUITY ANDDIFFERENTIABILITY_Miscellaneous

If

sin−1( )4 sin 2x+ 3 cos2x5

cos−1( )5cos x − 4sin x

√41−2 −1 1 2

g(x) = f(x)sinx,wheref(x) ( − ∞, ∞)f( − π) = 1. ∣∣g − π∣∣

f(x) = (log)x(ln x), thenf′ (x) x = e

1e

f(x − y) = f(x). g(y) − f(y)

. g(x)

18 and

forall .Ifrighthandedderivativeatx=0existsforf(x)findthederivativeofg(x)atx=0

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19

CENGAGE_MATHS_DPP DAILY PRACTICE PROBLEMS_CONTINUITY ANDDIFFERENTIABILITY_Miscellaneous

If then isa.0b.1c.-1d.noneofthese

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20

CENGAGE_MATHS_DPP DAILY PRACTICE PROBLEMS_CONTINUITY ANDDIFFERENTIABILITY_Miscellaneous

The right hand derivative of is (where [.] denotes thegreatestintegerfunction) b. c. d.noneofthese

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21

CENGAGE_MATHS_DPP DAILY PRACTICE PROBLEMS_CONTINUITY ANDDIFFERENTIABILITY_Miscellaneous

If

,provethat denotesthederivativew.r.t

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22

CENGAGE_MATHS_DPP DAILY PRACTICE PROBLEMS_CONTINUITY ANDDIFFERENTIABILITY_Miscellaneous

Let are differentiable functions. If andthederivativesoftheirpairwiseproductsat are

thencomputethevalueof .

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23

CENGAGE_MATHS_DPP DAILY PRACTICE PROBLEMS_CONTINUITY ANDDIFFERENTIABILITY_Miscellaneous

Ifforacontinuousfunction

g(x− y) = g(x). g(y) + f(x)

. f(y)x ∈ R

xexy − y = sin2 x atx = 0dy

dx

f(x) = [x]tanπxatx = 70 7π −7π

y = + x√x2 + 1

+ (log)e√x + √x2 + 1

x2

212

2y = xy ′ + (log)ey′ ,wherey' x.

f, g and h f(0) = 1; g(0) = 2; h(0) = 3x = 0

(fg)' (0) = 6; (gh)' (0)

= 4 and (hf)' (0) = 5(fgh)' (0)

,then isequaltoa.1b.2c.0d.noneofthese

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24

CENGAGE_MATHS_DPP DAILY PRACTICE PROBLEMS_CONTINUITY ANDDIFFERENTIABILITY_Miscellaneous

If

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25

CENGAGE_MATHS_DPP DAILY PRACTICE PROBLEMS_CONTINUITY ANDDIFFERENTIABILITY_Miscellaneous

b. c. d.noneofthese

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CENGAGE_MATHS_DPP DAILY PRACTICE PROBLEMS_CONTINUITY ANDDIFFERENTIABILITY_Miscellaneous

Let

f, f(0) = f(1) = 0, f ′ (1)

= 2andy(x) = f(ex)ef (x )

y ′ (0)

y = xlog x ^ ((log(log.x))),

then isdy

dx(1nx∞x−1) + 21nx1n(1nx))y

x(log x)log ( log x ) (2 log(logx) + 1)

y

x

[(1nx)2 + 21n(1nx)]y

x1nx[2 log(logx) + 1]

y

x

log ylogx

[cos−1(x√x− √(1 − x)(1 − x2))]=

d

dx

−1

√1 − x2

1

2√x − x2−

−1

√1 − x2

1

2√x − x2+

1

√1 − x2

1

2√x − x2

1

√1 − x20 1/4 −1/4

26

If

1

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27

CENGAGE_MATHS_DPP DAILY PRACTICE PROBLEMS_CONTINUITY ANDDIFFERENTIABILITY_Miscellaneous

Supposethat isdifferentiableinvertiblefunctionGiventhat and isinverseof .Let

Which of the following is/are correct? b. c. d.

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28

CENGAGE_MATHS_DPP DAILY PRACTICE PROBLEMS_CONTINUITY ANDDIFFERENTIABILITY_Miscellaneous

If

(b) (d)

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29

CENGAGE_MATHS_DPP DAILY PRACTICE PROBLEMS_CONTINUITY ANDDIFFERENTIABILITY_Miscellaneous

Let

where[x] denotes the greatest integer function. then the correct statements are (A)Limit exists for x=-1 (B) f(x) has removable discontonuity at x =1 (C) f(x) has nonremovablediscontinuityatx=2(D)f(x)isdiscontinuousatallpositiveintegers

g(x) = ef (x )andf(x+ 1) = x

+ f(x) ∀x ∈ R.

n ∈ I + , then

− =

g ′ (n + )12

g(n + )12

g ′ ( )12

g( )12

2(1 + + + + )12

13

1n

2(1 + + + )13

15

12n − 1

n

f(x) f ′ (x) ≠ 0andh ′ (x) = f(x).f(1) = f ′ (1) = 1, h(1) = 0 g(x) f(x)

G(x) = x2g(x) − xh(g(x))∀x∈ R.

G ′ (1) = 2 G ′ (1) = 3 G1 = 2G1 = 3

y = cos−1 √ ,

then isequa < o

√1 + x2 + 1

2√1 + x2

dy

dx

, x ∈ R1

2(1 + x2), x > 0

12(1 + x2)

, x < 0−1

2(1 + x2)

, x < 01

2(1 + x2)

f(x) = x[ ] + x[x] if x ≠ 0;

0 if x = 0

1x

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30

CENGAGE_MATHS_DPP DAILY PRACTICE PROBLEMS_CONTINUITY ANDDIFFERENTIABILITY_Miscellaneous

Let If and at then at

isgivenby1(b) (c) (d)noneofthese

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31

CENGAGE_MATHS_DPP DAILY PRACTICE PROBLEMS_CONTINUITY ANDDIFFERENTIABILITY_Miscellaneous

is a strictly monotonic differentiable function with If is the

inverseof then a. b. c. d.

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32

CENGAGE_MATHS_DPP DAILY PRACTICE PROBLEMS_CONTINUITY ANDDIFFERENTIABILITY_Miscellaneous

Suppose be a differentiable function such thatwith Thenthevalueof is b.

c. d.

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CENGAGE_MATHS_DPP DAILY PRACTICE PROBLEMS_CONTINUITY ANDDIFFERENTIABILITY_Miscellaneous

Inaquestionastudentwasgiventofindthederivativeoftheproductoftwofunctions

y = x3 − 8x + 7andx = f(t). = 2dy

dtx = 3 t = 0,

dx

dt

t = 0192

219

f f ′ (x) = .1

√1 + x3g

f, gx =2x2

2√1 + x3

2g2(x)

2√1 + g2(x)g2(x)

32

x2

√1 + x3

f :R→R

+

3f(x + y) = f(x)f(y) ∀x, y ∈ R f(1) = 6. f(2) 6 912 15

′ 3

33 The student by mistake thought for his questionandhegotthecorrectanswer.Giventhat Thenwhichof thefollowing is

false? b. c. d.noneofthese

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34

CENGAGE_MATHS_DPP DAILY PRACTICE PROBLEMS_CONTINUITY ANDDIFFERENTIABILITY_Miscellaneous

AnonzeropolynomialwithrealcoefficienthasthepropertythatIf istheleadingcoefficientof thenthevalueof is____

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35

CENGAGE_MATHS_DPP DAILY PRACTICE PROBLEMS_CONTINUITY ANDDIFFERENTIABILITY_Miscellaneous

If is an increasing function from such that exists then

is b. c. d.cannotbedetermined

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36

CENGAGE_MATHS_DPP DAILY PRACTICE PROBLEMS_CONTINUITY ANDDIFFERENTIABILITY_Miscellaneous

if satisfies then is:

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CENGAGE_MATHS_DPP DAILY PRACTICE PROBLEMS_CONTINUITY ANDDIFFERENTIABILITY_Miscellaneous

Let

fandg. (fg) ′ = f ' g' f(x) = x3

g(4) = 1.

g(5) =18

f ′ (x) < 0 f(0) < 0

f(x) = f ′ (x).f

′(x).

a f(x), 1/2a)

' f' R→R f x > 0andf −1

d2(f −1(x))

dx2

< 0 > 0 = 0

x = , y = +1 + t

t33

2t22t

f(x) ⋅ { }3

= 1 +dy

dx

dy

dxf(x)

2

37

hasexactlytwopointsofcontinuitythenthevalueof are b. c. d.

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38

CENGAGE_MATHS_DPP DAILY PRACTICE PROBLEMS_CONTINUITY ANDDIFFERENTIABILITY_Miscellaneous

If for and for then

b. has a removable discontinuity at c. has an irremovablediscontinuityat d.

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39

CENGAGE_MATHS_DPP DAILY PRACTICE PROBLEMS_CONTINUITY ANDDIFFERENTIABILITY_Miscellaneous

Let

If iscontinuousat thenthevalueof is b. c. d.9

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40

CENGAGE_MATHS_DPP DAILY PRACTICE PROBLEMS_CONTINUITY ANDDIFFERENTIABILITY_Miscellaneous

Statement1:Minimumnumberofpointsofdiscontinuityofthefunction

,where [.]denotes thegreatest integer functionand iszero.Statement 2: can be continuous at a point of discontinuity, say of

if

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41

CENGAGE_MATHS_DPP DAILY PRACTICE PROBLEMS_CONTINUITY ANDDIFFERENTIABILITY_Miscellaneous

Let ,(where[.]denotesthegreatestintegerless

thanorequalto ).Thenthenumberofpoints,where isdiscontinuousisa.oneb.zeroc.threed.infinite

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f(x) = { , ξsrationalb,

ξsrational

21 + x2

b (0, 3] [0, 1] (0, 2] φ

f(x) = {sin( )tan[ ]a − x

2πx

2ax > a

[cos( )]πx

2a

a − xx < a,

f(a− ) < 0 f x = a f

x = a f(a+ ) < 0

f(x) = { + , 0 < |x|

≤ 1 , x = 0

α cot xx

β

x2

13

f(x) x = 0 α2 + β2 1 2 5

f(x) = (g(x)[2x − 1] ∀x

∈ ( − 3, − 1)g(x) = ax3 + x2 + 1

f(x) x = c1[2x − 1] g(c1) = 0.

f(x) = [tanx[cotx]], x[ , ]π

12π

12x f(x)

42

CENGAGE_MATHS_DPP DAILY PRACTICE PROBLEMS_CONTINUITY ANDDIFFERENTIABILITY_Miscellaneous

Let beanyfunctionwhich issuchthat is rational for irrationalxandthat isiirrationalforrationalx,thenin[a,b]

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43

CENGAGE_MATHS_DPP DAILY PRACTICE PROBLEMS_CONTINUITY ANDDIFFERENTIABILITY_Miscellaneous

If ;then

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44

CENGAGE_MATHS_DPP DAILY PRACTICE PROBLEMS_CONTINUITY ANDDIFFERENTIABILITY_Miscellaneous

A curve in the xy-plane is parametrically given by is the parameter. For what value(s) of is

b. c. d.

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CENGAGE_MATHS_DPP DAILY PRACTICE PROBLEMS_CONTINUITY ANDDIFFERENTIABILITY_Miscellaneous

f : [a, b] → R f(x)f(x)

f(x) =100

∏n=1

(x − n)n( 101 −n ) =f(101)

f' (101)

x = t + t3andy = t2, wheret ∈ R t

= ?dy

dx

12

13

2 3 1

45 Numberofpointsofdiscontinuityof initsdomainisequalto(where[.]denotesthegreatestintegerfunction)a.0b.1c.2d.3

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46

CENGAGE_MATHS_DPP DAILY PRACTICE PROBLEMS_CONTINUITY ANDDIFFERENTIABILITY_Miscellaneous

If

when such that are continuous functionsat thenthevalueof is b. c. d.

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47

CENGAGE_MATHS_DPP DAILY PRACTICE PROBLEMS_CONTINUITY ANDDIFFERENTIABILITY_Miscellaneous

Suppose is continuously differentiable function with

and satisfies and then is

b. c. d.

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48

CENGAGE_MATHS_DPP DAILY PRACTICE PROBLEMS_CONTINUITY ANDDIFFERENTIABILITY_Miscellaneous

If then .

Findthevalueof

f(x) = [sin−1 x] − [x]

g(x)

= ( lim )m−→∞

xmf(x) + h(x) + 32xm + 4x + 1

x ≠ 1andg(1) = e3 f(x), g(x)andh(x)x = 1 5f(1) − 2h(1) 7 6 9 8

∣∣∣

f ' (x) f(x)

f' ' (x) f ' (x)

∣∣∣= 0

f ′ (x) ≠ 0 f(0) = 1 f ' (0) = 2 ( lim )x→0

f(x) − 1

x1/2 1 2 0

f(x) =

∣∣∣∣∣

(x − a)4 (x − a)3 1

(x − b)4 (x − b)3 1

(x − c)4 (x − c)3 1

∣∣∣∣∣

f ' (x) = λ

∣∣∣∣∣

(x − a)4 (x − a)3 1

(x − b)4 (x − b)3 1

(x − c)4 (x − c)3 1

∣∣∣∣∣

λ

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49

CENGAGE_MATHS_DPP DAILY PRACTICE PROBLEMS_CONTINUITY ANDDIFFERENTIABILITY_Miscellaneous

Let

for all real If is differentiable and exists for all real permissiblevalueof andisequalto Then ispositiveforallreal isnegativeforallreal hasrealrootsNothingcanbesaidaboutthesignof

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50

CENGAGE_MATHS_DPP DAILY PRACTICE PROBLEMS_CONTINUITY ANDDIFFERENTIABILITY_Miscellaneous

Afunction satisfiestheequation

Ifdifferentiableon

b. c. d.

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51

CENGAGE_MATHS_DPP DAILY PRACTICE PROBLEMS_CONTINUITY ANDDIFFERENTIABILITY_Miscellaneous

Let be a continuous function and is true If

thenthevalueof isequalto6(b)0(c) (d)

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52

CENGAGE_MATHS_DPP DAILY PRACTICE PROBLEMS_CONTINUITY ANDDIFFERENTIABILITY_Miscellaneous

Let and If

then b. c. d.

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CENGAGE_MATHS_DPP DAILY PRACTICE PROBLEMS_CONTINUITY AND

=

+ xy

f(x + y) − f(x)2

f(y) − a

2

xandy. f(x) f ′ (0)a √5a − 1 − a2

. f(x) x f(x)x f(x) = 0

f(x)

f :R−−−→1, ∞

f(xy) = f(x)f(y) − f(x) − f(y)+ 2.

R − {0}andf(2) = 5, f ′ (x)

=.λthenλ =

f(x) − 1x

2 ′f(1) 3f ′ (1) f ′ (1)12

f ′ (1)

f :R→R f(x) = f(2x) ∀x ∈ R.

f(1) = 3, ∫ 1

− 1f(f(x))dx 3f(3) 2f(0)

f(x) = whenx ≠ 0g(x)

xf(0) = 0. g(0) = g ′ (0) = 0andg0 = 17

f(0) = 3/4 −1/2 17/3 17/2

53

DIFFERENTIABILITY_Miscellaneous

Let

forallreal and beadifferentiablefunction.If theprovethat

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54

CENGAGE_MATHS_DPP DAILY PRACTICE PROBLEMS_CONTINUITY ANDDIFFERENTIABILITY_Miscellaneous

Let beacontinuousfunctionsuchthat

Also Then equal representsthegreatestintegerfunction b.c. d.

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55

CENGAGE_MATHS_DPP DAILY PRACTICE PROBLEMS_CONTINUITY ANDDIFFERENTIABILITY_Miscellaneous

iscontinuousfromrightatthepoint then equals b. c. d.noneofthese

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56

CENGAGE_MATHS_DPP DAILY PRACTICE PROBLEMS_CONTINUITY ANDDIFFERENTIABILITY_Miscellaneous

The derivative of the function represented parametrically asisa.-1b.1c.0d.doesnotexist

f(x + y) = f(x) + f(y) + 2xy− 1

xandy f(x) f ′ (0) = cosα,f(x) > 0 ∀x ∈ R.

f : ( − ∞, ∞)−−−→0, ∞

f(x + y) = f(x) + f(y)

+ f(x)f(y), ∀x ∈ R.f' (0) = 1. [f(2)] ([.] ) 5

6 7 8

f(x) = {(x2 + e )−1k, x

= 2, x ≠ 2

12−x

x = 2, k 0 1/4 −1/4

x = 2t = |t|, y = t3 + t2|t|a = 0

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57

CENGAGE_MATHS_DPP DAILY PRACTICE PROBLEMS_CONTINUITY ANDDIFFERENTIABILITY_Miscellaneous

If and then show that

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58

CENGAGE_MATHS_DPP DAILY PRACTICE PROBLEMS_CONTINUITY ANDDIFFERENTIABILITY_Miscellaneous

If

and iscontinuousat thenthevalueof isa.2b.3c.-3d.7

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59

CENGAGE_MATHS_DPP DAILY PRACTICE PROBLEMS_CONTINUITY ANDDIFFERENTIABILITY_Miscellaneous

Whichofthefollowingfunctionsis/arediscontinuousat

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x = sec θ − cosθ y = secn θ − cosn θ

(x2 + 4)( )2

= n2(y2 + 4)dy

dx

f(x) = { 1x

= 0, x ≠ 0

1 − cos(1 − )cos x2

2mxn

f(0) = 1 x = 0 m + n

x = 1? f(x) =1

1 + 2tanx

g(x) = ( lim )x−→∞

11 + n ∈ s2(πx)

h(x) = 2−2 ^ ((( ))), x

≠ 1andh(1) = 1

11 − x

φ(x) = , x

≠ 1andφ(1) = 1

x − 1

|x − 1| + 2(x − 1)2

60

CENGAGE_MATHS_DPP DAILY PRACTICE PROBLEMS_CONTINUITY ANDDIFFERENTIABILITY_Miscellaneous

If

thevalueof is b. c. d.

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61

CENGAGE_MATHS_DPP DAILY PRACTICE PROBLEMS_CONTINUITY ANDDIFFERENTIABILITY_Miscellaneous

Afunction isdefinedas

iscontinuouson thenfindthevaluesofa,b,c.

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62

CENGAGE_MATHS_DPP DAILY PRACTICE PROBLEMS_CONTINUITY ANDDIFFERENTIABILITY_Miscellaneous

If

b. c. d.

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CENGAGE_MATHS_DPP DAILY PRACTICE PROBLEMS_CONTINUITY ANDDIFFERENTIABILITY_Miscellaneous

t(1 + x2) = xandx2 + t2

= ythenatx = 2,dy

dx

245

101125

488125

358125

f :R→R

f(x)

= ( lim )n−→∞

ax2 + bx + c + enx

1 + c. enxR

x + y = 3e2the (xy)

= 0f or x =

d

dx

e2 ee e 2e2

63and thenn=1b.2c.3d.4

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64

CENGAGE_MATHS_DPP DAILY PRACTICE PROBLEMS_CONTINUITY ANDDIFFERENTIABILITY_Miscellaneous

If

is continuous at then ([.] denotes the greatest integer function)

b. c. d.

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65

CENGAGE_MATHS_DPP DAILY PRACTICE PROBLEMS_CONTINUITY ANDDIFFERENTIABILITY_Miscellaneous

If

, then which of the following holds? (a) is continuous at (b) has anirremovable discontinuity at (c) has a removable discontinuity at

(d)Noneofthese

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xy = (x + y)n =dy

dx

y

x

f(x) = {sin( )(x − [x]), x

< 55(b − 1), x

= 5 , x > 5

π

2

ab2∣∣x2 − 11x + 24∣∣x − 3

x = 5, a, b ∈ R

a = , b =25108

65

a = , b =613

1729

a = , b =12

2536

a = , b =23100

65

f(x) = { , x

≤ , x >

2cos x − sin 2x

(π− 2x)2

π

2

e−cotx − 18x − 4π

π

2f x = π/2 f

x = π/2 fx = π/2

66

CENGAGE_MATHS_DPP DAILY PRACTICE PROBLEMS_CONTINUITY ANDDIFFERENTIABILITY_Miscellaneous

Let

(where[.]denotesthegreatestintegerfunction).Then(a) isContinuousonlyata finite number of points (b)Discontinuous at a finite number of points.(c)Discontinuousataninfinitenumberofpoints.(d)Discontinuousat

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67

CENGAGE_MATHS_DPP DAILY PRACTICE PROBLEMS_CONTINUITY ANDDIFFERENTIABILITY_Miscellaneous

let

and where is a rational function such that it is continuous

everywhere except when and

thenthevalueof

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68

CENGAGE_MATHS_DPP DAILY PRACTICE PROBLEMS_CONTINUITY ANDDIFFERENTIABILITY_Miscellaneous

Forwhichofthefollowingfunctions existssuchthat iscontinuousatx=0

b. c. d.

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f(x) = {8 , x < 0a[x], a ∈ R

− {0}, x ≥ 0,

1x

f(x)

x = 0

f(x) = x3 − x2 − 3x − 1, g(x)

= (x+ 1)a

h(x) =f(x)

g(x)h (1)

x = − 1, (2) limx→ ∞

h(x) = ∞

(3) limx→ −1

h(x) =12

h(1)

f(0) f(x)

f(x) =1

(log)e|x|f(x) = (cos( ))sin|x|

xf(x) = x sin( )π

x

f(x) =1

1 + 2cotx

69

CENGAGE_MATHS_DPP DAILY PRACTICE PROBLEMS_CONTINUITY ANDDIFFERENTIABILITY_Miscellaneous

Let

.Determinethevalueofp,ifpossible,sothatthefunctioniscontinuousat .

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70

CENGAGE_MATHS_DPP DAILY PRACTICE PROBLEMS_CONTINUITY ANDDIFFERENTIABILITY_Miscellaneous

Let beanyfunction.Also isdefinedby forall Then is a. Onto if is onto b. One-one if is one-one c. Continuous if is

continuousd.Noneofthese

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71

CENGAGE_MATHS_DPP DAILY PRACTICE PROBLEMS_CONTINUITY ANDDIFFERENTIABILITY_Miscellaneous

If

is continuous at then minimum value of is b. c.d.0

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f(x) = [ , x

< and p, x

= and

1 − sinπx1 + cos2πx

1212

√2x − 1

√4 + √2x − 1 − 2

x =12

f :R → R g :R → R g(x) = |f(x)|x. f f f

f(x)

= {( + cos( ))a

b

/x2, x ≠ 0e3, x = 0

sin(2x2)

a

3xb

x = 0∀ b ∈ R a −1/8 −1/4−1/2

72

CENGAGE_MATHS_DPP DAILY PRACTICE PROBLEMS_CONTINUITY ANDDIFFERENTIABILITY_Miscellaneous

Let be continuous functions satisfyingThenthevalueof is b. c. d.5

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73

CENGAGE_MATHS_DPP DAILY PRACTICE PROBLEMS_CONTINUITY ANDDIFFERENTIABILITY_Miscellaneous

The function satisfies for all real Given thatand ,thenthevalueof is b. c. d.8

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74

CENGAGE_MATHS_DPP DAILY PRACTICE PROBLEMS_CONTINUITY ANDDIFFERENTIABILITY_Miscellaneous

non-zerovalue.Then, (b) (d)noneofthese

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75

CENGAGE_MATHS_DPP DAILY PRACTICE PROBLEMS_CONTINUITY ANDDIFFERENTIABILITY_Miscellaneous

equals: (b) (d)

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76

CENGAGE_MATHS_DPP DAILY PRACTICE PROBLEMS_CONTINUITY ANDDIFFERENTIABILITY_Miscellaneous

If

a.1b. c.-2d.noneofthese

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CENGAGE_MATHS_DPP DAILY PRACTICE PROBLEMS_CONTINUITY AND

f(x) f :R→R

f(0) = 1andf(2x) − f(x) = x. f(3) 2 3 4

f :R→R f(x2)

.fx

= f ′ (x).f

′(x2) x.

f(1) = 1 f1 = 8 f ′ (1) + f1 2 4 6

(lim)x→0

, wherea, b, c

∈ R~{0}, eξstsandhas

xa sinb x

sin(xc)

a + c = b −1 0

d2x

dy2( )

−1d2y

dx2

−( )−1

( )−3d2y

dx2

dy

dx( )( )−2d2y

dx2

dy

dx

−( )( )−3d2y

dx2

dy

dx

f(1) = 3, f ′ (1) = 2, f 1 = 4,

then(f −1)' '(3) =

−12

77

DIFFERENTIABILITY_Miscellaneous

,Provethat

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78

CENGAGE_MATHS_DPP DAILY PRACTICE PROBLEMS_CONTINUITY ANDDIFFERENTIABILITY_Miscellaneous

If , then can be put in the form of

b. c.

d.

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79

CENGAGE_MATHS_DPP DAILY PRACTICE PROBLEMS_CONTINUITY ANDDIFFERENTIABILITY_Miscellaneous

If (b) (c) (d)

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CENGAGE_MATHS_DPP DAILY PRACTICE PROBLEMS_CONTINUITY AND

(a + bx)e = xy

x x3 = (x − y)2d2y

dx2

dy

dx

R =

[1 + ( )2]3 /2dy

dx

d2y

dx2

R2 /3

+1

( )2 /3d2y

dx2

1

( )2/ 3d2x

dy2

−1

( )2/3d2y

dx2

1

( )2/ 3d2x

dy2

+2

( )2/ 3d2y

dx2

2

( )2 /3d2x

dy2

1

( )2/ 3d2y

dx2

.1

( )2/ 3d2x

dy2

x = t2y = t3, then =d2y

dx2

32

3(4t)

32(t)

3t2

80

DIFFERENTIABILITY_Miscellaneous

Thefunction in doesnottakethevalue b.

c. d.

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81

CENGAGE_MATHS_DPP DAILY PRACTICE PROBLEMS_CONTINUITY ANDDIFFERENTIABILITY_Miscellaneous

Thenumberofpointsofdiscontinuityof where isdefinedas,`f(x)={1+x,0lt=xlt=23-x,22`

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82

CENGAGE_MATHS_DPP DAILY PRACTICE PROBLEMS_CONTINUITY ANDDIFFERENTIABILITY_Miscellaneous

Ifthefunction iscontinuousat ,thenthevalue

of is b. c. d.noneofthese

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83

CENGAGE_MATHS_DPP DAILY PRACTICE PROBLEMS_CONTINUITY ANDDIFFERENTIABILITY_Miscellaneous

Let

.Then iscontinuousbutnotdifferentiableat isbothcontinuousbutnot differentiable at is neither continuous not differentiable at

isaperiodicfunction.

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f(x) = − sinπx + 4x3

8[ − 4, 4] −4

10 18 12

g(x) = f(f(x)) f(x)

f(x) =(128a + ax)1/ 8 − 2

(32 + bx)1/ 5 − 2x = 0

a/b f(0)35

28/ 5f(0) f(0)645

f(x) = ( lim )x−→∞

n−1

∑r=0

x

(rx + 1){(r + 1)x + 1}f(x) x = 0 f(x)

x = 0 f(x) x = 0f(x)

84

CENGAGE_MATHS_DPP DAILY PRACTICE PROBLEMS_CONTINUITY ANDDIFFERENTIABILITY_Miscellaneous

Let

Iff(x)isdifferentiableforall then equals

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85

CENGAGE_MATHS_DPP DAILY PRACTICE PROBLEMS_CONTINUITY ANDDIFFERENTIABILITY_Miscellaneous

If

then

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CENGAGE_MATHS_DPP DAILY PRACTICE PROBLEMS_CONTINUITY ANDDIFFERENTIABILITY_Miscellaneous

f(x)

=⎧⎪⎪⎨⎪⎪⎩

limn→ ∞

,

x ∈ (0, 1) ∪ (1, 2) and 0, x = 1

ax(x − 1)( )n

+ (px2 + 2)cot (πx )4

( )n

+ 1cot (πx )

4

x ∈ (0, 2) (a2 + p2)

f(x) = {sinx, x ≠ nπ, n ∈ I2,otherwiseg(x) = {x2 + 1, x ≠ 0, 4, x

= 05, x = 2(lim)

x→0g{f(x)}is =

86

The number of points at which is not differentiable, where

,is b. c. d.

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87

CENGAGE_MATHS_DPP DAILY PRACTICE PROBLEMS_CONTINUITY ANDDIFFERENTIABILITY_Miscellaneous

If for all then the domain of is b.c. d.

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88

CENGAGE_MATHS_DPP DAILY PRACTICE PROBLEMS_CONTINUITY ANDDIFFERENTIABILITY_Miscellaneous

Let be a function with continuous second derivative and

Determine a function by Then which of the

following statements is correct? has a continuous first derivative has a firstderivative iscontinuousbut failstohaveaderivative hasafirstderivativebutthefirstderivativeisnotcontinuous

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89

CENGAGE_MATHS_DPP DAILY PRACTICE PROBLEMS_CONTINUITY ANDDIFFERENTIABILITY_Miscellaneous

Let be a function defined on with Assume that iscontinuousat

then b. c. isdifferentiableat d. is

non-differentiableat

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g(x) =1

1 + 2f (x )

f(x) =1

1 + 1x

1 2 3 4

f(x) = x1 /3(x − 2)2/ 3x, f' x ∈ R − {0}

{x ∣ x⟩0} x ∈ R − {0, 2} x ∈ R

f f(0) = f ′ (0) = 0.

g g(x) = { , x ≠ 00, x = 0f(x)

xg g

g g g

f(x) ( − a, a) a > 0. f(x)

x = 0and( lim )x→0

= α, wherek ∈ (0, 1)

f(x) − f(kx)x

f ′ (0+ ) = 0 f ′ (0− ) =α

1 − kf(x) x = 0 f(x)

x = 0

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