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MANOJ CHAUHAN SIR(IIT-DELHI) EX. SR. FACULTY (BANSAL CLASSES)

KEY CONCEPTS (DIFFERENTIABILITY) THINGS TO REMEMBER : 1. Right hand & Left hand Derivatives ; f ( a + h )− f ( a ) if it exist h The right hand derivative of f ′ at x = a denoted by f ′(a+) is defined by :

By definition : f ′(a) = Limit h →0 (i)

f ( a + h )− f ( a ) , h provided the limit exists & is finite. The left hand derivative : of f at x = a denoted by f ′(a+) is defined by :

f ' (a+) = Limit h →0 +

(ii)

f ( a − h )− f ( a ) , −h Provided the limit exists & is finite. We also write f ′(a+) = f ′+(a) & f ′(a–) = f ′_(a). * This geomtrically means that a unique tangent with finite slope can be drawn at x = a as shown in the figure.

f ' (a–) = Limit h →0 +

(iii)

Derivability & Continuity : (a) If f ′(a) exists then f(x) is derivable at x= a ⇒ f(x) is continuous at x = a. (b)

If a function f is derivable at x then f is continuous at x. f ( x + h )− f ( x ) For : f ′(x) = Limit exists. h →0 h f (x + h )− f (x ) Also f ( x + h ) − f ( x ) = .h [ h ≠ 0 ] h Therefore : f ( x + h )− f ( x ) Limit [f ( x + h )− f ( x )] = Limit .h = f '( x ).0 = 0 h →0 h →0 h

Limit Therefore Limit h →0 [f ( x + h ) − f ( x )] = 0 ⇒ h →0 f (x+h) = f(x) ⇒ f is continuous at x.

Note : If f(x) is derivable for every point of its domain of definition, then it is continuous in that domain. The Converse of the above result is not true : “ IF f IS CONTINUOUS AT x , THEN f IS DERIVABLE AT x ” IS NOT TRUE. e.g. the functions f(x) = x & g(x) = x sin 1 x

; x ≠ 0 & g(0) = 0 are continuous at

x = 0 but not derivable at x = 0. NOTE CAREFULLY : (a) Let f ′+(a) = p & f ′_(a) = q where p & q are finite then : (i) p = q ⇒ f is derivable at x = a ⇒ f is continuous at x = a. (ii) p ≠ q ⇒ f is not derivable at x = a. It is very important to note that f may be still continuous at x = a. In short, for a function f : Differentiability ⇒ Continuity ; Continuity ⇒ / derivability ; Non derivibality ⇒ ; But discontinuity ⇒ Non derivability / discontinuous (b)

If a function f is not differentiable but is continuous at x = a it geometrically implies a sharp corner at x = a. ETOOS Academy Pvt. Ltd. : F-106, Road No. 2, Indraprastha Industrial Area, End of Evergreen Motors 2 (Mahindra Showroom), BSNL Office Lane, Jhalawar Road, Kota, Rajasthan (324005)

3.

(i) (ii)

DERIVABILITY OVER AN INTERVAL : f (x) is said to be derivable over an interval if it is derivable at each & every point of the interval f(x) is said to be derivable over the closed interval [a, b] if : for the points a and b, f ′(a+) & f ′(b −) exist & for any point c such that a < c < b, f ′(c+) & f′(c −) exist & are equal.

NOTE : 1. If f(x) & g(x) are derivable at x = a then the functions f(x) + g(x), f(x) − g(x) , f(x).g(x) will also be derivable at x = a & if g (a) ≠ 0 then the function f(x)/g(x) will also be derivable at x = a. 2. If f(x) is differentiable at x = a & g(x) is not differentiable at x = a , then the product function F(x) = f(x). g(x) can still be differentiable at x = a e.g. f(x) = x & g(x) = x. 3. If f(x) & g(x) both are not differentiable at x = a then the product function ; F(x) = f(x). g(x) can still be differentiable at x = a e.g. f(x) = x & g(x) = x. 4. If f(x) & g(x) both are non-deri. at x = a then the sum function F(x) = f(x) + g(x) may be a differentiable function. e.g. f(x) = x & g(x) = −x. 5. If f(x) is derivable at x = a ⇒ / f ′(x) is continuous at x = a. x 2 sin x1

e.g. f(x) = 

0

6.

if x ≠ 0 if x = 0

A surprising result : Suppose that the function f (x) and g (x) defined in the interval (x1, x2) containing the point x0, and if f is differentiable at x = x0 with f (x0) = 0 together with g is continuous as x = x0 then the function F (x) = f (x) · g (x) is differentiable at x = x0 e.g. F (x) = sinx · x2/3 is differentiable at x = 0.

EXERCISE–I Q.1

Discuss the continuity & differentiability of the function f(x) = sinx + sinx , x ∈ R. Draw a rough sketch of the graph of f(x).

Q.2

Examine the continuity and differentiability of f(x) = x + x − 1 + x − 2 x ∈ R. Also draw the graph of f(x).

Q.3

Given a differentiable function f (x) defined for all real x, and is such that f (x + h) – f (x) ≤ 6h2 for all real h and x. Show that f (x) is constant.

Q.4

 1  A function f is defined as follows : f(x) = 1+ | sin x |  π 2  2+ x − 2

(

)

for for

−∞ 1) is an odd natural number. Find f(10).

Q.25

A derivable function f : R+ → R satisfies the condition f (x) – f (y) ≥ ln

x + x – y for every x, y ∈ R+. y 100 1 If g denotes the derivative of f then compute the value of the sum ∑ g  . n =1  n 

EXERCISE–II

Fill in the blanks : Q.1

f (3 + h 2 ) − f ((3 − h 2 ) If f(x) is derivable at x = 3 & f ′(3) = 2 , then Limit = _______. 2

Q.2

If f(x) = sin x & g(x) = x3 then f[g(x)] is ______ & ______ at x = 0. (State continuity and derivability)

Q.3

Let f(x) be a function satisfying the condition f(− x) = f(x) for all real x. If f ′(0) exists, then its value is _____. x , x≠0  1 + e1 / x , the derivative from the right, f′(0+) = _____ & the derivative For the function f(x) =   0 , x=0 from the left, f′(0−) = _______.

Q.4

Q.5

Q.6

Q.7

h →0

2h

The number of points at which the function f(x) = max. {a − x, a + x, b}, − ∞ < x < ∞, 0 < a < b cannot be differentiable is ______. Select the correct alternative : (only one is correct) if x1 (A) derivable and continuous at x = 0 (B) derivable at x = 1 but not continuous at x = 1 (C) neither derivable nor continuous at x = 1 (D) not derivable at x = 0 but continuous at x = 1 For what triplets of real numbers (a, b, c) with a ≠ 0 the function x ≤1 x is differentiable for all real x ? 2 ax + bx + c otherwise

f(x) = 

(A) {(a, 1−2a, a)  a ε R, a ≠ 0 } (C) {(a, b, c)  a, b, c ε R, a + b + c = 1 } Q.8

(B) {(a, 1−2a, c)  a, c ε R, a ≠ 0 } (D) {(a, 1−2a, 0)  a ε R, a ≠ 0}

A function f defined as f(x) = x[x] for − 1 ≤ x ≤ 3 where [x] defines the greatest integer ≤ x is : (A) continuous at all points in the domain of f but non-derivable at a finite number of points (B) discontinuous at all points & hence non-derivable at all points in the domain of f (x) (C) discontinuous at a finite number of points but not derivable at all points in the domain of f (x) (D) discontinuous & also non-derivable at a finite number of points of f (x).

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Q.9

Q.10

[x] denotes the greatest integer less than or equal to x. If f(x) = [x] [sin πx] in (−1,1) then f(x) is : (A) continuous at x = 0 (B) continuous in (−1, 0) (C) differentiable in (−1,1) (D) none   log a [x] + [− x] a Given f(x) =     0

(

)

x

  [x ] +2[ − x ]  − 5    x   a  1   x   3+a  

for x ≠ 0 ; a > 1

where [ ] represents the integral

for x = 0

part function, then : (A) f is continuous but not differentiable at x = 0 (B) f is cont. & diff. at x = 0 (C) the differentiability of 'f' at x = 0 depends on the value of a (D) f is cont. & diff. at x = 0 and for a = e only. Q.11

x + {x} + x sin{x} for x ≠ 0 where {x} denotes the fractional part function, then : for x = 0 0

If f(x) = 

(A) 'f' is continuous & diff. at x = 0 (C) 'f' is continuous & diff. at x = 2 Q.12

The set of all points where the function f(x) = (A) (− ∞ , ∞)

(B) [ 0, ∞ )

(B) 'f' is continuous but not diff. at x = 0 (D) none of these x 1+ x

is differentiable is :

(C) (− ∞, 0) ∪ (0, ∞) (D) (0, ∞)

(E) none

Q.13

Let f be an injective and differentiable function such that f (x) · f (y) + 2 = f (x) + f (y) + f (xy) for all non negative real x and y with f '(0) = 0, f '(1) = 2 ≠ f (0), then (A) x f '(x) – 2 f (x) + 2 = 0 (B) x f '(x) + 2 f (x) – 2 = 0 (C) x f '(x) – f (x) + 1 = 0 (D) 2 f (x) = f '(x) + 2

Q.14

Let f (x) = [n + p sin x], x ∈ (0, π), n ∈ I and p is a prime number. The number of points where f (x) is not differentiable is (A) p – 1 (B) p + 1 (C) 2p + 1 (D) 2p – 1. Here [x] denotes greatest integer function.

Q.15

Consider the functions f (x) = x2 – 2x and g (x) = – | x | Statement-1: The composite function F (x) = f (g ( x ) ) is not derivable at x = 0. because Statement-2: F ' (0+) = 2 and F ' (0–) = – 2. (A) Statement-1 is true, statement-2 is true and statement-2 is correct explanation for statement-1. (B) Statement-1 is true, statement-2 is true and statement-2 is NOT the correct explanation for statement-1. (C) Statement-1 is true, statement-2 is false. (D) Statement-1 is false, statement-2 is true.

Q.16

2 2 Consider the function f (x) = x − x − 1 + 2 | x | −1 + 2 | x | −7 .

Statement-1: f is not differentiable at x = 1, – 1 and 0. because Statement-2: | x | not differentiable at x = 0 and | x2 – 1 | is not differentiable at x = 1 and – 1. (A) Statement-1 is true, statement-2 is true and statement-2 is correct explanation for statement-1. (B) Statement-1 is true, statement-2 is true and statement-2 is NOT the correct explanation for statement-1. (C) Statement-1 is true, statement-2 is false. (D) Statement-1 is false, statement-2 is true. ETOOS Academy Pvt. Ltd. : F-106, Road No. 2, Indraprastha Industrial Area, End of Evergreen Motors 6 (Mahindra Showroom), BSNL Office Lane, Jhalawar Road, Kota, Rajasthan (324005)

Select the correct alternative : (More than one are correct) Q.17

f(x) = x[x] in − 1 ≤ x ≤ 2 , where [x] is greatest integer ≤ x then f(x) is : (A) continuous at x = 0 (B) discontinuous x = 0 (C) not differentiable at x = 2 (D) differentiable at x = 2

Q.18

f(x) =1 + x.[cosx] in 0 < x ≤ π/2 , where [ ] denotes greatest integer function then , (A) It is continuous in 0 < x < π/2 (B) It is differentiable in 0 < x < π/2 (C) Its maximum value is 2 (D) It is not differentiable in 0 < x< π/2

Q.19

f(x) = (sin–1x)2. cos(1/x) if x ≠ 0 ; f(0) = 0 , f(x) is : (A) continuous no where in −1 ≤ x ≤ 1 (B) continuous every where in −1 ≤ x ≤ 1 (C) differentiable no where in −1 ≤ x ≤ 1 (D) differentiable everywhere in −1 < x < 1

Q.20

f(x) = x + sinx in  −

Q.21

 π π ,  . It is :  2 2

(A) Continuous no where (C) Differentiable no where

(B) Continuous every where (D) Differentiable everywhere except at x = 0

If f(x) = 2 + sin−1 x, it is : (A) continuous no where (C) differentiable no where in its domain

(B) continuous everywhere in its domain (D) Not differentiable at x = 0

Q.22

If f(x) = x². sin (1/x) , x ≠ 0 and f(0) = 0 then , (A) f(x) is continuous at x = 0 (B) f(x) is derivable at x = 0 (C) f′(x) is continuous at x = 0 (D) f′′(x) is not derivable at x = 0

Q.23

A function which is continuous & not differentiable at x = 0 is : (A) f(x) = x for x < 0 & f(x) = x² for x ≥ 0 (B) g(x) = x for x < 0 & g(x) = 2x for x ≥ 0 (C) h(x) = xx x ∈ R (D) K(x) = 1+x, x ∈ R

Q.24

If sin–1x + y = 2y then y as a function of x is : (A) defined for -1 ≤ x ≤ 1 (B) continuous at x = 0 (C) differentiable for all x

Q.25

(D) such that Min [ f (t ) / 0 ≤ t ≤ x] for 0 ≤ x ≤ π2

Let f(x) = Cosx & H(x) =  π

 −x

for

2

(A) H (x) is continuous & derivable in [0, 3] (C) H(x) is neither cont. nor deri. at x = π/2

π 2

0. cos2x + cos4x

Prove that if | a1 sin x + a2sin 2x + .......+ ansin nx | ≤ | sin x | for x ∈ R, then | a1 + 2a2 + 3a3 + ...... + nan | ≤ 1

The function f : R → R satisfies f (x2) · f ''(x) = f '(x) · f '(x2) for all real x. Given that f (1) = 1 and f '''(1) = 8, compute the value of f '(1) + f ''(1). x  d2y dy Q.7(a) Show that the substitution z = ln  tan  changes the equation + cot x + 4 y cos ec2 x = 0 to 2 2  d x dx (d2y/dz2) + 4 y = 0. (b) If the dependent variable y is changed to 'z' by the substitution y = tan z then the differential equation

Q.6

2

2 d2y 2(1 + y)  dy  d 2z  dz  2 1 = + cos z + k     , then find the value of k. is changed to 2 = dx 2 1 + y 2  dx  dx  dx 

[1+ (dy dx ) ] Show that R =

2 3/ 2

Q.8

2

d y dx

can be reduced to the form R2/3 =

2

1

(d y dx ) 2

2 2/3

+

1

(d x dy ) 2

2 2/3

.

Also show that, if x=a sin2θ(1+cos2θ) & y=acos2θ (1– cos2θ) then the value of R equals to 4a cos3θ. Q.9 Q.10

sin x if x ≠ 0 and f (0) = 1. Define the function f ' (x) for all x and find f '' (0) if it exist. x Suppose f and g are two functions such that f, g : R → R,

Let f (x) =

f (x) = ln 1 + 1 + x 2    then find the value of Q.11

g (x) = ln  x + 1 + x 2   

  1  '  f    + g' ( x ) at x = 1.   x  x≤0 then prove that x>0

x eg(x)

xe x  Let f (x) =  x + x 2 − x3

(a) (b)

and

f is continuous and differentiable for all x. f ' is continuous and differentiable for all x.

 1  3  x (1 − x ) sin  x 2  if 0 < x ≤ 1   Q.12 f : [0, 1] → R is defined as f (x) =  , then prove that  0 if x = 0

(a)

f is differentiable in [0, 1]

(b)

f is bounded in [0, 1] (c)

f ' is bounded in [0, 1]

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15

Q.13

 x + y f (x) + f (y)  = (k ∈ R , k ≠ 0, 2). Show that  k  k

Let f(x) be a derivable function at x = 0 & f  f (x) is either a zero or an odd linear function.

Q.14

f ( y) − a f ( x + y) − f ( x ) = + xy for all real x and y. If f (x) is differentiable and f ′(0) exists 2 2 for all real permissible values of 'a' and is equal to 5a − 1 − a 2 . Prove that f (x) is positive for all real x.

Let

Q.15

Column-I

Column-II

(A)

1 3  ln (1 + x ) ·sin x , if x > 0 f (x) =   0, if x ≤ 0

(P)

continuous everywhere but not differenti able at x = 0

(B)

1 2  ln (1 + x ) ·sin x , if x > 0 g (x) =   0, if x ≤ 0

(Q)

differenti able at x = 0 but derivative is discontinu ous at x = 0

(C)

 sin x   ln1 + 2 , if x > 0   u (x) =   0, if x ≤ 0

(R)

differentiable and has continuous derivative

(D)

v (x) = Lim t →0

(S)

continuous and differentiable at x = 0

2x 2 tan −1  2  π t 

Q.16

( x−a ) 4 If f (x) = ( x−b) 4 ( x−c) 4

( x−a )3 1 ( x−a ) 4 ( x−b)3 1 then f ′ (x) = λ . ( x−b) 4 ( x−c)3 1 ( x−c) 4

Q.17

cos( x+x 2 ) sin( x+x 2 ) −cos( x+x 2 ) If f(x) = sin( x−x 2 ) cos( x−x 2 ) sin( x−x 2 ) then find f'(x). sin2 x 0 sin2 x 2

Q.18

If α be a repeated root of a quadratic equation f(x) = 0 & A(x) , B(x) , C(x) be the polynomials of

( x−a ) 2 1 ( x−b) 2 1 . Find the value of λ. ( x−c) 2 1

A( x ) B( x ) C( x ) degree 3, 4 & 5 respectively , then show that A(α ) B(α) C(α ) is divisible by f(x), where dash A ' (α) B' (α ) C' (α)

denotes the derivative. Q.19

a +x b +x c+x Let f (x) = l + x m + x n + x . Show that f ′′ (x) = 0 and that f (x) = f (0) + k x where k denotes p+x

q+x

r+x

the sum of all the co-factors of the elements in f (0). Q.20

If Y = sX and Z = tX, where all the letters denotes the functions of x and suffixes denotes the differentiation w.r.t. x then prove that X X1 X2

Y Y1 Y2

Z s Z1 = X3 1 s2 Z2

t1 t2

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16

EXERCISE–III Evalute the following limits using L’Hospital’s Rule or otherwise : x + ln  x 2 + 1 − x    Lim 3 x →0 x

Q.1

 1 1− x2  Lim  −  −1 x →0 x2   x sin x

Q.3

Lim  1 − 1  x →0  x 2 sin 2 x  

Q.5

Lim

Q.7

Find the value of f(0) so that the function f(x)= 1 −

x →0

Q.2

Q.4

1 + sin x − cos x + ln (1 − x ) x·tan2 x

Q.6

Lim x (x

x →0

x

)

−1

+

sin x Lim sin x − (sin x ) x → π2 1 − sin x + ln (sin x )

x

2 e

2x

−1

, x ≠ 0 is continuous at x = 0 & examine the

differentiability of f(x) at x = 0. Q.8 Q.9

Lim x →0

sin(3x 2 ) ln.cos(2x 2 −x )

a sin x − bx + cx 2 + x 3 If Lim exists & is finite, find the values of a, b, c & the limit. x →0 2x 2.ln (1 + x ) − 2x 3 + x 4

x 6000 − (sin x ) 6000 Q.10 Evaluate: Lim x →0 x 2 ·(sin x ) 6000

1 − cos x ·cos 2x ·cos 3x........ cos nx has the value equal to 253, find the value of n x →0 x2 (where n ∈ N).

Q.11

If Lim

Q.12

Given a real valued function f(x) as follows:

(

)

x 2 + 2 cos x − 2 1 sin x − ln e x cos x for x < 0; f (0) = & f (x) = for x > 0. Test the 12 x4 6x 2 continuity and differentiability of f (x) at x = 0.

f (x) =

Q.13

Let a1 > a2 > a3 ............ an > 1; p1 > p2 > p3......... > pn > 0 ; such that p1 + p2 + p3 + ...... + pn = 1

(

Also F (x) = p1a1x + p 2 a 2x + ....... + p n a nx (a) Lim+ F( x ) x →0

Q.14 If Lim x →0

(b) Lim F( x ) x →∞

)

1x

. Compute (c) Lim F( x ) x→ − ∞

1 − cos 3x ·cos 9x ·cos 27 x......... cos 3n x = 310, find the value of n. 1 1 1 1 1 − cos x ·cos x ·cos x......... cos n x 3 9 27 3

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17

Q.15

Column-I contains function defined on R and Column-II contains their properties. Match them. Column-I Column-II n

(A)

π     1 + tan 2n  Lim  π  equal n →∞    1 + sin 3n  

(B)

Lim

x →0

1

+

(1 + cosec x )

1 ln (sin x )

equals

(P)

e

(Q)

e2

(R)

e–2/π

(S)

eπ/6

1x

(C)

2  Lim  cos −1 x  x →0  π 

equals

EXERCISE–IV

Q.1

x 2 −x , then find the domain and the range of f . Show that f is one-one. Also find the function If f (x) = 2 x +2x d f −1 (x ) and its domain. [ REE '99, 6 ] dx

Q.2(a) If x2 + y2 = 1, then : (A) y y′′ − 2 (y′)2 + 1 = 0 (C) y y′′ − (y′)2 − 1 = 0

(B) y y′′ + (y′)2 + 1 = 0 (D) y y′′ + 2 (y′)2 + 1 = 0 [ JEE 2000, Screening, 1 out of 35 ]

(b) Suppose p (x) = a0 + a1 x + a2 x2 + ...... + an xn . If p (x) ≤ ex − 1 − 1 for all x ≥ 0 prove that a1 + 2 a2 + ...... + n an ≤ 1 . [ JEE 2000 (Mains) 5 out of 100 ] Q.3(a) If ln (x + y) = 2xy, then y ' (0) = (A) 1 (B) –1  −1  x + c  b sin  2  ,   1  (b) f (x) =   2ax / 2 −1 e ,  x

−

(C) 2

(D) 0

[ JEE 2004 (Scr.)]

1