7_Continuity And Differentiability.pdf

CONTINUTITY AND DIFFERENTIABILITY

1

CONTINUITY AND DIFFERENTIABILITY 3.1 (a)

BASIC CONCEPTS AND IMPORTANT RESULTS Continuity of a real function at a point A function f is said to be left continuous or continuous from the left at x = c iff (i) f(c) exists

Lt f(x) exists and (ii) x®c -

Lt f(x) = f(c). (iii) x®c -

A function f is said to be right continuous or continuous from the right at x = c iff (i) f(c) exists

Lt f(x) exists and (ii) x®c +

Lt f(x) = f(c). (iii) x®c +

A function f is said to be continuous at x = c iff (i) f(c) exists

(ii) xLt f(x) exists and ®c

(iii) xLt f(x) = f(c). ®c

Hence, a function is continuous at x = c iff it is both left as well as right continuous at x = c. When xLt f(x) exists but either f(c) does not exist or xLt f(x) ¹ f(c), we say that f has a ®c ®c removable discontinuity; otherwise, we say that f has non-removable discontinuity. (b)

Continuity of a function in an interval A function f is said to be continuous in an open interval (a, b) iff f is continuous at every point of the interval (a, b) ; and f is said to be continuous in the closed interval [a, b] iff f is continuous in the open interval (a, b) and it is continuous at a from the right and at b from the left. Continuous function. A function is said to be a continuous function iff it is continuous at every point of its domain. In particular, if the domain is a closed interval, say [a, b], then f must be continuous in (a, b) and right continuous at a and left continuous at b. The set of all point where the function is continuous is called its domain of continuity. The domain of continuity of a function may be a proper subset of the domain of the function.

3.2

PROPERTIES OF CONTINUOUS FUNCTIONS Property 1. Let f, g be two functions continuous at x = c, then (i) af is continuous at x = c, " a Î R (ii) f + g is continuous at x = c (iii) f – g is continuous at x = c (iv) fg is continuous at x = c (v)

f g

is continuous at x = c, provided g(c) ¹ 0.

Property 2. Let D1 and D2 be the domains of continuity of the functions f and g respectively then (i) af is continuous on D1 for all a Î R (ii) f + g is continuous on D1 Ç D2 (iii) f – g is continuous on D1 Ç D2 (iv) fg is continuous on D1 Ç D2 (v)

f g

is continuous on D1 Ç D2 except those points where g(x) = 0.

Property 3. A polynomial function is continuous everywhere. In particular, every constant function and every identity function is continuous. Property 4. A rational function is continuous at every point of its domain. Property 5. If f is continuous at c, then | f | is also continuous at x = c. In particular, the function | x | is continuous for every x Î R. Property 6. Let f be a continuous one-one function defined on [a, b] with range [c, d], then the inverse function f–1 : [c, d] ® [a, b] is continuous on [c, d] Property 7. If f is continuous at c and g is continuous at f(c), then gof is continuous at c. Property 8. All the basic trigonometric functions i.e. sin x, cos x, tan x, cot x, sec x and cosec x are continuous. Property 9. All basic inverse trigonometric functions i.e. sin–1x, cos–1x, tan–1x, cot–1x, sec–1x, cosec–1 x are continuous (in their respective domains). Property 10. Theorem. If a function is differentiable at any point, it is necessarily continuous at that point. The converse of the above theorem may not be true i.e. a function may be continuous at a point but may not be derivable at that point.

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2 3.3 (a)

CONTINUTITY AND DIFFERENTIABILITY DERIVATIVE OF VARIOUS FUNCTIONS Derivative of composite functions Theorem. If u = g(x) is differentiable at x and y = f(u) is differentiable at u, then y is dy dx

differentiable at x and

=

dy du

.

du dx

.

If g is differentiable at x and f is differentiable at g(x), then the composite function h(x) = f(g(x)) is differentiable at x and h’(x) = f’(g(x)). g’(x). Chain Rule. The above rule is called the chain rule of differentiation, since determining the derivative of y = f(g(x)) at x involves the following chain of steps : (i) First, find the derivative of the outer function f at g(x). (ii) Second, find the derivative of the inner function g at x. (iii) The product of these two derivatives gives the required derivative of the composite function fog at x . (i)

(iii) (b)

(c)

dy dx

dy dx

=

.

dy dt dx dt

dx dy

dx dt

, provided

¹ 0.

(ii)

dx

1

dy dx

= dx , provided dy ¹ 0. dy

=1

d dx

(iv)

x

(| x |) = | x | , x ¹ 0.

Derivatives of inverse trigonometric functions 1

(i)

d dx

(sin–1 x) =

(ii)

d dx

(cos–1 x) = –

(iii)

d dx

(tan–1 x) =

(iv)

d dx

(cot–1 x) = –

(v)

d dx

(sec–1 x) =

(vi)

d dx

(cosec–1 x) = –

1 - x2

, x Î (–1, 1) i.e. | x | < 1

1 1 - x2 1

1+ x2

, x Î (–1, 1) i.e. | x | < 1

, for all x Î R

1 1+ x2

, for all x Î R

1 x x2 - 1

,x>1

1 x x2 - 1

,x>1

Derivatives of algebraic and trigonometric functions (i)

d dx

(xn) = nxn – 1

(ii)

d dx

(xx) = xx log ex

(iii)

d dx

(sin x) = cos x

(iv)

d dx

(cos x) = – sin x

(v)

d dx

(tan x) = sec2 x

(vi)

d dx

(cot x) = – cosec2 x

(vii)

d dx

(cosec x) = – cosec x cot x.

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3 (d)

(e)

CONTINUTITY AND DIFFERENTIABILITY Derivatives of exponential and logarithmic functions d dx

(ex) = ex , for all x Î R

(vi)

d dx

(loga | x |) = x log a , x ¹ 0, a > 0, a ¹ 1.

(ii)

d dx

(ax) = ax log a, a > 0, a ¹ 1, x Î R (v)

d dx

(log | x |) =

(iii)

d dx

(log x) =

d dx

(loga x) = x log a , x > 0, a > 0, a ¹ 1

1 ,x>0 x

(iv)

1

d dx

(un) = un

d dx

(n log u).

Derivatives of functions in parametric form If x and y are two variables such that both are explicitly expressed in terms of a third variable, say t, i.e. if x = f(t) and y = g(t), then such functions are called parametric functions and dy dx

(g)

1 ,x¹0 x

Logarithmic differentiation If u, n are differentiable functions of x, then

(f)

1

(i)

=

dy dt dx dt

, provided

dx dt

¹ 0.

Derivative of second order If a function f is differentiable at a point x, then its derivative f’ is called the first derivative or derivative of first order of the function f. If f’ is further differentiable at the same point x, then its derivative is called the second derivative or derivative of the second order of f at that point and is denoted by f’’. If the function f is defined by y = f(x), then its first and second derivatives are denoted by f ’(x) and f’’(x) or by

dy dx

and

d2 y dx2

or by y1 and y2 or by y’ and y’’ respectively..

3.4 (i)

ROLLE’S THEOREM AND LAGRANGE’S MEAN VALUE THEOREM Rolle’s theorem If a function f is (i) continuous in the closed interval [a, b] (ii) derivable in the open interval (a, b) and (iii) f(a) = f(b), then there exists atleast one real number c in (a, b) such that f’(c) = 0. Thus converse of Rolle’s theorem may not be true.

(ii)

Lagrange’s mean value theorem If a function f is (i) continuous in the closed interval [a, b] and (ii) derivable in the open interval (a, b), then there exists atleast one real number c in (a, b) such that f ’(c) = The converse of Lagrange’s mean value theorem may not be true.

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f (b) - f (a) b-a

CONTINUTITY AND DIFFERENTIABILITY

4

SOLVED PROBLESM Ex.1 Sol.

Is the function defined by f(x) = x2 – sin x + 5 continuous at x = p ? Here, f(p) = (p)2 – sin p – 5 = p2 – 5 lim

x ®p +

lim 2 f(x)= hlim ® 0 f(p+h)= h ® 0 [(p+h) –sin(p+h)–5] 2 2 = hlim ® 0 [(p + h) + sin h – 5] = p – 5

lim lim 2 and x lim ® p - f(x)= h ® 0 f(p–h) = h ® 0 [(p–h) –sin (p–h)–5] 2 2 = hlim ® 0 [(p – h) – sin h – 5] = p – 5

f(x) = x lim Since, xlim ® p - f(x) = f(p), ®p + the function f is continuous at x = p. Ex.2 Discuss the continuity of the cosine, cosecant, secant and cotangent functions. Sol. Continuity of f(x) = cos x Let a be an arbitrary point of the domain of the function f(x) = cos x. Then, f (a) = cos a lim

x ®a +

lim f(x) = hlim ® 0 f(a + h) = h ® 0 cos (a + h)

= hlim ® 0 [cos a cos h – sin a sin h] = cos a × 1 – sin a × 0 = cos a and

lim

x ®a -

lim f(x) = hlim ® 0 f(a – h) = h ® 0 cos (a – h)

= hlim ® 0 [cos a cos h + sin a sin h] = cos a × 1 + sin a × 0 = cos a lim Since, x lim ®a + f(x) = x ®a - f(x)=f(a), the function is continuous at x = a.

As a is an arbitrary point of the domain, the function is continuous on the domain of the functions, Proceed as above and prove yourself the continuity of other trigonometric Ex.3

Find all points of discontinuity of f, where ì sinx ï , if x < 0

f(x) = í x

ïî x + 1 , if x ³ 0

Sol.

The point of discontinuity of f can at most be x = 0. Let us examine the continuity of f at x = 0. lim lim Here, x lim ®0 + f(x)= h ® 0 f(0+h)= h ® 0 [(0+h)+1]=1

and

lim

x ®0 -

lim sin( 0 - h) = lim - sinh = 1 f(x) = hlim ® 0 f(0 – h) = h ® 0 h ®0

0 -h

-h

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CONTINUTITY AND DIFFERENTIABILITY

5 Also, f(0) = 0 + 1 = 1 Since,

lim

x ®0 +

f(x) = x lim ®0 - f(x) = f(0), f is continuous at x = 0.

Hence, there is no point of discontinuity of f. Ex.4

Determine if f defined by 1 ì 2 ïx sin , if x ¹ 0 x ïî 0, if x = 0

f(x) = í

Sol.

is a continuous function. It is sufficient to examine the continuity of the function f at x = 0. Here f (0) = 0 Also,

lim

x ®0 +

f(x) = hlim ® 0 f(0 + h)

1 ù é lim éh2 sin 1 ù = 0 (0 + h)2 sin = hlim = ú ® 0 êë h ® 0 êë 0 + hû h úû

and

lim

x ®0 -

f(x) = hlim ® 0 f(0 – h)

æ 1 öù 1 ù lim é 2 é (0 - h)2 sin ÷ú = 0 êh sin ç = hlim = ® 0 êë 0 - h úû h ® 0 ë è - h øû

é ù 1 êQ sin £ 1ú h ë û

lim Hence, x lim ®0 + f(x) = x ®0 - f(x) = f(0)

Ex.5

So, f is continuous at x = 0 This implies that f is a continuous function at all x Î R. Examine the continuity of f, where f is defined by ìsin x - cos x , if x ¹ 0 if x = 0

f(x) = íî - 1, Sol.

Here, f(0) = –1

lim lim Also, x lim ®0 + f(x)= h ® 0 f(0+h)= h ® 0 [sin(0+h)–cos(0+h)]

= hlim ® 0 [sin h – cos h] = –1 lim lim and x lim ®0 - f(x)= h ® 0 f(0–h)= h ® 0 [sin(0–h)–cos(0–h)]

= hlim ® 0 [–sin h – cos h]

[Q sin (–h) = –sin h]

= –0 – 1 = –1

and cos (–h) = cos h]

lim Hence, x lim ®0 + f(x) = x ®0 - f(x) = f(0)

So, f is continuous at x = 0; and hence continuous at all x Î R. Ex.6

Find the value of k so that the following function f is continuous at the indicated point : ì kx + 1 , if x £ 5

(i)

f(x) = í3 x - 5 , if x > 5 at x = 5 î

(ii)

ì 2 f(x) = í kx , if x £ 2 at x = 2 î3 ,

if x > 2

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6 Sol.

(i)

CONTINUTITY AND DIFFERENTIABILITY Since f is given to be continuous at x = 5, we have lim

x ®5 +

f(x) = x lim ®5 - f(x) = f(5)

lim

= hlim ® 0 f(5 – h) = f(5)

Þ

h ® 0 f(5 + h)

Þ

h ® 0 [3(5+h)–5]= h ® 0

Þ

10 = 5k + 1

(ii)

Since f is given to be continuous at x = 2, we have

lim

lim

lim

x ® 2+

[k(5–h)+1] = 5k + 1

Þ k=

9 5

f(x) = x lim ® 2- f(x) = f(2)

lim

Þ

h ® 0 f(2 + h)

Þ

h ® 0 (3)

lim

Þ

= hlim ® 0 f(2 – h) = f(2)

2 = hlim ® 0 [k(2 – h) ] = 4k

3 = 4k

Þ k=

3 4

Ex.7 Find the values of a and b such that the function defined by ì5, ï

if x £ 2

ïî 21,

if x ³ 10

f(x) = íax + b, if 2 < x £ 10 Sol.

is a continuous function. Since the function f is continuous, it is continuous at x = 2 as well as at x = 10. lim

f(x) = x lim ® 2- f(x) = f(2)

So,

x ® 2+

i.e.,

h ® 0 f(2 + h)

i.e., and

lim

= hlim ® 0 f(2 – h) = f(2)

2a + b = 5 (......1) lim

x ®10 +

lim f(x) = x ®10 - f(x) = f(10)

lim i.e., hlim ® 0 f(10 + h) = h ® 0 f(10 – h) = f(10)

i.e., 21 = 10a + b (......2) From (1) and (2), we find that a = 2 and b = 1 Ex.8 Show that the function defined by g( x) = x – [ x] is dis con tin uou s at all integral points. Here, [x] denotes the greatest integer less than or equal to x. Sol. The function f(x) = x – [x] can be written as ì x - (k - 1), if k - 1 < x < k if k < x < k + 1, where k is an arbitrary integer..

f(x) = íx - k, î

lim lim Now, x lim ®k + f(x)= h ® 0 f(k + h)= h ® 0 [(k+h) –k]=0 lim lim and x lim ®k - f(x)= h ® 0 f(k–h)= h ® 0 [(k–h)– (k–1)]=1

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CONTINUTITY AND DIFFERENTIABILITY

7

lim Since, x lim ®k + f(x) ¹ x ®k - f(x), the function f is not continuous at x = k.

Since k is an arbitrary integer, we can easily conclude that the function is discontinuous at all integral points. Ex.9

Verify LMV Theorem for the function ì 3 f(x) = íx + 2, when x £ 1 on [ -1, 2 ]. î 3x

Sol.

, when x > 1

Both x3 + 2 and 3x are polynomial functions. So, f (x) is continuous and differentiable everywhere except at x = 1. Here,

lim f ( x ) = 3.1 = 3

x ®1+

lim f ( x ) = 13 + 2 = 3

x ® 1-

As lim f(x) = lim f(x) = f(1),f(x) iscontinuous at x =1.

x®1+

x®1-

Obviously, then f(x) is continuous on [–1, 2]. Again to test the differentiability of f(x) at x = 1, we have 3 3 f ( x ) - f (1) lim ( x + 2) - (1 + 2) L f ‘(1) = xlim = ®1 x ®1

x -1

x -1

x3 - 1 2 = xlim = xlim ®1®1- (x + x + 1) = 3 x -1

f ( x ) - f (1) R f ‘(1) = xlim ®1+ x -1

3 x - 3.1 = xlim ®1+

= xlim ®1+ (3) = 3

x -1

As L f ‘(1) = R f ‘(1), the function f (x) is differentiable at x = 1. Hence, f is differentiable in (–1, 2). Thus, both the conditions required for the applicability of the LMV Theorem are satisfied and hence, there exists at least one c Î (–1, 2) such that f ‘(c) =

f (2) - f (-1) 2 - ( -1)

Þ f ‘(c) =

6 -1 3

=

5 3

Now, in x > 1, f ‘(x) = 3. So, f ‘(c) cannot be In x £ 1. Þ

5 3

in this interval.

f ‘(x) = 3x2 f ‘(c) = 3c2

Obviously, 3c2 = Both

5 3

5 3

and –

gives c2 = 5 3

5 9

or c = ±

5 3

lie in (–1, 2). Thus, LMV is verified for f(x) and [–1, 2].

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CONTINUTITY AND DIFFERENTIABILITY 8 Ex.10 Verify Rolle’s theorem for the function f (x) = x (x – 3)2 in the closed interval 0 £ x £ 3. Sol.

(i)

Here,

f(x) = x (x – 3)2 = x (x2 – 6x + 9) = x3 – 6x2 + 9x

Since f(x) is a polynomial function of x, it is continuous in [0, 3] (ii)

f ‘(x) = 3x2 – 12x + 9 exists uniquely in the open interval (0, 3)

(iii)

f(0) = (0)3 – 6(0)2 + 9(0) = 0 – 0 + 0 = 0 f(3) = (3)2 – 6(3)2 + 9(3) = 27 – 54 + 27 = 0

f(0) = f(3) Thus, all the three conditions are satisfied, Hence, Rolle’s Theorem is applicable. Let us now solve i.e.

f ‘(c) = 0

3c – 12c + 9 = 0 2

3(c2 – 4c + 3) = 0 (c – 3) (c – 1) = 0 c = 3, 1 SInce, c = 1 Î (0, 3), the Rolle’s Theorem is verified for the function. f(x) = x(x – 3)2 in the closed interval [0, 3]. Ex.11 Verify Rolle’s Theorem for the function f(x) = (x – a)m (x – b)n in [a, b] ; m, n being positive integers. Sol.

Here, f(x) is a polynomial function of degree (m + n). So, it is a continuous function in [a, b]. f ‘(x) = (x – a)m– 1 (x – b)n – 1 [m (x – b) + n (x – a)] exists uniquely in (a, b). So, it is derivable m (a, b). Further, f(a) = 0 and f(b) = 0. So, f(a) = f(b) Thus, all the three conditions of Rolle’s Theorem are satisfied. Hence, Rolle’s Theorem is applicable. Let us now solve f ‘(c) = 0 Þ (c–a)m – 1 (c – b)n – 1 [m (c – b) + n (c–a)] = 0 Since c =

mb + na m+n

Þ

c = a or c = b or c =

mb + na m+n

Î (a, b), the Rolle’s Theorem is verified. ì1 + x, if x £ 2

Ex.12 Show that the function f(x) = íî5 - x , if x > 2 is continuous at x = 2, but not differentiable at x = 2. Sol.

At x = 2, lim

lim f(x) = hlim ® 0 f(2 + h) = h ® 0 [5–(2+ h)] = 3

lim

lim f(x) = hlim ® 0 f(2 – h) = h ® 0 [1+(2–h)] = 3

x ® 2+ x ® 2-

Also, Since,

f(2) = 1 + 2 = 3 lim

x ® 2+

f(x) = x lim ® 2- f(x) = f(2), f(x) is continuous at x = 2.

f (2 - h) - f (2) Next, Lf ‘(2) = hlim ®0 -h

(1 + 2 - h) - (1 + 2) = hlim =1 ®0 = -h

Rf ‘(2)= hlim ®0

f (2 + h) - f ( 2) lim 5 - (2 + h) - (1 + 2) = h®0 = h h

= –1

Since, Lf ‘(2) ¹ Rf ‘(2), the function f is not differentiable at x = 2.

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CONTINUTITY AND DIFFERENTIABILITY

9 ì1 - x,

if x < 1

Ex.13 Show that the function f(x) = í x2 - 1, if x ³ 1 is continuous at x = 1, but not differentiable î Sol.

thereat. The function is continuous at x = 1, because lim

x ®1+

f(x) = xlim ®1- f(x) = f(1) as shown below :

lim

lim lim 2 2 f(x) = hlim ® 0 f(1 + h) = h ® 0 [(1 + h) – 1] = h ® 0 (h + 2h) = 0 ;

lim

lim lim f(x) = hlim ® 0 f(1 – h) = h ® 0 [1 – (1 – h)] = h ® 0 (h) = 0

x ®1+ x ®1-

and

f(1) = (1)2 – 1 = 1 – 1 = 0

f (1 + h) - f (1) Further,Rf ‘(1) = hlim ®0 h

[(1 + h ) 2 - 1] - [0 ] = hlim =2 ®0 h

f(1 - h) - f(1) [(1 - h ) 2 - 1] - [0 ] = hlim Lf ‘(1) = hlim ®0 ®0 h

h

æ h ö lim ç ÷ = hlim ® 0 è - h ø = h ® 0 (–1) = –1 Since, Rf ‘(1) ¹ Lf ‘(1), the function is not differentiable at x = 1. Ex.14 Show that the function f defined as

ì 3x - 2, if 0 < x £ 1 ï f(x) = í2x2 - x, if 1 < x £ 1 ï 5 x - 4, if x > 2 î

is continuous at x = 2, but not differentiable thereat. Sol.

lim lim At x=2, x lim ®2+ f(x)= h ® 0 f(2+h)= h ® 0 [5(2+h)–4]=6

lim

x ®2-

lim 2 f(x) = hlim ® 0 f(2–h)= h ® 0 [2(2–h) –(2–h)]

2 = hlim ® 0 [2(4 – 4h + h ) – (2 – h)] 2 = hlim ® 0 [6 – 7h + 2h ] = 6

and

f(2) = 2 (2)2 – 2 = 8 – 2 = 6

lim Since x lim ® 2+ f(x) = x ® 2- f(x)=f(2), the function f is continuous at x = 2. f (2 - h) - f (2) Next, Lf ‘(2) = hlim ®0 -h

2(2 - h)2 - ( 2 - h) - [5(2) - 4] 6 - 7h + 2h2 - 6 = hlim = =7 ®0 -h

-h

f (2 + h) - f (2) [5(2 + h) - 4] - [5(2) - 4] [5(2 + h) - 4] - [5(2) - 4] = hlim = hlim f ‘(2) = hlim ®0 ®0 ®0 h

h

é 6 + 5h - 6 ù = hlim ú =5 ® 0 êë h û

Since, Lf ’(2) = Rf ‘(2), the function f is not differentiable at x = 2.

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h

CONTINUTITY AND DIFFERENTIABILITY

10

UNSOLVED PROBLEMS EXERCISE – I ì 1 ï e x -1 ï , when x ¹ 0 í 1 ï ex +1 ï 0 , when x = 0 î

Q.1

Show that the function f(x) =

Q.2

ï Show that the function f(x) = í x sin x , when x ¹ 0

Q.3

Is the following function continuous at the origin ?

ì

f(0) = Q.4

Q.5

Q.6

1

ïî 0

f(x) =

cos ax - cos bx x2 b2 - a2 2

is continuous at x = 0

, when x = 0

, when x ¹ 0

, when x = 0

If the function defined by f(x) =

if f(x) =

is discontinuous at x = 0.

ì cos2 x - sin 2 x - 1 , ï í x2 + 1 - 1 ï k , î

ì ex - 1 ï , when x ¹ 0 í log(1 + 2x) ï k, when x = 0 î when x ¹ 0

is continuous at x = 0, find k.

when x = 0

Determine the value of k so that the function f(x) =

ì ï 1 - cos 4x , when x < 0 ï x2 ï k , when x = 0 is continuous at x = 0. í ï x , when x > 0 ï ïî 16 + x - 4

Q.7

Discuss the continuity of the function f(x) at x = 0 if f(x) =

Q.8

Let f(x) = íax + b, if 3 < x < 5

ì ï

2, if x £ 3

ïî

a , if x ³ 5

is continuous at x = 0, find the value of k.

ì x- | x | ï , when x ¹ 0 í 2 ïî 2 , when x = 0

find the value of a and b, so that f(x) is continuous. x -1 ì , x ¹1 ïï 2x 2 - 7 x + 5 í -1 ï , x =1 ïî 3

at x = 1

Q.9

Find the derivative of f(x) =

Q.10

Find the value of ‘a’ and ‘b’, so that the function f(x) = íbx + 2

Q.11

Differentiate the following w.r.t x :

Q.12

ì x 2 + 3x + a , if x £ 1 is differentiable at each x Î R. , if x > 1 î

1 - cos x 1 + cos x

(i)

log

(iv)

xö x æ log ç sin + cos ÷

(i)

If y =

è

2

1- x 1+ x

(ii)

2ø

loga æç x + x 2 + a2 ö÷ è

(v)

, prove that (1 – x2)

dy dx

ø

æ 1 + x sin x ö ÷ è 1 - x sin x ø

log ç

+y=0

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(iii)

log (sec x + tan x)

(vi)

log çç

æ 2 ö 2 ç x +a +x÷ 2 2 ÷÷ è x +a -xø

CONTINUTITY AND DIFFERENTIABILITY

11

Q.13

(ii)

2 æ x +1+ x -1ö ÷ , prove that dy = x + x - 1 If y = çç ÷ 2

(iii)

If y =

(iv)

If y =

(i)

, show that 2x If y = x + + y = 2 x (ii) x dx

è x +1 - x -1 ø cos x + sin x cos x - sin x

dx

dy dx

, show that

sec x + tan x sec x - tan x 1

pö 4ø

æ

= sec2 ç x + ÷ è

dy dx

, show that

x -1

= sec x (sec x + tan x)

dy

If y = x sin y, prove that x

dy dx

=

y (1 - x cos y )

(iii) Q.14

If y = x + x + x + ........ , prove that

Q.15

Given that cos

Q.16

If x = tan–1

Q.17

Differentiate : (i) (iii)

-1

dy

If x 1+ y + y 1+ x = 0, prove that = (1 + x )2 , x ¹ y dx

x 2

2t 1- t

2

æ

. cos

x 4

. cos

and y = sin–1

......= 2t

1+ t2

1

= ( 2y - 1)

1 sin x , prove that 2 x 2 dy dx

, show that

sec2

î

ö ÷÷ è x + 3x + 3 ø

+ tan–1 çç æ

1

2

ö ÷ 2 ÷ è 1- x ø

tan–1 çç

x

+

(ii)

þ

æ

1

2

x 2

æ

ö ÷÷ è x + 5x + 7 ø

+ tan–1 çç æ ç

1 2

4

1

2

x

ö ÷

+ tan–1 ç 2 ÷ è 1+ 1- x ø

(ii)

æ

2

æ 3 cos x - 4 sin x ö ÷ 5 è ø

Differentiate : sin–1 çç 2 è 1+ x

Q.20

Differentiate : (i)

Q.21

Discuss the continuity of the function f(x) = íî2x + 1, if x ³ 0

Q.22

If a function f(x) is defined as f(x) = í x - 4

Q.23 Q.24

Discuss the continuity of the function f(x) = | x | + |x – 1| in the interval [–1, 2] Show that f(x) = | x | is not differential at x = 1.

Q.25

Let f(x) = í2 - x , if x < 0 , show that f(x) is not derivable at x = 0. î

Q.26

2x

cos–1 ç

Q.19

ö ÷ ÷ ø

x2

+ ............ to n terms.

Differentiate : (i)

1

1

ö ÷÷ è 1 + 15 x ø

tan–1 çç

Q.18

æ

x 4

sec2 + ......= cosec2x –

= 1.

sin–1 ìíx 1 - x + x 1 - x 2 üý

ö ÷÷ è x + x + 1ø

tan–1 çç

x 8

dy dx

æ 1 + x2 - 1ö ç ÷ ÷÷ . x è ø

+ tan–1 ç ç

æ 1 + x + 1- x 2 è

sin–1 çç

ö ÷ ÷ ø

(ii)

æ

-1 ö

x-x ÷ cos–1 çç -1 ÷ èx+x

ø

ì2x - 1, if x < 0

ì| x-4| ï , x¹4 ïî 0

, x=4

show that f is everywhere continuous except at x = 4.

ì2 + x , if x ³ 0

Show that the function f(x) =

ì 2 æ 1ö ï x sinç ÷ , if x ¹ 0 í èxø ï 0 , if x = 0 î

is differential at x = 0 and f ‘(0) = 0.

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CONTINUTITY AND DIFFERENTIABILITY

12

BOARD PROBLES EXERCISE – II dy dx

log x

Q.1

If xy = ex – y, prove that

Q.2

If xp yq = (x + y)p + q, prove that

Q.3

Find

Q.4

If y = ae2x + be–x, prove that

Q.5

If y = A cos nx + B sin nx show that

Q.6

Discuss the continuity of the function f(x) at x = 0 if f(x) = í2x + 1 , x ³ 0 î

[C.B.S.E. 2002]

Q.7

Show that the function f(x) = 2x – | x | is continuous at x = 0.

[C.B.S.E. 2002]

Q.8

ï If the function f(x) = í 11

=

(1 + log x )2

dy dx

y x

=

.

[C.B.S.E. 2000]

.

[C.B.S.E. 2000]

æ 2ö

d2 y

x when y = log çç x ÷÷ .

dx2

[C.B.S.E. 2000]

èe ø

d2 y dx

2

dy dx

–

– 2y = 0.

d2 y dx2

[C.B.S.E. 2000]

+ n2y = 0.

[C.B.S.E. 2001] ì2 x - 1 , x < 0

ì 3ax + b

, x >1 , x = 1 is continuous. at x = 1, find the values of a and b. ïî 5ax - 2b , x < 1

[C.B.S.E. 2002] 1 - sin 2x 1 + sin 2x

dy dx

æp

ö

+ sec2 ç 4 - x ÷ = 0. è ø

Q.9

If y =

Q.10

dy æ p x ö If y = log çç tanæç + ö÷ ÷÷ , show that – sec x = 0.

Q.11

è

è4

, prove that

[C.B.S.E. 2002] [C.B.S.E. 2002]

dx

2 øø

Verify Lagrange’s mean value theorem for the following functions in the given intervals. [C.B.S.E. 2002] Also find ‘c’ of this theorem : (i) f(x)=x2 +x–1 in [0, 4] (ii) f(x)= x 2 - 4 on [2, 4]

Q.12

If y = ex (sin x + cos x), prove that

Q.13

Differentiate the following w.r.t. x

d2 y dx2

–2

dy dx

æ 1 - cos x ö

(ii) dy dx

Q.14

If x = a(t + sin t), y = a(1 – cos t), find

Q.15

Differentiate the following functions w.r.t x : æ 1 + sin x ö

at t =

æ 1 - sin x ö

(i) tan–1 çç 1 - sin x ÷÷ . (ii) cot–1 çç 1 + sin x ÷÷ è è ø ø æ

5x + 12 1 - x 2 çç 13 è

(iv) sin

ö ÷ ÷÷ ø

[C.B.S.E. 2002] [C.B.S.E. 2003]

(i) log çç 1 + cos x ÷÷ è ø

–1 ç

+ 2y = 0.

(v) tan

–1

log (x + 1+ x 2 )

p . 2

[C.B.S.E. 2003] [C.B.S.E. 2004] (iii)

æ 2 2 ç 1+ x - 1 - x çç 2 2 è 1 + x + 1- x

ö æ 1 + x 2 - 1÷ tan–1 çç ÷÷ x ç è

ö ÷ ÷÷ ø

ø

æ 1+ x - 1 - x ö

(vi) tan–1 çç 1 + x + 1 - x ÷÷ è ø

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CONTINUTITY AND DIFFERENTIABILITY

13 æx xö a2 ç sin -1 ÷ a2 - x 2 + ç2 2 ÷ø 2 è

d dx

= a2 - x 2 .

Q.16

Prove that

Q.17

If y = (sin x)x + (cos x)tan x, find

Q.18

Find

Q.19

Differentiate tan–1 çç

Q.20

If f(x) = ç

Q.21

Find

Q.22

æ x ö d2 y 1 If y = x log ç a + bx ÷ , prove that 2 = è ø x dx

Q.23

dy dx

, when x = a

1- t2 1+ t

2

æ 2x ö ÷÷ è 1 - x2 ø

æ3+ xö ÷ è 1+ x ø

dy dx

[C.B.S.E. 2004]

dy dx

,y=

[C.B.S.E. 2004] 2bt

[C.B.S.E. 2004]

1+ t2 æ 2x ö ÷÷ . è 1 + x2 ø

w.r.t. sin–1 çç

[C.B.S.E. 2004]

2 + 3x

, find f ’(0).

[C.B.S.E. 2005]

æ 1 + t2 ö 2t ÷ if : x = a çç 2 ÷, y= 2 è 1- t ø

[C.B.S.E. 2005]

1- t

If the function f defined by f(x) =

æ a ö ç ÷ è a + bx ø

2

.

ì ï 1 - cos 4x , x0 ï 16 + x - 4 î

[C.B.S.E. 2005]

is continuous at x = 0, find the value of a.

[C.B.S.E. 2006] 1

If y = x +

Q.25

÷ Differentiate w.r.t. x : tan–1 çç ÷ è 1 + sin x - 1 - sin x ø

Q.26

If y x 2 + 1 = log ( x 2 + 1 – x), prove that (x2 + 1) + xy + 1 = 0. dx

Q.27

If x = a sin 2t (1 + cos 2t) and y = b cos 2t (1 – cos 2t), show that ç

x

, then show that 2x

dy dx

Q.24

+y=2 x .

[C.B.S.E. 2006]

æ 1 + sin x + 1 - sin x ö

[C.B.S.E. 2006] dy

[C.B.S.E. 2006]

b æ dy ö = ÷ p dx a è øt =

.

4

[C.B.S.E. 2006] d2 y

Q.28

If y = cosec x + cot x, show that sin x .

Q.29

Verify LMV ; find ‘c’ f(x) = x2 + 2x + 3 in [4, 6] ì x 2 - 25 ï , x¹5 í x-5 ïî k , x =5

Q.30

If f(x) =

Q.31

If y = 3e2x + 2e3x, prove that

Q.32

If y = A emx + B enx, prove that

dx2

= y2.

[C.B.S.E. 2006] [C.B.S.E. 2006]

is continuous at x = 5, find the value of k. d2 y dx

2

–5

d2 y dx2

dy dx

+ 6y = 0.

– (m + n)

dy dx

+ mny = 0.

www.thinkiit.in

[C.B.S.E. 2007]

[C.B.S.E. 2007] [C.B.S.E. 2007]

CONTINUTITY AND DIFFERENTIABILITY

14 d2 y

dy dx

Q.33

If y = sin (log x), prove that x2

Q.34

For what value of k is the following function continuous at x = 2 ?

dx

2

+x

+ y = 0.

[C.B.S.E. 2007]

ì 2x + 1 , x < 2 ï k , x=2 ïî 3 x - 1 , x > 2

f(x) = í Q.35

Discuss the continuity of the following function at x = 0 : f(x) =

Q.36

Q.37

[C.B.S.E. 2008]

ì x 4 + 2x3 + x 2 ï , x¹0 . í tan-1 x ï 0 , x =0 î

ì 1 - sin 3 x ï ï 3 cos2 x ï a Let f(x) = í ï ï b(1 - sin x ) ï ( p - 2x )2 î

[C.B.S.E. 2008]

p 2 p p , if x = . If f(x) be a continuous function at x = , find a and b. 2 2 p , if x > 2 , if x <

[C.B.S.E. 2008] If f(x), defined by the following, is continuous at x = 0, find the values of a, b and c.

f(x) =

ì ï sin( a + 1)x + sin x , if x < 0 ï x ï c , if x = 0 í ï 2 ï x + bx - x , if x > 0 ï bx 3 / 2 î

[C.B.S.E. 2008]

ny

dy

Q.38

= If y = (x + x 2 + a 2 )n, prove that dx

Q.39

If x 1 + y + y 1 + x = 0 , find

Q.40

If y = x 2 + 1 – log çç x + 1 +

Q.41

If x = a ç cos q + log tan ÷ and y = a sin q, find the value of

Q.42

If y = (log (x + x 2 + 1 ))2, show that (1 + x2)

Q.43

If sin y = x sin (a + y), prove that

Q.44

If (cos x)y = (sin y)x, find

Q.45

If y =

Q.46

Differentiate the following function w.r.t. x : xsin x + (sin x)cos x

æ1 è

x + a2

.

[C.B.S.E. 2008]

.

[C.B.S.E. 2008]

1 ö÷ dy 2 ÷ , find dx x ø

.

[C.B.S.E. 2008]

qö 2ø

æ è

sin -1 x 1- x

dy dx

2

2

dy dx

dy dx

=

d2 y dx2

+x

dy dx

dy dx

at q =

– 2 = 0.

sin 2 (a + y ) . sin a

.

, show that (1 – x2)

p . 4

[C.B.S.E. 2008] [C.B.S.E. 2008] [C.B.S.E. 2009] [C.B.S.E. 2009]

d2 y dx

2

– 3x

dy dx

–y=0

www.thinkiit.in

[C.B.S.E. 2009] [C.B.S.E. 2009]

CONTINUTITY AND DIFFERENTIABILITY

15 sin x æ sin x + cos x log x ö ç ÷

Q.47 x

è x

Q.48 Find

dy dx

ø

+ (sin x)

cos x

(cos x cot x – sin x logsin x)

[C.B.S.E. 2009]

if (x2 + y2)2 = xy..

[C.B.S.E. 2009] d2 y

dy dx

Q.49

If y = 3 cos (log x) + 4 sin (log x), then show that x2 .

Q.50

If y = cos–1 ç ç

Q.51

If y = cosec–1 x, x > 1, then show that x (x2 – 1)

Q.52

If xy = ex – y, show that

Q.53

æ ö If x = tan ç log y ÷ , show that (1 + x2)

Q.54

dy y =- . If x = asin-1 t , y = acos-1 t , show that dx x

[C.B.S.E. 2012]

Q.55

Differentiate tan–1 ê

é 1 + x2 - 1 ù ú x ëê ûú

[C.B.S.E. 2012]

Q.56

2 2 2 dy dx dy If x = a (cos t + t sin t) and y = a (sin t – t cos t), 0 < t < 2p , find 2 , 2 and 2 .

æ 3x + 4 1 - x 2 ç 5 è

1 èa

dy dx

ö dy ÷ ÷÷ , find dx ø

=

dx2

+x

+y=0

.

[C.B.S.E. 2009] [C.B.S.E. 2010]

log x {log (xe)}2

ø

d2 y dx2

+ (2x2 – 1)

dy dx

=0

.

[C.B.S.E. 2010] [C.B.S.E. 2011]

d2y dx2

+ (2x – a)

dy dx

=0

[C.B.S.E. 2011]

with respect to x.

dt

dt

dx

[C.B.S.E. 2012] dy (1 + log)2 = dx log y

.

Q.57

If yx = ey – x, prove that

Q.58

Differentiate the following with respect to x :

[C.B.S.E. 2013] [C.B.S.E. 2013]

æ 2 x +1.3x ö

sin–1 çç 1 + (36)x ÷÷ è

Q.59

ø

Find the value of k, for which

f(x) =

[C.B.S.E. 2013]

ì 1 + kx - 1 - kx , if - 1 £ x < 0 ïï x í 2x + 1 ï , if 0 £ x < 1 ïî x -1

is continuous at x = 0. OR If x = a cos3 q and y = a sin3 q, then find the value of

d2 y dx 2

at q =

p 6

.

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CONTINUTITY AND DIFFERENTIABILITY

16

ANSWER KEY EXERCISE – 1 (UNSOLVED PROBLEMS) 3. yes

4. 1/2

5. –4

9. –2/9

10. 3 and 5

11. (i) cosec x (ii)

(v)

2( x cos x + sin x) 2

1 - x sin x

1

2

(vi)

2

2

x +a

2

6. 8

1

17. (i)

1- x

2

7. discontinuous 1

(iii) sec x (iv)

log a x 2 + a2

+

1 2 x-x

(ii)

2

8. 7/5 and –17/2

5 1 + 25 x

2

–

æp

1 2

xö

tan ç 4 - 2 ÷ è ø

3 1 + 9x2

1

(iii) 1 + ( x + x )2 – 1+ x2

18. (i)

3 2 1- x

2

-1

(ii) 1

19. 2(1 + x 2 )

20. (i)

-1 2 1- x

(ii)

2

-2 1+ x2

EXERCISE – 2 (BOARD PROBLEMS) 3.

-2 x2

6. Discontinuous

8. 3,2

11. (i) 2 (ii) 6

1

13. (i) cosec x (ii)

1+ x2

1 2

14. 1

15. (i)

19. 1

21.

1+ t2 2at

35. continuous 36.

1 ,4 2

(ii)

1 2

1

(iii) 2(1 + x 2 )

23. 8

(iv)

3 2

log sin y + y tan x

48.

1- x

25. –1/2

37. – , any real number,

44. log cos x - cot y

1

y - 4x 3 - 4xy 2 2

3

4x y + 4y - x

50.

x

(v)

2

1- x

29. 5

1 2

-1

40.

1- x

1

2

(vi)

-1 2 1- x

30. 10

39. (1 + x )2 -1

4

55. 2(1 + x2 )

3 56. at cos t, at sin t and sec t

at

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x2 + 1 x

2

18.

- b(1 - t 2 ) 2at

32. 5/2 34. 5

41. 1