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CONTINUITY and DIFFERENTIABILITY

Top Definitions

1.A function f(x) is said to be continuous at a point c if.

2.A real function f is said to be continuous if it is continuous at every point in the domain of f.

3.If f and g are real-valued functions such that (f o g) is defined at c, then

(f g)(x) – f(g(x)).

If g is continuous at c and if f is continuous at g(c), then (f o g) is continuous at c.

4.A function f is differentiable at a point c if Left Hand Derivative (LHD) = Right Hand Derivative (RHD),

5.If a function f is differentiable at every point in its domain, then

6.If LHD ≠ RHD, then the function f(x) is not differentiable at x = c.

7.Geometrical meaning of differentiability:

The function f(x) is differentiable at a point P if there exists a unique tangent at point P. In other words, f(x) is differentiable at a point P if the curve does not have P as its corner point.

8.A function is said to be differentiable in an interval (a, b) if it is differentiable at every point of (a, b).

9.A function is said to be differentiable in an interval [a, b] if it is differentiable at every point of [a, b].

10.Chain Rule of Differentiation: If f is a composite function of two functions u and v such that f = v(t) and

11.Logarithm of a to the base b is x, i.e., log_ba = x if b^x = a, where b > 1 is a real number. Logarithm of a to base b is denoted by log_ba.

12.Functions of the form x = f(t) and y = g(t) are parametric functions.

13.Rolle’s Theorem: If f : [a, b] → R is continuous on [a, b] and differentiable on (a, b) such that f(a) = f(b), then there exists some c in (a, b) such that f’(c) = 0.

14.Mean Value Theorem: If f : [a, b] → R is continuous on [a, b] and differentiable on (a, b), then thereexists some c in (a, b) such that

Top Concepts

1.A function is continuous at x = c if the function is defined at x = c and the value of the function at x = c equals the limit of the function at x = c.

2.If function f is not continuous at c, then f is discontinuous at c and c is called the point of discontinuity of f.

3.Every polynomial function is continuous.

4.The greatest integer function [x] is not continuous at the integral values of x.

5.Every rational function is continuous.

Algebra of continuous functions:

i.Let f and g be two real functions continuous at a real number c, then f + g is continuous at x = c.

ii.f – g is continuous at x = c.

iii.f . g is continuous at x = c.

iv.

(\frac{f}{g})

is continuous at x = c, [provided g(c) ≠ 0].

v.kf is continuous at x = c, where k is a constant.

6.Consider the following functions:

i.Constant function

ii.Identity function

iii.Polynomial function

iv.Modulus function

v.Exponential function

vi.Sine and cosine functions

The above functions are continuous everywhere.

7.Consider the following functions:

i.Logarithmic function

ii.Rational function

iii.Tangent, cotangent, secant and cosecant functions

The above functions are continuous in their domains.

8.If f is a continuous function, then |f| and

\frac{1}{f}

are continuous in their domains.

9.Inverse functions sin^-1x, cos^-1x, tan^-1x, cot^-1 x, cosec^-1 x and sec^-1x are continuous functions on their respective domains.

10. The derivative of a function f with respect to x is f’(x) which is given by

11. If a function f is differentiable at a point c, then it is also continuous at that point.

12. Every differentiable function is continuous, but the converse is not true.

13. Every polynomial function is differentiable at each x

\in

R .

14. Every constant function is differentiable at each x

\in

R .

15. The chain rule is used to differentiate composites of functions.

16. The derivative of an even function is an odd function and that of an odd function is an even function.

17. Algebra of Derivatives

If u and v are two functions which are differentiable, then

18. Implicit Functions

If it is not possible to separate the variables x and y, then the function f is known as an implicit function.

19. Exponential function: A function of the form y = f(x) = b^x, where base b > 1.

1.Domain of the exponential function is R, the set of all real numbers.

2.The point (0, 1) is always on the graph of the exponential function.

3.The exponential function is ever increasing.

20. The exponential function is differentiable at each x

\in

R .

21. Properties of logarithmic functions:

i.Domain of log function is R⁺.

ii.The log function is ever increasing.

iii.For ‘x’ very near to zero, the value of log x can be made lesser than any given real number.

22. Logarithmic differentiation is a powerful technique to differentiate functions of the form f(x) = [u(x)]^v^(x). Here both f(x) and u(x) need to be positive.

23. To find the derivative of a product of a number of functions or a quotient of a number of functions, take the logarithm of both sides first and then differentiate.

24. Logarithmic Differentiation

y = a^x

Taking logarithm on both sides

log y = log a^x.

Using the property of logarithms

log y = x log a

Now differentiating the implicit function

25. The logarithmic function is differentiable at each point in its domain.

26. Trigonometric and inverse-trigonometric functions are differentiable in their respective domains.

27. The sum, difference, product and quotient of two differentiable functions are differentiable.

28. The composition of a differentiable function is a differentiable function.

29. A relation between variables x and y expressed in the form x = f(t) and y = g(t) is the parametric form with t as the parameter. Parametric equation of parabola y² = 4ax is x = at², y = 2at.

30. Differentiation of an infinite series: If f(x) is a function of an infinite series, then to differentiate the function f(x), use the fact that an infinite series remains unaltered even after the deletion of a term.

31. Parametric Differentiation:

Differentiation of the functions of the form x = f(t) and y = g(t):

32. Let u = f(x) and v = g(x) be two functions of x. Hence, to find the derivative of f(x) with respect g(x), we use the following formula:

33. If y = f(x) and

\frac{d y}{d x} = f ‘ (x)

and if f’(x) is differentiable, then

34. If x = f(t) and y = g(t), then

Top Formulae

1.Derivative of a function at a point

2.Properties of logarithms

3.Derivatives of Functions

4.Differentiation of constant functions

1.Differentiation of a constant function is zero, i.e.

2.If f(x) is a differentiable function and c is a constant, then cf(x) is a differentiable function such that

5.Some useful results in finding derivatives

Important Questions

Multiple Choice questions-

1. The function

Diagram Description automatically generated with low confidence

is continuous at x = 0, then the value of ‘k’ is:

(a) 3

(b) 2

(d) 1.5.

2. The function f(x) = [x], where [x] denotes the greatest integer function, is continuous at:

(a) 4

(b)-2

(d) 1.5.

3. The value of ‘k’ which makes the function defined by

MCQ Questions for Class 12 Maths Chapter 5 Continuity and Differentiability with Answers 2

continuous at x = 0 is

(a) -8

(b) 1

(d) None of these.

4. Differential coefficient of sec (tan^-1 x) w.r.t. x is

Diagram, schematic Description automatically generated

5. If y = log ( $\frac{1 - x_{2}}{1 + x_{2}}$ ) then $\frac{d y}{d x}$ is equal to:

Schematic Description automatically generated

Text Description automatically generated

7. If u = sin-1 ( $\frac{2 x}{1 + x_{2}}$ ) and u = tan-1 ( $\frac{2 x}{1 - x_{2}}$ ) then $\frac{d y}{d x}$ is

(a) 12

(b) x

(d) 1

8. If x = t², y = t³, then $\frac{d^{2} y}{{d x}^{2}}$ is

(a) $\frac{3}{2}$

(b) $\frac{3}{4 t}$

(d) $\frac{3 t}{2}$

9. The value of ‘c’ in Rolle’s Theorem for the function f(x) = x³ – 3x in the interval [0, √3] is

(a) 1

(b) -1

(d) $\frac{1}{3}$

10. The value of ‘c’ in Mean Value Theorem for the function f(x) = x (x – 2), x ∈ [1, 2] is

(a) $\frac{3}{2}$

(b) $\frac{2}{3}$

(d) $\frac{3}{4}$

Very Short Questions:

1.If y = log (cos ex), then find

\frac{d y}{d x}

(Delhi 2019)

2.Differentiate cos {sin (x)₂} w.r.t. x. (Outside Delhi 2019)

3.Differentiate sin²(x²) w.r.t. x². (C.B.S.E. Sample Paper 2018-19)

4.Find

\frac{d y}{d x}

, if y + siny = cos or.

6.Is it true that x = e^logx for all real x? (N.C.E.R.T.)

7.Differentiate the following w.r.t.x : 3^{x + 2}. (N.C.E.R.T.)

8.Differentiate log (1 + θ) w.r.t. sin^-1θ.

9.If y = x^x, find

\frac{d y}{d x}

10.

Short Questions:

1.Discuss the continuity of the function: f(x) = |x| at x = 0. (N.C.E.R.T.)

2.If f(x) = x + 1, find

\frac{d}{d x}

(fof)(x). (C.B.S.E. 2019)

3.Differentiate tan⁻¹ (

\frac{c o s x - s i n x}{c o s x + s i n x}

) with respect to x. (C.B.S.E. 2018 C)

4.Differentiate: tan^-1 (

\frac{1 + c o s x}{s i n x}

) with respect to x. (C.B.S.E. 2018)

5.Write the integrating factor of the differential equation:

Chart Description automatically generated (tan^-1 y – x) dy = (1 + y²) dx. (C.B.S.E. 2019 (Outside Delhi))

Long Questions:

1.Find the value of ‘a’ for which the function ‘f’ defined as:

Class 12 Maths Important Questions Chapter 5 Continuity and Differentiability 20

s continuous at x = 0 (CBSE 2011)

2.Find the values of ‘p’ and ‘q’ for which:

Class 12 Maths Important Questions Chapter 5 Continuity and Differentiability 23

is continuous at x = 2 (CBSE 2016)

3.Find the value of ‘k’ for which

Class 12 Maths Important Questions Chapter 5 Continuity and Differentiability 26

is continuous at x = 0 (A.I.C.B.S.E. 2013)

4.For what values of ‘a’ and ‘b\ the function ‘f’ defined as:

Class 12 Maths Important Questions Chapter 5 Continuity and Differentiability 29

is continuous at x = 1. (CBSE 2011)

Assertion and Reason Questions-

1. Two statements are given-one labelled Assertion (A) and the other labelled Reason (R). Select the correct answer to these questions from the codes(a), (b), (c) and (d) as given below.

a)Both A and R are true and R is the correct explanation of A.

b)Both A and R are true but R is not the correct explanation of A.

c)A is true but R is false.

d)A is false and R is true.

e)Both A and R are false.

Assertion(A): A picture containing text, close Description automatically generated is continuous at x = 0.

Reason (R):Both h(x) = x², A picture containing text, close Description automatically generated are continuous at x = 0.

2. Two statements are given-one labelled Assertion (A) and the other labelled Reason (R). Select the correct answer to these questions from the codes(a), (b), (c) and (d) as given below.

a)Both A and R are true and R is the correct explanation of A.

b)Both A and R are true but R is not the correct explanation of A.

c)A is true but R is false.

d)A is false and R is true.

e)Both A and R are false.

Assertion (A):The function is continuous everywhere.

Reason (R):f(x) is periodic function.

Case Study Questions-

1.If a relation between x and y is such that y cannot be expressed in terms of x, then y is called an implicit function of x. When a given relation expresses y as an implicit function of x and we want to find

\frac{d y}{d x}

, then we differentiate every term of the given relation w.r.t. x, remembering that a tenn in y is first differentiated w.r.t. y and then multiplied by

\frac{d y}{d x}

Based on the ab:ve information, find the value of $\frac{d y}{d x}$ in each of the following questions.

1.If y = f(u) is a differentiable function of u and u = g(x) is a differentiable function of x, then y = f(g(x)] is a differentiable function of x and

This rule is also known as CHAIN RULE.

Based on the above information, find the derivative of functions w.r.t. x in the following questions.

Answer Key-

Multiple Choice questions-

1.Answer: (b) 2

2.Answer: (d) 1.5.

3.Answer: (d) None of these.

4.Answer:

5.Answer:

6.Answer:

7.Answer: (d) 1

8.Answer: (b)

\frac{3}{4 t}

9.Answer: (a) 1

10.Answer: (a)

\frac{3}{2}

Very Short Answer:

1.Solution:

We have: y = log (cos e^x)

= – e^x tan e^x

2.Solution:

Let y = cos {sin (x)²}.

∴ $\frac{d y}{d x}$ = – sin {sin (x)²}. $\frac{d y}{d x}$ {sin (x)²}

= – sin {sin (x)²}. cos(x)² $\frac{d y}{d x}$ (x²)

= – sin {sin (x)²}. cos(x)²2x

= -2x cos(x)² sin {sin(x)²}.

3.Solution:

Let y = sin²(x²).

∴ $\frac{d y}{d x}$ = 2 sin (x²) cos (x²) = sin (2x²).

4.Solution:

We have: y + sin y = cos x.

Differentiating w.r,t. x, we get:

$\frac{d y}{d x}$ + cos y. $\frac{d y}{d x}$ = – sin x

(1 + cos y) $\frac{d y}{d x}$ = -sin x

Hence, $\frac{d y}{d x}$ = − $\frac{sin . x}{1 + cos . y}$

where y ≠ (2n + 1)π, n ∈ Z.

5.Solution:

Put 3x = sin θ.

y = sin^-1 (2 sin θ cos θ)

= sin^-1 (sin 2θ) = 2θ

= 2 sin^-1 3x

6.Solution:

The given equation is x = e^logx

This is not true for non-positive real numbers.

[ ∵ Domain of log function is R+]

Now, let y = e^logx

If y > 0, taking logs.,

log y = log (e^logx) = log x.log e

= log x . 1 = log x

⇒ y = x.

Hence, x = e^logx is true only for positive values of x.

7.Solution:

Let y = 3^{x + 2}.

$\frac{d y}{d x}$ = ^{3x + 2}.log3. $\frac{d}{d x}$ (x + 2)

= 3^{x + 2} .log3.(1 + 0)

= 3^{x + 2}. log 3 = log 3 (3^{x + 2}).

8.Solution:

Let y = log (1 + θ) and u = sin^-1θ.

Class 12 Maths Important Questions Chapter 5 Continuity and Differentiability 1

Class 12 Maths Important Questions Chapter 5 Continuity and Differentiability 2

9.Solution:

Here y = x^x…(1)

Taking logs., log y = log x^x

⇒ log y = x log x.

Differentiating w.r.t. x, we get:

$\frac{1}{y}$ ⋅ $\frac{d y}{d x}$ = x 1x + logx. (1)

= 1 + log x.

Hence, $\frac{d y}{d x}$ = y (1 + log x) dx

= x^x (1 + log x). [Using (1)]

10.Solution:

The given series can be written as:

Squaring, y² = 2^x + y

⇒ y²– y = 2^x.

Diff. w.r.t. x, (2y -1) $\frac{d y}{d x}$ = 2^x log 2.

Short Answer:

1.Solution:

Class 12 Maths Important Questions Chapter 5 Continuity and Differentiability 3

Hence ‘f’ is continuous at x = 0.

2.Solution:

We have : f(x) = x + 1 …(1)

∴fof(x) = f (f(x)) =f(x)+ 1

= (x + 1) + 1 = x + 2.

∴ $\frac{d}{d x}$ (fof)(x).) = $\frac{d}{d x}$ (x + 2) = 1 + 0 = 1.

3.Solution:

Class 12 Maths Important Questions Chapter 5 Continuity and Differentiability 4

Class 12 Maths Important Questions Chapter 5 Continuity and Differentiability 5

4.Solution:

Class 12 Maths Important Questions Chapter 5 Continuity and Differentiability 6

5.Solution:

The given differential equation is:

(tan^-1 y – x) dy = (1 + y²) dx

Diagram, schematic Description automatically generated

6.Solution:

Class 12 Maths Important Questions Chapter 5 Continuity and Differentiability 7

Class 12 Maths Important Questions Chapter 5 Continuity and Differentiability 8

7.Solution:

Class 12 Maths Important Questions Chapter 5 Continuity and Differentiability 9

8.Solution:

Class 12 Maths Important Questions Chapter 5 Continuity and Differentiability 10

Long Answer:

1.Solution:

Class 12 Maths Important Questions Chapter 5 Continuity and Differentiability 21

Class 12 Maths Important Questions Chapter 5 Continuity and Differentiability 22

Also f(0) = a sin π/2 (0+1)

= a sin π/2 = a(1) = a

For continuity,

⇒ a = 1/2 = a

Hence, a = ½

2.Solution:

Class 12 Maths Important Questions Chapter 5 Continuity and Differentiability 24

Class 12 Maths Important Questions Chapter 5 Continuity and Differentiability 25

Hence p = 1/2 and q = 4

3.Solution:

Class 12 Maths Important Questions Chapter 5 Continuity and Differentiability 27

Class 12 Maths Important Questions Chapter 5 Continuity and Differentiability 28

⇒ k = -1 = -1

Hence k = -1

4.Solution:

lim_x→1 – f(x) = lim_x→1 − (3ax +b)

= lim_h→0 (3a (1-h) + b]

= 3a(1 – 0) + b

= 3a + b

lim_x→1 + f(x) = lim_x→1 + (5ax − 2b)

= lim_h→0 [5a (1+h) – 2b]

= 5a (1+0) – 2b

= 5a – 2b

Also f(1) = 11

Since ‘f’ is continuous at x = 1,

∴ lim_x_→₁ − f(x) = lim_x_→₁ + f(x) = f(1)

⇒ 3a + b = 5a – 2b = 11.

From first and third,

3a + b = 11 …………… (1)

From last two,

5a – 2b = 11 …………… (2)

Multiplying (1) by 2,

6a + 2b = 22 ………….. (3)

Adding (2) and (3),

11a = 33

⇒ a = 3.

Putting in (1),

3(3) + b = 11

⇒ b = 11 – 9 = 2.

Hence, a = 3 and b = 2.

Case Study Answers-

1.Answer :

2.Answer :

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