NCERT Solutions for Class 12 Maths Chapter 5 Continuity and Differentiability

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NCERT Solutions for Class 12 Mathematics Chapter 5 Continuity and Differentiability is included in this article. Continuity of a function refers to the characteristic of a function as a result of which, the graphical form of that function is a continuous wave. A differentiable function is a function whose derivative is present at each point in its domain.

Chapter 5 Continuity and Differentiability will carry a weightage of 8 to 17 marks in the CBSE Class 12 examination. Around 3-4 short answer questions can come from Mean Value Theorem, Rolle’s Theorem, Limits, Euler’s Number, Quotient Rule.

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NCERT Solutions for Class 12 Mathematics Chapter 5 Continuity and Differentiability

NCERT Solutions

NCERT Solution

NCERT Solutions Class 12 Mathematics Chapter 5 Important Topics

Continuity and Differentiability is an important topic in the board examination as per CBSE Class 12 exam pattern. In NCERT Class 12 Mathematics Chapter 5, derivative of composite functions, chain rule, derivative of inverse trigonometric functions, derivative of implicit functions, the concept of exponential and logarithmic functions, derivatives of logarithmic and exponential functions, logarithmic differentiation are discussed. The important topics that are covered in the Continuity and Differentiability chapter are:

Mean Value Theorem

As per Mean Value Theorem, if a function f is continuous on the closed interval [a,b] and differentiable on the open interval (a,b), then a point c will exist in the interval (a,b) such that f'(c) is equal to the function's average rate of change over [a,b].

Mean value theorem

Mean value theorem

Let us assume that f(x) is a function satisfying below conditions:

f(x) is Continuous in [a,b]
f(x) is Differentiable in (a,b)

Then, there exists a number c, s.t. a < c < b and

f(b) – f(a) = f ‘(c) (b – a)

Rolle’s Theorem

Rolle’s theorem is the special case of Lagrange’s mean-value theorem of differential calculus and it states that if a function f is continuous on the closed interval [a, b] and differentiable on the open interval (a, b) in a way that f(a) = f(b).

Rolle's theorem

Rolle's theorem

Suppose a function f is defined in the closed interval [a, b] in such a way that it satisfies the following conditions:

i) The function f is continuous on the closed interval [a, b]

ii)The function f is differentiable on the open interval (a, b)

iii) Now if f (a) = f (b) , then there exists at least one value of x, let us assume this value to be c, which lies between a and b i.e. (a < c < b ) in such a way that f‘(c) = 0 .

Limits

Limits, which are important in calculus and mathematical analysis, can be defined as a value that a function approaches the output for the given input values and are used to define integrals, derivatives, and continuity. The "lim" denotes the limit, and the right arrow denotes the fact that function f(x) approaches the limit L as x approaches c.

Limits

Limits

Mathematical limits are unique real numbers. Consider the limit of a real-valued function "f" and a real number "c," which is generally defined as:

lim_x→cf(x)=L

It says, “The limit of f of x as x approaches c equals L.”

Euler’s Number

Euler’s Number ‘e’ is a numerical constant that is found in many contexts and is the base for natural logarithms. The value of e is 2.718281828459045…so on, where the digits go on forever in a series that never ends or repeats (similar to pi). The Euler’s number is the limit of (1 + 1/n)ⁿ as n approaches infinity, an expression that arises in the study of compound interest. It can be expressed as the sum of infinite numbers as well.

Euler's Number

Euler's Number

Quotient Rule

The quotient Rule in Calculus is defined as a method for determining the derivative (differentiation) of a function in the form of the ratio of two differentiable functions. It is a formal rule, that follows the definition of the limit of the derivative and is used in the differentiation problems in which one function is divided by the other function.

The Quotient rule

The Quotient rule

NCERT Solutions For Class 12 Maths Chapter 5 Exercises

The detailed solutions for all the NCERT Solutions for Continuity and Differentiability under different exercises are as follows:

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NCERT Solutions for Class 12 Maths Chapter 5 Continuity and Differentiability

NCERT Solutions for Class 12 Mathematics Chapter 5 Continuity and Differentiability

NCERT Solutions Class 12 Mathematics Chapter 5 Important Topics

Mean Value Theorem

Rolle’s Theorem

Limits

Euler’s Number

Quotient Rule

NCERT Solutions For Class 12 Maths Chapter 5 Exercises

CBSE CLASS XII Related Questions

1.
Solve system of linear equations, using matrix method.
x-y+2z=7
3x+4y-5z=-5
2x-y+3z=12

2.
If A=\(\begin{bmatrix}2&-1&1\\-1&2&-1\\1&-1&2\end{bmatrix}\)verify that A³-6A²+9A-4 I=0 and hence find A^-1

3.
Let f: R→R be defined as f(x) = 3x. Choose the correct answer.

4.
If A'= \(\begin{bmatrix} 3 & 4 \\ -1 & 2 \\ 0 &1 \end{bmatrix}\)\(\begin{bmatrix} -1 & 2 & 1 \\ 1 &2 & 3\end{bmatrix}\) , then verify that
(i) \((A+B)'=A'+B' \)
(ii) \((A-B)'=A'-B'\)

5.
If (i) A=\(\begin{bmatrix} \cos\alpha & \sin\alpha\\ -\sin\alpha & \cos\alpha \end{bmatrix}\),then verify that A'A=I
(ii) A= \(\begin{bmatrix} \sin\alpha & \cos\alpha\\ -\cos \alpha & \sin\alpha \end{bmatrix}\),then verify that A'A=I

6.
Find the inverse of each of the matrices,if it exists. \(\begin{bmatrix} 2 & 3\\ 5 & 7 \end{bmatrix}\)

Similar Mathematics Concepts

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NCERT Solutions for Class 12 Maths Chapter 5 Continuity and Differentiability

NCERT Solutions for Class 12 Mathematics Chapter 5 Continuity and Differentiability

NCERT Solutions Class 12 Mathematics Chapter 5 Important Topics

Mean Value Theorem

Rolle’s Theorem

Limits

Euler’s Number

Quotient Rule

NCERT Solutions For Class 12 Maths Chapter 5 Exercises

CBSE CLASS XII Related Questions

1. Solve system of linear equations, using matrix method. x-y+2z=7 3x+4y-5z=-5 2x-y+3z=12

2. If A=\(\begin{bmatrix}2&-1&1\\-1&2&-1\\1&-1&2\end{bmatrix}\)verify that A3-6A2+9A-4 I=0 and hence find A-1

3. Let f: R→R be defined as f(x) = 3x. Choose the correct answer.

4. If A'= \(\begin{bmatrix} 3 & 4 \\ -1 & 2 \\ 0 &1 \end{bmatrix}\)\(\begin{bmatrix} -1 & 2 & 1 \\ 1 &2 & 3\end{bmatrix}\) , then verify that (i) \((A+B)'=A'+B' \)(ii) \((A-B)'=A'-B'\)

5. If (i) A=\(\begin{bmatrix} \cos\alpha & \sin\alpha\\ -\sin\alpha & \cos\alpha \end{bmatrix}\),then verify that A'A=I(ii) A= \(\begin{bmatrix} \sin\alpha & \cos\alpha\\ -\cos \alpha & \sin\alpha \end{bmatrix}\),then verify that A'A=I

6. Find the inverse of each of the matrices,if it exists. \(\begin{bmatrix} 2 & 3\\ 5 & 7 \end{bmatrix}\)

Similar Mathematics Concepts

Comments

SUBSCRIBE TO OUR NEWS LETTER

1.
Solve system of linear equations, using matrix method.
x-y+2z=7
3x+4y-5z=-5
2x-y+3z=12

2.
If A=\(\begin{bmatrix}2&-1&1\\-1&2&-1\\1&-1&2\end{bmatrix}\)verify that A³-6A²+9A-4 I=0 and hence find A^-1

3.
Let f: R→R be defined as f(x) = 3x. Choose the correct answer.

4.
If A'= \(\begin{bmatrix} 3 & 4 \\ -1 & 2 \\ 0 &1 \end{bmatrix}\)\(\begin{bmatrix} -1 & 2 & 1 \\ 1 &2 & 3\end{bmatrix}\) , then verify that
(i) \((A+B)'=A'+B' \)
(ii) \((A-B)'=A'-B'\)

5.
If (i) A=\(\begin{bmatrix} \cos\alpha & \sin\alpha\\ -\sin\alpha & \cos\alpha \end{bmatrix}\),then verify that A'A=I
(ii) A= \(\begin{bmatrix} \sin\alpha & \cos\alpha\\ -\cos \alpha & \sin\alpha \end{bmatrix}\),then verify that A'A=I

6.
Find the inverse of each of the matrices,if it exists. \(\begin{bmatrix} 2 & 3\\ 5 & 7 \end{bmatrix}\)