NCERT Solutions for Class 12 Maths Chapter 5: Continuity and Differentiability

Mathematics is a crucial subject for students, especially for those pursuing science and commerce streams in Class 12. One of the essential chapters in the NCERT Class 12 Mathematics curriculum is Chapter 5: Continuity and Differentiability. This chapter builds on the concepts introduced in earlier classes, like functions and calculus, and delves deeper into the nuances of continuity and differentiability. Understanding these concepts is crucial for students aiming to excel in competitive exams such as JEE, NEET, and others. The NCERT Solutions for Class 12 Maths Chapter 5 serve as an essential tool for students, helping them grasp these challenging topics more efficiently.

In this article, we will explore the significance of this chapter, break down the key concepts, and provide insights into how NCERT Solutions can assist students in mastering Continuity and Differentiability. This detailed guide will help students understand not only the chapter itself but also how to approach its problems with clarity.

Overview of Chapter 5: Continuity and Differentiability

Before diving into the solutions, it is essential to understand the key topics covered in Chapter 5 of the NCERT Class 12 Maths book. The chapter builds on the basic idea of functions and calculus from earlier classes. It is divided into two major parts:

Continuity: This part deals with the concept of whether a function is continuous or not. A function is said to be continuous at a point if there is no abrupt change in its value at that point.
Differentiability: Differentiability explores whether a function has a derivative at a point. A function is differentiable at a point if it is continuous and if its derivative exists at that point.

The chapter also touches upon key subtopics like:

Exponential and logarithmic functions
Chain rule of differentiation
Implicit differentiation
Logarithmic differentiation
Derivatives of functions in parametric forms
Second-order derivatives
Mean Value Theorem
Rolle’s Theorem

Each of these subtopics plays a crucial role in both board exams and competitive exams.

1. Continuity

Continuity is a fundamental concept in calculus that describes the behavior of functions. A function $f(x)$ is said to be continuous at a point $x = a$ if:

$f(a)$ is defined.
The limit $\lim_{x \to a} f(x)$ exists.
$\lim_{x \to a} f(x) = f(a)$ .

In simpler terms, there should be no sudden jumps or breaks in the graph of the function at point $a$ . The NCERT textbook explains the concept of continuity using graphs and mathematical examples, making it easier for students to grasp.

NCERT Solutions for Continuity

The NCERT Solutions for Class 12 Maths Chapter 5 cover several exercises based on continuity. The problems involve checking whether a function is continuous at a given point or over a given interval. These solutions are designed to help students understand the process of evaluating the continuity of different types of functions, such as polynomial, rational, trigonometric, and piecewise functions.

One typical problem involves showing that a function is continuous at all points, except where the denominator is zero (for rational functions). Students are guided step-by-step through the process of calculating limits and confirming whether the function meets the criteria for continuity.

2. Differentiability

Differentiability is closely linked to the concept of continuity, but it takes things a step further. While a function may be continuous at a point, it is not necessarily differentiable there. A function $f(x)$ is said to be differentiable at a point $x = a$ if:

The derivative $\frac{d}{dx}f(x)$ exists at $x = a$ .
A function is differentiable if it is smooth at that point, meaning there are no sharp edges or cusps in the graph of the function at $x = a$ .

One important result is that if a function is differentiable at a point, it must be continuous there. However, the converse is not always true—continuity does not necessarily imply differentiability.

NCERT Solutions for Differentiability

The NCERT Solutions for differentiability involve finding the derivative of functions using first principles as well as applying standard differentiation techniques. The exercises guide students through the process of determining whether a function is differentiable at a given point by evaluating its limits from the left and right-hand sides.

For instance, problems may ask students to show that a function is differentiable everywhere except at certain points where the function has a sharp corner or discontinuity. The solutions help students apply both conceptual and algebraic approaches to determine differentiability.

3. The Chain Rule

The chain rule is a crucial differentiation technique that is particularly useful when dealing with composite functions. If a function $y = f(g(x))$ , then the derivative of $y$ with respect to $x$ can be found using the chain rule as:

\frac{dy}{dx} = \frac{dy}{dg} \cdot \frac{dg}{dx}

The NCERT textbook introduces the chain rule using simple examples, gradually progressing to more complex functions involving trigonometric, exponential, and logarithmic functions.

NCERT Solutions for Chain Rule

The NCERT Solutions provide numerous problems based on the chain rule, ranging from basic to advanced levels. These problems help students develop a solid understanding of how to apply the chain rule to differentiate composite functions. The step-by-step solutions ensure that students understand the logic behind each step, making the chain rule easier to apply in different contexts.

4. Implicit Differentiation

Implicit differentiation is used when functions are not given in the form $y = f(x)$ . Instead, both $x$ and $y$ are intermixed in an equation. Implicit differentiation involves differentiating both sides of an equation with respect to $x$ , treating $y$ as a function of $x$ , and then solving for $\frac{dy}{dx}$ .

NCERT Solutions for Implicit Differentiation

The NCERT Solutions guide students through the process of differentiating equations implicitly. For example, given an equation like $x^2 + y^2 = r^2$ , students are shown how to differentiate both sides with respect to $x$ , apply the chain rule to terms involving $y$ , and solve for $\frac{dy}{dx}$ . These problems are especially useful for dealing with circular, elliptical, and hyperbolic equations.

5. Logarithmic Differentiation

Logarithmic differentiation is a powerful technique used to differentiate functions that are products, quotients, or powers of functions. By taking the logarithm of both sides of an equation and then differentiating, the process becomes more manageable, especially when dealing with complicated expressions.

NCERT Solutions for Logarithmic Differentiation

The NCERT Solutions include problems where logarithmic differentiation is the most efficient approach. These problems involve functions like $y = x^x$ or $y = (x^2 + 1)^5$ , where applying direct differentiation would be cumbersome. The solutions simplify these problems by applying logarithms first and then differentiating, making the process clearer for students.

6. Derivatives of Functions in Parametric Form

When functions are given in parametric form, i.e., when both $x$ and $y$ are defined in terms of a third variable $t$ , the derivative of $y$ with respect to $x$ can be found using the formula:

\frac{dy}{dx} = \frac{\frac{dy}{dt}}{\frac{dx}{dt}}

NCERT Solutions for Parametric Differentiation

The NCERT Solutions for this section walk students through problems where they must differentiate parametric equations. These problems often involve trigonometric and algebraic functions, and the solutions provide clear instructions on how to apply the formula to find the derivative.

7. Second-Order Derivatives

Once students have mastered first-order derivatives, the chapter progresses to second-order derivatives. The second-order derivative, denoted as $\frac{d^2y}{dx^2}$ , represents the rate of change of the first derivative. It provides insights into the concavity and inflection points of a function's graph.

NCERT Solutions for Second-Order Derivatives

The NCERT Solutions provide several problems involving second-order derivatives. These problems help students understand how to calculate second-order derivatives and interpret their geometric meaning. The solutions guide students step-by-step through the process, ensuring a solid grasp of this advanced concept.

8. Mean Value Theorem and Rolle’s Theorem

The chapter concludes with two important theorems in calculus: the Mean Value Theorem (MVT) and Rolle’s Theorem. Both theorems provide critical insights into the behavior of functions over a closed interval.

Rolle’s Theorem states that if a function is continuous and differentiable over a closed interval $[a, b]$ and $f(a) = f(b)$ , then there exists at least one point $c$ in the open interval $(a, b)$ where $f'(c) = 0$ .
Mean Value Theorem generalizes Rolle’s Theorem, stating that if a function is continuous on $[a, b]$ and differentiable on $(a, b)$ , then there exists a point $c \in (a, b)$ where:

f'(c) = \frac{f(b) - f(a)}{b - a}

NCERT Solutions for MVT and Rolle’s Theorem

The NCERT Solutions provide examples and proofs of these theorems, along with problems that apply these concepts to real-world functions. The solutions clarify the conditions under which these theorems can be applied and help students understand their significance in calculus.

Conclusion

NCERT Solutions for Class 12 Maths Chapter 5: Continuity and Differentiability play an indispensable role in preparing students for their board exams and competitive exams. This chapter introduces students to essential concepts that are foundational to higher-level calculus, making it vital to understand them thoroughly.

By practicing the exercises and going through the step-by-step solutions, students can develop a deeper understanding of continuity, differentiability, and related theorems. These solutions offer a structured approach to learning, helping students tackle problems with confidence and ease.

Incorporating these NCERT Solutions into daily study routines will not only help students excel in their Class 12 board exams but also lay a strong foundation for future studies in mathematics, engineering, and other related fields.