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Continuity and Differentiability

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Class 12 Maths Chapter 5
Continuity and Differentiability
Important Questions

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Preparing for the Class 12 Maths exams, particularly Chapter 5 focusing on Continuity and Differentiability, stands as a cornerstone for a solid mathematical groundwork. This chapter navigates through the concepts of continuity and differentiability, unraveling the intricacies of functions and their behavior. To excel in this chapter, acquainting oneself with pivotal questions becomes imperative. Here are class 12 Continuity and Differentiability important questions and answers These questions are meticulously crafted to gauge the depth of understanding fundamental concepts. Embracing a diverse array of question styles aids in navigating uncertainties, ensuring comprehensive readiness for the impending exams.

Introduction

The concepts of continuity and differentiability are fundamental in calculus and form the core of many mathematical analyses, especially in the field of real analysis. Let’s delve into a brief introduction of these concepts.In simple terms, a function is continuous at a point if there is no interruption in the graph of the function at that point. Class 12 Continuity and Differentiability important questions and answers help students in understanding the chapter in more detailed way. Continuity and differentiability are key to understanding how functions behave. Continuity deals with the smoothness of the function’s graph, while differentiability concerns how a function changes at a particular point. Differentiability implies continuity, but a continuous function may not necessarily be differentiable at every point.

What are Continuity and Differentiability?

A function is said to be continuous at a point if there is no abrupt change or break in its value at that point. In simpler terms, you can draw the graph of a continuous function without lifting your pen from the paper.A function is differentiable at a point if it has a defined derivative at that point. In other words, the function has a tangent at that point and does not have sharp turns or cusps. Differentiability implies a certain “smoothness” of the function’s graph at that point.

Class 12 Continuity and Differentiability Important Questions and Answers

Q1. Find the continuity of f(x) = x at x = k, k be any positive value :
Options
(a) f(x) is continuous at x = k
(b) f(x) is not continuous at x = k
(c) f(x) is continuous at x = 0
(d) None of the above

Ans. (a) f(x) is continuous at x = k
Explanation:
The given function is, f(x) = x
At x = k, f(k) = k
\lim\limits_{x \rightarrow k} f(x) = \lim\limits_{x \rightarrow k} (x) = k\\[4.5 bp] ∴ \lim\limits_{x \rightarrow k}f(x) = f(k)\\[4.5 bp] ∴ f(x) is continuous at x = k.

Q2. If y = ax^2+ b,\text{ then } \dfrac{dy}{dx} at x = 2 is equal to

Options
(a) 2a
(b) 3a
(c) 4a
(d) None of these

Ans. (c) 4a

Explanation:
Given that, y = ax^2+b\\[4.5 bp] \text{Then, } \dfrac{dy}{dx} = 2ax\\[4.5 bp] \text{At x }= 2, \dfrac{dy}{dx} = 2(a)(2) = 4a

Q3. The function f(x) = \dfrac{[ln(1 + ax) - ln( 1 - bx)]}{x}, not defined at x = 0. The value should be assigned to f at x = 0, so that it is continuous at x = 0.

Explanation:
For f(x) to be continuous at x = 0,
f(0) = \lim\limits_{x \rightarrow 0}f(x)\\[4.5 bp] \text{Therefore, f(0)} = \lim\limits_{x \rightarrow 0} \dfrac{[ln(1+ax) - ln(1-bx)]}{x} \\[4.5 bp] \text{By using }\lim\limits_{x \rightarrow 0} \dfrac{[ln(1+x)]}{x} = 1, we get
f(0) = a + b.

Q4. If x^y. y^x = 16, \text{ then } \dfrac{dy}{dx} at (2, 2) is

Explanation:
Given: x^y. y^x = 16\\ Now, take log on both sides, we get
\text{log x}^y +\text{ log }y^x = \text{log 16}\\ y log x + x log y = log 16
Differentiate with respect to x, we get
\left(\dfrac{y}{x}\right)\text{ + log x }\left(\dfrac{dy}{dx}\right) + \left(\dfrac{y}{x}\right) \left(\dfrac{dy}{dx}\right) + \text{log y = 0}\\[4.5 bp] \text{Hence, }\dfrac{dy}{dx} = – \left(\dfrac{y}{x}\right)\left[\dfrac{\text{(y + x log y)}}{\text{(x + y log x)}}\right]\\[4.5 bp] \text{Therefore, } \dfrac{dy}{dx} at (2, 2) = -1.

Q5. If y = \sqrt{[sin x + y]},\text{ then } \dfrac{dy}{dx}

Explanation:
\text{ y }= \sqrt{[sin x + y]} \\ Now, take square on both sides,
y^2 = \text{sin x + y}\\[4.5 bp] Differentiating with respect to x, we get
\text{2y} \left(\dfrac{dy}{dx} \right)\text{ = cos x }+ \dfrac{dy}{dx}\\[4.5 bp] \text{Hence, 2y }\left(\dfrac{dy}{dx} \right) – \left(\dfrac{dy}{dx} \right) \text{= cos x}\\[4.5 bp] \left(\dfrac{dy}{dx} \right) (2y-1) = \text{cos x}\\[4.5 bp] \text{Hence, }\dfrac{dy}{dx} = \dfrac{\text{cos x}}{\text{(2y-1)}}.

CBSE Class 12 Maths Chapter wise Important Questions

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Conclusion

In conclusion, continuity and differentiability are foundational concepts in calculus that have far-reaching implications in mathematics and its various applications. Exploring class 12 Continuity and Differentiability important questions and answers, becomes essential for a comprehensive grasp of this fundamental aspect of mathematics.. To enhance understanding, platforms like Oswal.io provide a wealth of resources, including question-answer sets, comprehensive class 12 questions and answers.Utilising these resources ensures students acquire the knowledge necessary to excel in their studies and examinations.

Frequently Asked Questions

Ans: Yes, functions like polynomials, sine, and cosine functions are continuous everywhere. However, some functions are only continuous over a certain interval or at certain points.
Ans: To determine if a function is continuous at a point, check that the function is defined at that point, the limit as.
Ans: No, not all continuous functions are differentiable. For example, the absolute value function is continuous everywhere but is not differentiable at.
Ans: A function is differentiable at a point if it has a defined derivative (slope) at that point. This means the function is smooth (without sharp corners or cusps) at that point.
Ans: To check differentiability at a point, you calculate the derivative at that point. If the derivative exists and is finite, the function is differentiable there.
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