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Class 12 Continuity And Differentiability Ncert Solutions

By on Saturday, March 29, 2025
Class 12 Continuity And Differentiability Ncert Solutions

Continuity and Differentiability form a crucial part of the Class 12 Mathematics curriculum. These concepts not only build the foundation for advanced calculus but also play a significant role in competitive exams like JEE, NEET, and other entrance tests. Understanding the fundamentals of continuity and differentiability is essential to solving complex mathematical problems efficiently.

This content provides a comprehensive guide to Class 12 Continuity and Differentiability NCERT Solutions, covering essential formulas, step-by-step methods, and tips for mastering the chapter.

What is Continuity?

In mathematics, a function is said to be continuous if its graph can be drawn without lifting the pen from the paper. More formally, a function is continuous at a point if there is no abrupt jump, hole, or break at that point.

Definition of Continuity at a Point

A function f(x) is continuous at x = a if:

lim_{x to a^-} f(x) = lim_{x to a^+} f(x) = f(a)

This means the left-hand limit (LHL), right-hand limit (RHL), and the value of the function at that point must all be equal.

Types of Continuity:

  1. Pointwise Continuity: Continuity at a specific point.
  2. Continuous Function: A function continuous over its entire domain.
  3. Piecewise Continuous: Functions that are continuous within specific intervals.

What is Differentiability?

Differentiability refers to the ability to find the derivative of a function at a given point. If a function has a well-defined derivative at every point in its domain, it is said to be differentiable.

Differentiability Condition:

A function f(x) is differentiable at x = a if:

lim_{h to 0} frac{f(a + h) – f(a)}{h}

exists and is finite. If the function is differentiable at every point in its domain, it is called a differentiable function.

Key Differences Between Continuity and Differentiability

Aspect Continuity Differentiability
Definition No breaks or jumps in the graph. Derivative exists at the point.
Mathematical Condition LHL = RHL = f(a) Derivative limit exists and is finite.
Dependency Required for differentiability. Continuity does not ensure differentiability.
Graphical Indicator Smooth curve without breaks. Smooth curve without sharp corners.

Note: Every differentiable function is continuous, but not every continuous function is differentiable.

Important Formulas in Continuity and Differentiability

  1. Derivative of Basic Functions:

    • frac{d}{dx}(x^n) = nx^{n-1}
    • frac{d}{dx}(sin x) = cos x
    • frac{d}{dx}(cos x) = -sin x
    • frac{d}{dx}(e^x) = e^x
    • frac{d}{dx}(ln x) = frac{1}{x}

  2. Product Rule:

    frac{d}{dx}(uv) = ufrac{dv}{dx} + vfrac{du}{dx}

  3. Quotient Rule:

    frac{d}{dx}left(frac{u}{v}right) = frac{vfrac{du}{dx} – ufrac{dv}{dx}}{v^2}

  4. Chain Rule:

    frac{d}{dx}f(g(x)) = f'(g(x)) cdot g'(x)

  5. Logarithmic Differentiation:
    Useful when dealing with complex functions:

    y = u^v Rightarrow ln y = v ln u

Step-by-Step Solutions for Common NCERT Problems

Example 1: Check Continuity at a Point

Problem: Is the function f(x) = x^2 continuous at x = 3 ?

Solution:

  1. Find f(3) = 3^2 = 9

  2. Calculate LHL and RHL:

    lim_{x to 3^-} x^2 = 9
    lim_{x to 3^+} x^2 = 9

  3. Since LHL = RHL = f(3) , the function is continuous at x = 3 .

Example 2: Find Derivative Using Chain Rule

Problem: Find frac{d}{dx}(sin^2 x)

Solution:

Let u = sin x , then sin^2 x = u^2 .

By chain rule:

frac{d}{dx}(u^2) = 2u cdot frac{du}{dx}
= 2 sin x cdot cos x = sin 2x

Example 3: Prove Differentiability at a Point

Problem: Is f(x) = |x| differentiable at x = 0 ?

Solution:

  1. Left-hand derivative at x = 0 :

    lim_{h to 0^-} frac{|0 + h| – |0|}{h} = lim_{h to 0^-} frac{-h}{h} = -1

  2. Right-hand derivative at x = 0 :

    lim_{h to 0^+} frac{|0 + h| – |0|}{h} = lim_{h to 0^+} frac{h}{h} = 1

Since the left-hand and right-hand derivatives are not equal, the function is not differentiable at x = 0 , even though it is continuous.

Tips to Master Continuity and Differentiability

  1. Understand Graphs: Visualizing functions helps in identifying points of discontinuity or non-differentiability.
  2. Practice Limits: Many problems require evaluating limits. Strengthening this skill is essential.
  3. Use Derivative Rules Wisely: Apply the chain rule, product rule, and quotient rule where necessary.
  4. Solve NCERT Exercises: NCERT provides a wide range of problems that cover all difficulty levels.
  5. Analyze Special Functions: Pay attention to modulus, greatest integer functions, and piecewise functions, which often have points of discontinuity or non-differentiability.

Common Mistakes to Avoid

  • Ignoring Domain Restrictions: Always check the domain before testing continuity or differentiability.
  • Incorrect Application of Derivative Rules: Ensure you’re using the right rules for complex functions.
  • Overlooking Absolute Value Functions: These often have sharp corners, making them non-differentiable at certain points.

Mastering Class 12 Continuity and Differentiability requires a clear understanding of the concepts and consistent practice. By focusing on core definitions, using the right formulas, and solving a variety of problems, students can build a strong foundation in calculus.

This chapter not only prepares students for board exams but also sharpens problem-solving skills essential for competitive exams. By practicing regularly and avoiding common mistakes, you can confidently tackle even the most challenging problems related to continuity and differentiability.

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