Class 12 CBSE - Mathematics - Continuity and Differentiability | Xylem CBSE 11 & 12

Xylem Class 12 CBSE・2 minutes read

Continuity is ensuring no breaks in a function, with differentiability established when left-hand and right-hand derivatives are equal. Various functions and equations are explored to illustrate the distinction between continuity and differentiability in calculus.

Insights

  • Understanding continuity in functions involves the absence of breaks, with a function being continuous if no breaks exist, as exemplified by the function FX being continuous at specific points.
  • The distinction between continuity and differentiability is significant, with differentiability established when left-hand and right-hand derivatives are equal at a point, showcased through examples like the function f(x) = mod(x) that is continuous but not differentiable at every point.

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Recent questions

  • What is continuity in mathematics?

    Continuity refers to the absence of breaks in a function, making it continuous if there are no interruptions in its behavior.

  • How is differentiability determined in calculus?

    Differentiability is established when the left-hand derivative equals the right-hand derivative at a specific point, indicating a smooth transition in the function's behavior.

  • Can a function be continuous but not differentiable?

    Yes, a function like f(x) = mod(x) can be continuous at every point but not differentiable due to sharp corners or cusps in its graph.

  • What is the significance of left-hand limit and right-hand limit in calculus?

    Left-hand limit, right-hand limit, and the function value play a crucial role in determining the continuity and differentiability of a function at a specific point.

  • How are derivatives calculated for common functions in calculus?

    Derivatives of common functions like sin(x), cos(x), tan(x), and log(x) are provided, along with rules like the chain rule for differentiation to find the rate of change of a function at a given point.

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Summary

00:00

Understanding Continuity and Differentiability in Functions

  • Continuity is defined as the absence of any breaks in a function, with the function being continuous if there are no breaks.
  • The function FX is continuous at 2.7, lying between 2 and 3.
  • The function is also continuous at -4.7, lying between -4 and -5.
  • Left-hand limit, right-hand limit, and the function value are crucial in determining continuity and differentiability.
  • Differentiability is established when the left-hand derivative equals the right-hand derivative at a point.
  • The function f(x) = mod(x) is an example of a function that is continuous but not differentiable at every point.
  • The function f(x) = K cos(x) / -2x if x ≠ 2 and 3 if x = 2 is explored for differentiability at x = π/2.
  • The function f(x) = sin^2(λx) / x^2 if x ≠ 0 and K if x = 0 is analyzed for differentiability at x = 0.
  • The concept of differentiability is distinct from continuity, with examples illustrating the difference.
  • Derivatives of common functions like sin(x), cos(x), tan(x), and log(x) are provided, along with the chain rule for differentiation.

01:44:44

Calculus Equations and Derivatives Explained

  • The equation sin e x / cos e x is equivalent to tan e x.
  • The derivative of 2x F dash of x e root of x is 2x.
  • The function F of x² is equal to a x e x into 2x.
  • The equation sin x / cos x + sin x / cos x is equal to tan x.
  • The equation 1 - tan x / 1 + tan x simplifies to tan < by 4 - x.
  • The equation x + y / x - y = k leads to Dy by DX = 2y / 2x.
  • Implicit functions are crucial in calculus exams.
  • The derivative of y + sin y = cos x with respect to x is -sin x / cos x.
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