Class 12 CBSE - Mathematics - Continuity and Differentiability | Xylem CBSE 11 & 12
Xylem Class 12 CBSE・2 minutes read
A continuous function has no breaks, while differentiability refers to equal left-hand and right-hand derivatives at a point, with examples like mod(x) showcasing non-differentiable functions. Various derivative formulas for exponential, logarithmic, trigonometric functions are presented, emphasizing that continuity does not guarantee differentiability.
Insights
- Continuity in a function means there are no breaks in its graph, with a function being continuous if it has no gaps. Continuous functions have equal left-hand limit, right-hand limit, and function value at a point.
- Differentiability refers to when a function's left-hand derivative equals the right-hand derivative at a specific point. Continuity does not always guarantee differentiability, as seen in functions like f(x) = |x|, showcasing examples of non-differentiable functions.
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Recent questions
What is continuity in a function?
Absence of breaks in a function.
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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 2.7, and at -4.7, lying between -4 and 5.
- A continuous function is one where the left-hand limit, right-hand limit, and the function value at a point are equal.
- Differentiability is the property of a function where the left-hand derivative equals the right-hand derivative at a point.
- The function f(x) = mod(x) is not differentiable at x = 1, showcasing an example of a non-differentiable function.
- The function f(x) = K cos(x) / -2x if x ≠ 2 and 3 if x = 2 is explored to find the value of K.
- The function f(x) = sin^2(Lambda x) / x^2 if x ≠ 0 and K if x = 0 is analyzed to determine the value of K.
- The concept of differentiability is discussed, emphasizing that continuity does not always imply differentiability.
- The function f(x) = 2x^2 - x is examined to find the values of a and b for the function to be differentiable in the interval [0, 2].
- Various derivative formulas are presented, including those for exponential, logarithmic, trigonometric, and inverse trigonometric functions.
01:44:44
Mathematical Equations and Derivatives Simplified
- The equation sin e x / cos e x is equivalent to tan e x.
- The derivative of 2x F dash of x eot f f x e root of x s root square 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 the context of CBC exams.
- The derivative of y + sin y = cos x with respect to X is -sinx / cosx.
