This chapter covers important concepts of continuity and differentiability of functions. Understanding these topics is essential for further studies in calculus and mathematical analysis.
Continuity and Differentiability - Quick Look Revision Guide
Your 1-page summary of the most exam-relevant takeaways from Mathematics Part - I.
This compact guide covers 20 must-know concepts from Continuity and Differentiability aligned with Class 12 preparation for Mathematics. Ideal for last-minute revision or daily review.
Complete study summary
Essential formulas, key terms, and important concepts for quick reference and revision.
Key Points
Definition of Continuity.
A function f is continuous at c if limits from both sides equal f(c).
Continuity at an interval.
A function is continuous on [a, b] if continuous at every point in the interval.
Polynomials are continuous.
All polynomial functions are continuous over the entire set of real numbers.
Types of discontinuity.
Functions can be continuous, removable, jump, or infinite discontinuities.
Limit definition.
A function f has a limit L at c if the values of f approach L as x approaches c.
Left and Right Hand Limits.
Left limit: lim x→c⁻ f(x); Right limit: lim x→c⁺ f(x).
Continuous function graphs.
You can draw continuous function graphs without lifting the pen.
Derivatives introduction.
A derivative measures the rate of change of a function at a point.
Differentiability implies continuity.
If f is differentiable at c, then it is continuous at c.
Chain Rule for differentiation.
If f = g(h(x)), then df/dx = dg/dh * dh/dx.
Mean Value Theorem.
If f is continuous on [a, b] and differentiable on (a, b), there exists c in (a, b) such that f'(c) = (f(b) - f(a)) / (b - a).
Product Rule.
If u and v are functions, then (uv)' = u'v + uv'.
Quotient Rule.
If u and v are functions, then (u/v)' = (u'v - uv') / v².
Differentiation of sin(x) and cos(x).
d/dx(sin x) = cos x and d/dx(cos x) = -sin x.
Differentiability at points.
A function is not differentiable if there's a corner, cusp, or vertical tangent.
Inverse Functions.
For sin⁻¹(x), dy/dx = 1/√(1 - x²) where |x| < 1.
Exponential Functions.
Derivative of e^x is e^x, remains unchanged.
Logarithmic Functions.
Derivative of log(x) is 1/x, defined for x > 0.
Second Derivative.
The second derivative provides information about concavity of the function.
Applications of derivatives.
Used to find tangents, normals, and in optimization problems.
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