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 – Formula & Equation Sheet
Essential formulas and equations from Mathematics Part - I, tailored for Class 12 in Mathematics.
This one-pager compiles key formulas and equations from the Continuity and Differentiability chapter of Mathematics Part - I. Ideal for exam prep, quick reference, and solving time-bound numerical problems accurately.
Key concepts & formulas
Essential formulas, key terms, and important concepts for quick reference and revision.
Formulas
Definition of Continuity: f is continuous at c if lim (x → c) f(x) = f(c)
f represents a function, c is a point in its domain. This definition formalizes what it means for a function to be continuous at a point.
Derivative Definition: f'(c) = lim (h → 0) [f(c+h) - f(c)] / h
This expression defines the derivative of a function at point c, capturing the rate of change of the function.
Chain Rule: If y = f(g(x)), then dy/dx = (dy/dg) * (dg/dx)
This rule is essential for differentiating composite functions, where dy/dg is the derivative of the outer function and dg/dx is the derivative of the inner function.
Sum Rule: (f + g)' = f' + g'
The derivative of a sum of functions is the sum of their derivatives, facilitating differentiation of composite functions.
Product Rule: (uv)' = u'v + uv'
This formula is used to find the derivative of a product of two functions u and v.
Quotient Rule: (u/v)' = (u'v - uv') / v²
This formula provides a method to differentiate the quotient of two functions, where u' and v' are the derivatives of u and v, respectively.
Limit Definition of Derivative: f'(c) = lim (x → c) [f(x) - f(c)] / (x - c)
This expression is another formulation used to define the derivative of a function at a specific point c.
Differentiability Implies Continuity: If f is differentiable at c, then f is continuous at c.
This statement establishes an important relationship between differentiability and continuity.
Derivative of sin x: (sin x)' = cos x
This standard result is crucial for differentiation and is widely applied in problems involving trigonometric functions.
Derivative of e^x: (e^x)' = e^x
This unique property of the natural exponential function shows that its derivative is the same as the function itself.
Equations
Continuity at Interval: A function f is continuous on [a, b] if it is continuous at every point in (a, b) and lim (x → a⁻) f(x) = f(a) and lim (x → b⁺) f(x) = f(b)
This equation establishes the conditions necessary for a function to be continuous over a closed interval.
First Derivative Test: If f'(x) changes from positive to negative at c, then f has a local maximum at c.
This test is useful for finding relative extrema of functions.
Second Derivative Test: If f''(x) > 0 at c, then f has a local minimum at c; if f''(x) < 0, then f has a local maximum.
This test provides a way to determine the concavity of the function and its relation to local extrema.
Differentiability Condition: A function is differentiable at c if it is continuous at c and f'(c) exists.
This condition emphasizes the connection between differentiability and continuity.
Limit at Infinity: lim (x → ±∞) f(x) = L means that f(x) approaches L as x goes to infinity.
This expression is vital for understanding horizontal asymptotes of functions.
Intermediate Value Theorem: If f is continuous on [a, b] and k is a number between f(a) and f(b), then there exists at least one c in (a, b) such that f(c) = k.
This theorem is important for proving the existence of roots within an interval.
Mean Value Theorem: If f is continuous on [a, b] and differentiable on (a, b), there exists at least one c in (a, b) such that f'(c) = (f(b) - f(a)) / (b - a).
This theorem links the average rate of change of a function to instantaneous rate of change.
Differentiability of Polynomial Functions: All polynomial functions are differentiable, and hence continuous, at all real numbers.
This serves as a fundamental concept for understanding the behavior of polynomial functions.
Derivative of cos x: (cos x)' = -sin x
This derivative is essential for problems involving trigonometric functions.
Chain Rule Application: If y = (g(x))^n, then dy/dx = n(g(x))^(n-1)g'(x)
This expresses the application of the chain rule for differentiating power functions.
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