Chapter 5 Arithmetic Progressions

Notes

Continuity

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

lim x→a f(x) = f(a)

Conditions:

1. f(a) exists

2. lim x→a f(x) exists

3. lim x→a f(x) = f(a)

• LHL = Left Hand Limit
• RHL = Right Hand Limit

• Limit exists when:

LHL = RHL

Important Continuous Functions

• Polynomial functions are continuous everywhere.

• Rational functions are continuous where denominator is not zero.

• sin x and cos x are continuous everywhere.

Differentiability

• A function is differentiable if derivative exists.

• Differentiability condition:

LHD = RHD

Important Relation

• Differentiable ⇒ Continuous

• Continuous does not always mean differentiable.

Example:

|x| is continuous at x = 0 but not differentiable at x = 0.

Basic Derivative Formulas

• d/dx(c) = 0

• d/dx(xⁿ) = n xⁿ⁻¹

• d/dx(sin x) = cos x

• d/dx(cos x) = -sin x

• d/dx(tan x) = sec²x

• d/dx(cot x) = -cosec²x

• d/dx(sec x) = sec x tan x

• d/dx(cosec x) = -cosec x cot x

• d/dx(eˣ) = eˣ

• d/dx(log x) = 1/x

Rules of Differentiation

Sum Rule:

d/dx(u + v) = u' + v'

Product Rule:

d/dx(uv) = uv' + vu'

Quotient Rule:

d/dx(u/v) = (v u' - u v')/v²

Chain Rule:

d/dx[f(g(x))] = f'(g(x)) × g'(x)

Inverse Trigonometric Derivatives

• d/dx(sin⁻¹x) = 1/√(1 - x²)

• d/dx(cos⁻¹x) = -1/√(1 - x²)

• d/dx(tan⁻¹x) = 1/(1 + x²)

• d/dx(cot⁻¹x) = -1/(1 + x²)

Logarithmic Differentiation

Useful for:

• xˣ
• (sin x)ˣ
• x^(sin x)

Steps:

1. Take log

2. Differentiate

3. Multiply by y

Common Mistakes

• Forgetting chain rule

• Writing derivative of cos x as sin x

• Forgetting denominator square in quotient rule

• Thinking continuous always means differentiable