module 2

MODULE II

1. Overview of Content Covered

   • Section 1.5: Continuity

   • Section 2.1: The Derivative of a Function

   • Section 2.2: Differentiation Rules

   • Section 2.3: Rates of Change

   • Section 2.5: The Chain Rule

   • Section 2.6: Implicit Differentiation and Rational Exponents

1.5 Continuity

• Definition of Continuity:

  • A function f is continuous at a point x=C if:

    • 1. f(C) exists

    • 2. lim (x -> C) f(x) exists

    • 3. lim (x -> C) f(x) = f(C)

• Types of Continuity:

  • Interior Points: Points within the interval of the domain.

  • Endpoints: Left and right endpoints require one-sided limits to determine continuity.

2.1 The Derivative of a Function

• Definition of Derivative: The derivative of f with respect to x is given by:

  • f'(x) = lim (h -> 0) [f(x+h) - f(x)] / h

• Notation: Various notations for derivatives include:

  • f'(x), dy/dx, d(f)/dx.

• Calculating Derivatives: Steps to differentiate include writing expressions, expanding, and simplifying differences.

2.2 Differentiation Rules

• Basic Rules:

  • Power Rule: For f(x) = x^n, f'(x) = nx^(n-1).

  • Sum Rule: If f and g are differentiable, then (f + g)' = f' + g'.

  • Product Rule: If f and g are differentiable, then (fg)' = f'g + fg'.

  • Quotient Rule: If f and g are differentiable at x, then (f/g)' = (f'g - fg')/g^2, g != 0.

2.3 Rates of Change

• Average Rate of Change:

  • The average rate over the interval [x0, x1] is given by (f(x1) - f(x0))/(x1 - x0).

• Instantaneous Rate of Change: It is the derivative f'(x).

2.5 The Chain Rule

• Chain Rule Definition:

  • If y = f(g(x)), then dy/dx = f'(g(x)) * g'(x).

• Application: Useful in differentiating composite functions.

2.6 Implicit Differentiation and Rational Exponents

• Implicit Differentiation: Used when functions are not easily expressed as y = f(x). Differentiate both sides and solve for dy/dx.

• Rational Exponents: The Power Rule extends to rational exponents, allowing differentiable functions of the form f(x) = x^(p/q) to be differentiated using the chain rule.