Lecture Notes on Continuity and Differentiability

Introduction

• Focus on 5th chapter: Continuity and Differentiability
• Topics include logarithmic functions, limits, differentiability

Logarithmic Functions

• Logarithmic differentiation example: f(y) = x²
• Techniques for differentiating log functions

Continuity

• Definition of continuity: A function is continuous if left-hand limit (LHL) = right-hand limit (RHL) = function value at that point
• Example function: f(x) = 2x + 3 (a simple linear function)
• Importance of checking limits from both sides (LHL, RHL)

Discontinuity

• Conditions under which a function is not continuous
• Example: f(x) being not continuous at integer values when f(x) = 1/x
• Checking continuity by calculating LHL and RHL

Finding Values for Continuity

• Solving for constants (A and B) to make a function continuous
• Example process using linear equations:
  • First equation: 10 = 3A + B
  • Second equation: 20 = 4A + B

Points of Discontinuity

• Identifying points where a function is discontinuous
• Example: Function f(x) with pieces 2x + 3 and 2x - 3
• Check limits as x approaches critical points

Differentiability

• Definition: A function is differentiable if it has a well-defined tangent (no sharp edges)
• Calculating derivatives: dy/dx
• Example derivatives:
  • Simple power functions
  • Exponential and logarithmic functions
  • Trigonometric functions

Product and Quotient Rule

• Product Rule: Differentiating products of functions
  • Formula: (u'v + uv')
  • Example: y = sin(x)cos(x)
• Quotient Rule: Differentiating ratios of functions
  • Formula: (v'u - uv')/v²

Chain Rule

• Differentiating composite functions: "inside-out"
• Example: sin²(5e^x)

Implicit Differentiation

• Differentiating equations not solved for y explicitly
  • Example: Differentiating both sides with respect to x
  • Using results in implicit functions

Inverse Functions

• Differentiating inverse trigonometric functions, e.g., cos⁻¹(x)

Logarithmic Differentiation

• Applying logarithmic differentiation to simplify complex derivatives
• Example: y = a^x

Homework

• Questions involving mod functions and cos(x)

Closing Remarks

• Importance of practice and familiarity with rules
• Encouragement to keep studying