Lecture Notes on Limits and Derivatives

Pre-Midterm Overview

• Instructor reassures that the midterm will be easy.
• Quick review of topics covered in the course:
  • Limits
  • Derivatives
  • Continuity

Limits

• Concept of Limits
  • Functions can have jumps, holes, or asymptotes.
  • Limit is the intended value a function approaches at a certain point.
• One-Sided and Two-Sided Limits
  • One-sided limits: approach from left or right.
  • Two-sided limits: if one-sided limits agree, the two-sided limit exists.
• Continuity
  • A function is continuous at a point if the limit at that point exists and equals the function's value.
  • Not all limits imply continuity.

Problems with Rational Functions

• Discontinuities and Holes
  • Occur where the denominator is zero.
  • Use algebra to manipulate functions to find limits (e.g., factoring, canceling terms).

Special Techniques

• Multiplying by the Conjugate
  • Useful for removing square roots in complex expressions.
  • Helps simplify limits that involve square roots.

Infinite Limits and Asymptotic Behavior

• Limits at infinity evaluate the behavior of functions as they grow large or negative.
• Factor highest degree terms to simplify expressions.
• Example:
  • Limit of rational functions as x approaches positive or negative infinity.
  • Beware of sign changes when x approaches negative infinity.
  • Importance of writing steps to avoid mistakes.

Derivatives

• Definition
  • Derivative as a limit of a difference quotient.
  • A function is differentiable if this limit exists.
  • Differentiability implies continuity.

Derivative Rules

• Basic Rules
  • Sum/Difference Rule
  • Linearity (constants can be factored out)
  • Power Rule
• Advanced Rules
  • Product Rule
  • Quotient Rule
  • Chain Rule
• Special Functions
  • Trig Functions, Exponentials, Logarithms, Hyperbolic Functions.
  • Differences between ordinary and hyperbolic trigonometric derivatives.

Complex Derivative Example

• Example Problem
  • Combination of multiple rules: product, quotient, chain.
  • Handling trigonometric, exponential, and logarithmic functions.
  • Break down into smaller parts, solve each part.

Implicit Differentiation

• Implicit Functions
  • Differentiate when y is a function of x but not explicitly solved.
  • Use pseudo formula to find derivative.
  • Careful with constants when differentiating implicitly.

Conclusion

• Reminder of office hours for further questions.
• Encouragement and good luck wishes for the midterm.