Unit 1 Overview: AP Calculus AB/BC

Introduction to Calculus

• Main Idea: Study of change and motion.
  • Key Question: Can change occur at an instant?
  • Example: Moving arrow demonstrating instantaneous change.

Rates of Change

• Average Rate of Change (AROC):
  • Relation: Slope of the secant line between two points.
  • Important Note: Undefined if denominator is zero.
• Instantaneous Rate of Change (IROC):
  • Relation: Slope of the tangent line at a point.

Limits and Limit Notation

• Definition: A value that a function approaches as the input gets closer to a certain value.
• Example: Limit of f(x) = x^2 as x approaches 2 is 4.
• Reading Limit Notation: Evaluating the function as it approaches a specific x-value.

Estimating Limits

• From Graphs:
  • One-sided limits: Approaching a value from one direction.
  • Two-sided limits: Approaching a value from both directions.
  • Conditions where limits may not exist: Oscillation, unbounded behavior, vertical asymptotes.
• From Tables:
  • Approach values from both sides; if limits differ, the limit does not exist.

Algebraic Properties and Manipulations

• Algebraic Properties: Provide rules to evaluate limits.
• Solving Limits Algebraically:
  1. Substitution - Replace variable with a specific value.
  2. Factoring - Simplify by canceling common factors.
  3. Common Denominator - Match denominators to simplify.
  4. Conjugate Method - Multiply by the conjugate to remove indeterminacy.

The Squeeze Theorem

• Used to find limits by comparing the function to two other functions with known limits.

Types of Discontinuities

• Jump Discontinuity: Sudden change in the function's value.
• Removable Discontinuity: A hole in the graph that can be "filled".
• Infinite Discontinuity: Vertical asymptotes.

Continuity

• At a Point:
  • Function value must exist.
  • Limit as x approaches the point must exist and equal the function's value.
• Over an Interval:
  • Function must be defined, have no breaks, and limits must match function values at all points.

Removing Discontinuities

• Modify a function at points of discontinuity to match the limit.

Asymptotes

• Vertical Asymptotes: Occur when the denominator is zero.
• Horizontal Asymptotes: Depends on the degrees of numerator and denominator.

Intermediate Value Theorem (IVT)

• For a continuous function over \[a,b\], any value between f(a) and f(b) must be taken at some point within the interval.

Key Terms

• Algebraic Properties of Limits: Rules for evaluating limits.
• AROC & IROC: Measures of change.
• Continuity: Lack of breaks, holes, or jumps in a function.
• Rational Functions: Functions as ratios of polynomials.
• Squeeze Theorem: Limits estimation by bounding.

These notes provide a comprehensive overview of Unit 1 in AP Calculus, covering essential concepts such as limits, continuity, types of discontinuities, and methods for solving limits.