AP Calculus AB/BC: Unit 1 Overview

Introduction to Calculus

• Calculus is the study of change and motion.
• Key question: "Can change occur at an instant?" Illustrated by the moving arrow analogy.

1.1 Introducing Calculus: Rates of Change

• Average Rate of Change (AROC): Slope of the secant line between two points.
  • Formula: (f(b) - f(a)) / (b - a)
• Instantaneous Rate of Change (IROC): Derivative, slope of the tangent line at a point.

1.2 Limits and Limit Notation

• Limit: Value a function approaches as the input approaches a particular value.
• Used to understand instantaneous rate of change.
• Limit notation: ( \lim_{x \to a} f(x) = L )
• Example: ( \lim_{x \to 2} x^2 = 4 )

1.3 Estimating Limits from Graphs

• One-sided Limits: Limit as x approaches a value from one direction.
• Two-sided Limits: Limit as x approaches a value from both directions.
• Limits may not exist if the function oscillates, becomes unbounded, or has a vertical asymptote.

1.4 Estimating Limits from Tables

• Approach the x-value from both directions to estimate the limit.
• If left and right limits differ, the limit does not exist.

1.5 Algebraic Properties of Limits

• Useful for evaluating limits.
  • Addition/Subtraction: ( \lim (f(x) + g(x)) = \lim f(x) + \lim g(x) )
  • Multiplication: ( \lim (f(x) \cdot g(x)) = \lim f(x) \cdot \lim g(x) )

1.6 Solving Limits Algebraically

• Methods include:
  • Substitution: Plug in the value of x.
  • Factoring: Factor and cancel common terms.
  • Common Denominator: Combine fractions with a common denominator.
  • Conjugates: Use to simplify expressions with roots.

1.8 Squeeze Theorem

• If ( f(x) < g(x) < h(x) ) and ( \lim f(x) = \lim h(x) = L ), then ( \lim g(x) = L ).

1.10 Types of Discontinuities

• Jump Discontinuity: Gap in a piecewise function.
• Removable Discontinuity: Hole in graph; limit exists but not continuous.
• Infinite Discontinuity: Vertical asymptote.

1.11 Continuity at a Point

• A function is continuous at ( x = a ) if:
  1. ( f(a) ) is defined.
  2. ( \lim_{x \to a} f(x) ) exists.
  3. ( \lim_{x \to a} f(x) = f(a) ).

1.12 Continuity over an Interval

• Continuous if function is defined, has no jumps/holes, and limits match values within the interval.

1.13 Removing Discontinuities

• Redefine the function at a point to make it continuous.

1.14 & 1.15 Asymptotes

• Vertical Asymptotes: Set denominator to zero in rational functions.
• Horizontal Asymptotes: Compare degrees of numerator and denominator.
• Intermediate Value Theorem (IVT): Continuous function over [a,b] takes all values between ( f(a) ) and ( f(b) ).

Key Terms

• Algebraic Properties of Limits
• Average Rate of Change (AROC)
• Calculus
• Common Denominator Method
• Conjugate Method
• Continuity at a Point
• Exponential Functions
• Factoring Method
• Infinite Discontinuity
• Instantaneous Rate of Change (IROC)
• Limits
• Rational Functions
• Removable Discontinuity
• Secant Line
• Squeeze Theorem
• Substitution Method
• Trigonometric Functions
• Types of Discontinuities
• Vertical Asymptote