AP Calculus AB Review Lecture

Overview

• Comprehensive review of topics for AP Calc AB exam.
• Topics covered:
  • Limits and Continuity
  • Definition of Differentiation
  • Composite, Implicit, and Inverse Differentiation
  • Contextual and Analytical Applications of Differentiation
  • Integration and Accumulation of Change
  • Differential Equations and Applications of Integration

Concepts of Change

• Pre-Calculus: Average rates of change (e.g., average speed).
• Calculus: Instantaneous rates of change.
  • Introduces limits to define the slope of a curve at a point.

Limits and Continuity

• Limits:

  • Define the slope of a curve and foundation of derivatives.
  • Can break limits across operations if individual limits exist.
  • Important for evaluating limits using graphs and tables.
  • Squeeze Theorem: Used when a function is bounded between two others.
  • Types of discontinuities: removable, jump, and infinite.

• Continuity:

  • Conditions for a function to be continuous at a point:
    1. Function is defined at that point.
    2. Limit exists at that point.
    3. Value of the function equals the limit.
  • Intermediate Value Theorem (IVT): Ensures a value exists between f(a) and f(b).

Differentiation

• Definition:

  • Limit definitions of the derivative.
  • Notations: f'(x), y', dy/dx.
  • Derivative is the slope of the tangent line.

• Differentiability:

  • Differentiability implies continuity.
  • Conditions: No sharp corners, cusps, or discontinuities.

• Basic Rules:

  • Constant, power, product, quotient, and chain rules.
  • Common Derivatives: Trig and exponential functions.

• Applications in Motion:

  • Velocity = derivative of position.
  • Acceleration = derivative of velocity.

• Related Rates and Tangent Approximations:

  • Solve related rates by differentiating implicit equations.
  • Tangent line approximations for function values near a point.

Li Tall’s Rule

• Used for indeterminate forms like 0/0.
• Differentiate numerator and denominator separately.

Theorems

• Mean Value Theorem: Instantaneous rate equals average rate over an interval.
• Extreme Value Theorem: Guarantees minimum and maximum values.
• First and Second Derivative Tests: To determine local extrema and concavity.

Optimization

• Solve using constraints and derivatives to find extrema.

Integration

• Riemann Sums: Approximating area under curves.

• Definite Integrals:

  • Properties: Constant multiplication, addition, subtraction.
  • Fundamental Theorems of Calculus:
    1. Derivatives and integrals are inverses.
    2. Area under a curve between bounds.

• Indefinite Integrals:

  • No bounds, results in antiderivative + constant.
  • U-Substitution: Simplify and evaluate integrals.

Differential Equations

• Slope Fields: Visual representation of differential equations.
• Solving:
  • Separate variables, integrate, solve for dependent variable.
  • Exponential growth modeled by dydt = ky.

Applications of Integration

• Average Value: Using definite integrals to find function’s average value.
• Position, Velocity, and Acceleration: Integrals relating motion concepts.
• Area Between Curves: Integrate top function minus bottom function.
• Volume of Solids:
  • Cross-sections and revolved solids.
  • Washer method for volumes with spaces between areas and axes.