AP Calculus BC Review

Overview

• Concepts of calculus include instantaneous rates of change and limits.
• Calculus uses limits to define the slope of a curve and the foundation of derivatives.

Limits and Continuity

• Limits: Define the slope of a curve at a point.

  • Rules allow breaking limits across operations if individual limits exist.
  • Graphical evaluation: y-value approached as x approaches a point.
  • Numerical evaluation: Using table of values.
  • Squeeze Theorem: Used when a function is bounded between two others.
  • Discontinuities: Removable, Jump, Infinite.

• Continuity:

  • A function is continuous at a point if it is defined, the limit exists, and the function equals the limit.
  • IVT: Intermediate Value Theorem for continuous functions.

Derivatives

• Definition: Derivative represents slope of tangent line.

  • Notation: f'(x), y', dy/dx.
  • Different rules for differentiation (constant, power, product, quotient, chain rule).
  • Derivatives of trigonometric and exponential functions.

• Applications:

  • Motion problems: Position, velocity, acceleration relationships.
  • Related rates: Finding the rate of change of one quantity in terms of another.
  • Tangent line approximations for estimating function values.

• Differentiability:

  • Implies continuity.
  • Non-differentiable at sharp corners, cusps, discontinuities.

Integration

• Definite and Indefinite Integrals:

  • Definite: Area under a function.
  • Fundamental Theorems of Calculus link derivatives and integrals.

• Techniques:

  • U-substitution, Integration by parts, Partial fractions.
  • Improper integrals: Evaluating integrals with unbounded limits.

• Applications:

  • Solving for average value, total distance, and areas between curves.
  • Calculating volumes by integration (washer method).

Differential Equations

• Solving Techniques:

  • Slope fields visualize first derivatives.
  • Euler's method for approximating solutions.
  • Steps: Separate variables, integrate, solve.

• Models:

  • Exponential growth and logistic models.

Vector Functions and Parametric Equations

• Parametric Functions:

  • Separate x and y in terms of t.
  • Calculating derivatives and arc length.

• Polar Coordinates:

  • Polar equations and conversions to Cartesian.
  • Calculating areas between polar curves.

Sequences and Series

• Definitions:

  • Sequences: List of numbers.
  • Series: Sum of numbers.

• Convergence:

  • Tests to determine if a series converges or diverges.

• Taylor and Maclaurin Series:

  • Taylor: Centered at any point.
  • Maclaurin: Centered at x = 0.
  • Error bound formulas and radius of convergence.

This review covers critical topics in AP Calculus BC, providing a comprehensive summary of differentiation, integration, applications, and advanced topics like sequences and series. Each section provides foundational understanding and techniques necessary for solving calculus problems effectively.