AP Calculus BC

Advanced Placement Calculus BC including series, parametric equations, and polar functions.

Limits and ContinuityDifferentiation and ApplicationsIntegration and Its Uses
Infinite Series and ConvergenceParametric Equations and MotionPolar Coordinates and Graphs
Analyzing Population GrowthDesigning Roller Coasters with Parametric EquationsSignal Processing and Polar Graphs
Mastering Multiple-Choice QuestionsEffective Free-Response Techniques
Limits and Continuit...Differentiation and ...Integration and Its ...Infinite Series and ...Parametric Equations...Polar Coordinates an...Analyzing Population...Designing Roller Coa...Signal Processing an...Mastering Multiple-C...Effective Free-Respo...
Basic Concepts

Differentiation and Applications

Rates of Change

Differentiation is the process of finding the derivative, which represents the rate of change of a function.

  • The derivative of \( f(x) \), written \( f'(x) \), tells how \( f(x) \) changes as \( x \) changes.
  • The power rule, product rule, quotient rule, and chain rule are essential techniques for finding derivatives.

Applications

  • Finding the slope of a tangent line at a point
  • Analyzing motion (velocity and acceleration)
  • Solving optimization problems

Real-World Relevance

Derivatives are everywhere—speedometers, economics, biology, and more!

Examples

  • The derivative of \( f(x) = x^2 \) is \( 2x \).

  • If a car's position is \( s(t) \), its velocity is \( s'(t) \).

In a Nutshell

Derivatives measure how things change and help solve real-world problems.

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Integration and Its Uses