AP Calculus AB

Advanced Placement Calculus AB covering limits, derivatives, and integrals.

Understanding LimitsIntroduction to DerivativesIntroduction to Integrals
Continuity and Types of DiscontinuitiesThe Chain Rule and Implicit DifferentiationFundamental Theorem of Calculus
Modeling Motion and VelocityApplications in Economics and BiologyEngineering and Technology Applications
Multiple Choice MasteryFree Response SuccessTime Management for the Exam
Understanding LimitsIntroduction to Deri...Introduction to Inte...Continuity and Types...The Chain Rule and I...Fundamental Theorem ...Modeling Motion and ...Applications in Econ...Engineering and Tech...Multiple Choice Mast...Free Response Succes...Time Management for ...
Advanced Topics

Continuity and Types of Discontinuities

Understanding Continuity

A function is continuous if you can draw its graph without lifting your pencil. This means there are no gaps, jumps, or holes in the graph.

Types of Discontinuities

  • Removable Discontinuity: A hole in the graph—often caused by a factor that cancels.
  • Jump Discontinuity: The graph jumps from one value to another.
  • Infinite Discontinuity: The graph goes off to infinity (vertical asymptote).

Why Does This Matter?

Calculus relies on continuity for derivatives and integrals to exist. Discontinuities can cause problems in calculations.

Identifying Discontinuities

  1. Look for division by zero.
  2. Check for places where the function isn't defined.
  3. Analyze the left and right limits.

Real-World Impact

Discontinuities can represent sudden changes, like a switch turning on, or an object changing direction.

Key Formula

\[f(x) = \frac{x^2-1}{x-1}\]

Examples

  • The graph of \( f(x) = \frac{x^2-1}{x-1} \) has a hole at \( x = 1 \).

  • A step function modeling elevator floors has jump discontinuities.

In a Nutshell

Continuity ensures smooth behavior; discontinuities reveal gaps, jumps, or infinite breaks.

Key Terms

Removable Discontinuity
A single point missing from an otherwise continuous curve.
Jump Discontinuity
A sudden jump in the value of a function.
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The Chain Rule and Implicit Differentiation
Continuity and Types of Discontinuities - AP Calculus AB Content | Practice Hub