College Board-Aligned Original Notes
AP Calculus AB Unit 2 Topic 2: Connecting differentiability and continuity
Use Connecting differentiability and continuity across graphical, numerical, algebraic, and verbal representations.
Unit 2: Differentiation: Definition and Fundamental Properties. College Board exam weighting listed for this unit: 10%-12% of exam score.
What to Know
- Check the conditions of a theorem or method before applying it.
- Show the setup before the calculation.
- Interpret the result in context, including units when the problem supplies them.
- Always connect this topic back to the larger unit: Differentiation: Definition and Fundamental Properties.
Detailed Notes
Connecting differentiability and continuity should be studied through multiple representations. A graph may show behavior quickly, an equation may make calculation possible, and a verbal interpretation explains what the result means.
In AP Calculus AB, AP questions often award credit for setup and reasoning, not just final answers. Write the expression, theorem, condition, or model before doing the computation.
When this topic appears in free response, check whether the question asks for a value, a rate, an interval, a comparison, or a justification. Use units and context to make the final answer precise.
Key Vocabulary
Limit
The value a function approaches as the input approaches a specified value.
Continuity
A property of a function with no break, hole, or jump at a point or on an interval.
Asymptote
A line that a graph approaches but may not reach.
Intermediate Value Theorem
A theorem guaranteeing that continuous functions take on every value between two function values.
Derivative
Instantaneous rate of change of a function.
Differentiability
The property of having a derivative at a point.
Quick Practice
How would you explain Connecting differentiability and continuity in one or two AP-style sentences?
Name the concept, apply it to a specific example or source, and explain the reasoning that connects the evidence to your answer.
Related Topics in This Unit
- Defining the derivative of a function at a point and as a function
- Determining derivatives for elementary functions
- Applying differentiation rules