AP Calculus AB · Unit 2: Differentiation: Definition and Fundamental Properties
2.4 Connecting Differentiability and Continuity: Determining When Derivatives Do and Do Not Exist
In short
You must understand the one-way implication that differentiability at a point guarantees continuity at that point, and that a point not in the domain of a function cannot be in the domain of its derivative. Conversely, you must recognize that continuity does not guarantee differentiability.
Last updated
What else do you need to know about Connecting Differentiability and Continuity: Determining When Derivatives Do and Do Not Exist?
The course specifically requires you to identify cases where a continuous function is not differentiable, such as when the left-hand and right-hand limits of the difference quotient are not equal (e.g., a corner or cusp like f(x) = |x| at x = 0) or when the tangent line is vertical and has no slope (e.g., f(x) = ∛x at x = 0).
Can you do these?
- I can state that if a function is differentiable at a point, it must be continuous at that point, and that a point outside the domain of a function is also outside the domain of its derivative.
- I can recognize and provide examples of continuous functions that fail to be differentiable at a point in their domain (e.g., corners, cusps, vertical tangents).
How is Connecting Differentiability and Continuity: Determining When Derivatives Do and Do Not Exist tested on the exam?
You are expected to provide reasons or rationales for solutions and conclusions (Skill 3.E). You may be assessed on their ability to connect graphical representations of functions to their differentiability and continuity, and must avoid misinterpreting the graph of a derivative as the graph of the original function. Justifications should rely on clear mathematical structures, such as referencing the limits of the difference quotient when explaining why a function is not differentiable at a corner or cusp.
Free practice questions with full explanations, and instant marking of your free-response answers.
Connecting Differentiability and Continuity: Determining When Derivatives Do and Do Not Exist: common questions
What do you need to know about Connecting Differentiability and Continuity: Determining When Derivatives Do and Do Not Exist for AP Calculus AB?
You must understand the one-way implication that differentiability at a point guarantees continuity at that point, and that a point not in the domain of a function cannot be in the domain of its derivative. Conversely, you must recognize that continuity does not guarantee differentiability.
How is Connecting Differentiability and Continuity: Determining When Derivatives Do and Do Not Exist tested on the AP Calculus AB exam?
You are expected to provide reasons or rationales for solutions and conclusions (Skill 3.E). You may be assessed on their ability to connect graphical representations of functions to their differentiability and continuity, and must avoid misinterpreting the graph of a derivative as the graph of the original function.
How do I know I have mastered Connecting Differentiability and Continuity: Determining When Derivatives Do and Do Not Exist?
Check you can do each of these: state that if a function is differentiable at a point, it must be continuous at that point, and that a point outside the domain of a function is also outside the domain of its derivative; recognize and provide examples of continuous functions that fail to be differentiable at a point in their domain (e.g., corners, cusps, vertical tangents).
Where does Connecting Differentiability and Continuity: Determining When Derivatives Do and Do Not Exist fit in AP Calculus AB?
Connecting Differentiability and Continuity: Determining When Derivatives Do and Do Not Exist is topic 2.4 in Unit 2: Differentiation: Definition and Fundamental Properties of AP Calculus AB.
Want help mastering this topic?
Work through Connecting Differentiability and Continuity: Determining When Derivatives Do and Do Not Exist one-on-one with an AP Guru tutor.
Book a free demo class