AP Calculus AB Flashcards: Connecting Differentiability and Continuity: Determining When Derivatives Do and Do Not Exist
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Review key ideas with interactive flashcards. This set includes 10 cards to help you master important concepts.
What is the fundamental relationship between differentiability and continuity?
If a function is differentiable at a point, then it must be continuous at that point. However, the reverse is not always true; a continuous function is not necessarily differentiable.
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What is the fundamental relationship between differentiability and continuity?
If a function is differentiable at a point, then it must be continuous at that point. However, the reverse is not always true; a continuous function is not necessarily differentiable.
A function g(x) is not continuous at x = 3. Can g(x) be differentiable at x = 3?
No, it cannot be differentiable at x = 3. Because differentiability implies continuity, a function must be continuous at a point to be differentiable there.
What does it mean for a derivative 'not to exist' at a point?
It means the function is not differentiable at that point. This can occur even if the function is continuous at the point.
If a function is differentiable at x = a, what can you conclude about its continuity at x = a?
If a function is differentiable at a point, it is guaranteed to be continuous at that same point.
State the one-way implication that connects differentiability and continuity.
Differentiability at a point implies continuity at that point.
If you successfully calculate a finite value for the derivative of a function f(x) at x=c, what does this immediately tell you about the behavior of f(x) at that point?
It tells you that the function is not only differentiable at x=c, but also that it must be continuous at x=c.
Which is considered a 'stronger' condition for a function at a point: differentiability or continuity?
Differentiability is a stronger condition because any differentiable function must be continuous, but not all continuous functions are differentiable.
If a function is continuous at x = b, can you conclude it is differentiable at x = b?
No, a function can be continuous at a point in its domain but still fail to be differentiable at that point.
A calculus student claims that since f(x) is continuous on the interval [-5, 5], its derivative f'(x) must also exist for every point in that interval. Is this claim correct?
The claim is incorrect. A continuous function may fail to be differentiable at one or more points in its domain.
Define Differentiability (in terms of its relationship to continuity).
Differentiability is a property of a function at a point that implies the function is also continuous at that point.