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AP Calculus AB: 3.1.4 Differentiability

Mathematics9 CardsCreated 8 months ago

This flashcard set explains the concept of differentiability, highlighting that a function must be smooth and continuous to be differentiable. It covers cases where differentiability fails due to sharp turns or discontinuities and provides examples to identify differentiability on intervals, along with interpretations of derivative values.

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Differentiability

  • Sometimes the derivative of a function is not defined. This may happen because the function is discontinuous or because it makes a sharp turn.

  • If a function is differentiable, then it is continuous.

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Key Terms

Term
Definition

Differentiability

  • Sometimes the derivative of a function is not defined. This may happen because the function is discontinuous or because it makes a sharp tu...

note

  • Differentiability refers to the existence of a derivative.

  • A function is said to be differentiable if its derivative exists ...

Is the function f (x) = 3x^ 2 − 27 differentiable on the open interval (2, 4)?

Yes, the function is differentiable on this interval.

is the function
f(x)=1/2x^2, x<1
x, x≥1
differentiable on the interval (0, ∞)?

No, the function is not differentiable on this interval.

Is the function f(x)=|x| differentiable on the open interval (−3,2)?

No, the function is not differentiable on that interval.

Suppose you are given a function f(x) and you are told that f′(a)=2. What do you know about the function f(x)?
I. The slope of the line tangent to f(x) at a is positive.
II. The instantaneous rate of f(x) at a is negative.
III. The function is defined at f(a).

I and III

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AP Calculus AB: 3.1.4 Differentiability
TermDefinition

Differentiability

  • Sometimes the derivative of a function is not defined. This may happen because the function is discontinuous or because it makes a sharp turn.

  • If a function is differentiable, then it is continuous.

note

  • Differentiability refers to the existence of a derivative.

  • A function is said to be differentiable if its derivative exists for all points in its domain. The graph of a differentiable function is smooth.

  • Some functions may not be differentiable everywhere. A function is differentiable at a point x if the derivative exists at x. A function is differentiable on an open interval ( a, b ) if the derivative exists at every point in the interval.

  • If a function is not differentiable at every point, then its graph may be broken, or may have sharp turns or other irregularities.

  • Because of its power and flexibility, many mathematicians rank the derivative as one of humanity’s greatest achievements.

Is the function f (x) = 3x^ 2 − 27 differentiable on the open interval (2, 4)?

Yes, the function is differentiable on this interval.

is the function
f(x)=1/2x^2, x<1
x, x≥1
differentiable on the interval (0, ∞)?

No, the function is not differentiable on this interval.

Is the function f(x)=|x| differentiable on the open interval (−3,2)?

No, the function is not differentiable on that interval.

Suppose you are given a function f(x) and you are told that f′(a)=2. What do you know about the function f(x)?
I. The slope of the line tangent to f(x) at a is positive.
II. The instantaneous rate of f(x) at a is negative.
III. The function is defined at f(a).

I and III

Suppose you are given a function f(x) and you are told that f′(a) is undefined. What do you know about the function f(x)?
I. The slope of the line tangent to f(x) is positive.
II. The instantaneous rate of f(x) at a is negative.
III. The function is defined at f(a).

None of the statements must be true.

Is the function f(x)=x^2 / x+3differentiable on the open interval (−4,4)?

No, the function is not differentiable on that interval.

Suppose you are given a function f(x) and you are told that f′(a) exists. What do you know about the function f(x)?
I. The slope of the line tangent to f(x) is positive.
II. The instantaneous rate of f(x) at a is negative.
III. The function is defined at f(a).

III only