AP Calculus AB · Topic 2.4

Connecting Differentiability and Continuity Practice

Part of Differentiation: Definition and Fundamental Properties.(FUN-1.D)

Practice questions

7

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Sample questions

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Sample 1difficulty 2/5
$At the V-shape corner, the function is$
- A
  $A local maximum$
- B
  $Differentiable$
- C
  $Discontinuous$
- D
  $Continuous but not differentiable$
  check_circle
Why
$f'$
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Sample 2difficulty 2/5
$f(x) = |x|$
- A
  $x = 0$
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- B
  $x = 1$
- C
  $Nowhere$
- D
  $Everywhere$
Why
$x = 0$
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Sample 3difficulty 2/5
$Which is true?$
- A
  $Continuous \Rightarrow differentiable$
- B
  $Differentiable \Leftrightarrow continuous$
- C
  $Differentiable \Rightarrow continuous$
  check_circle
- D
  $Neither implies the other$
Why
$|x|$
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Sample 4difficulty 2/5
$x = 2$
- A
  $Continuous and differentiable$
- B
  $Continuous but not differentiable$
  check_circle
- C
  $Neither$
- D
  $Differentiable but not continuous$
Why
$Corners \Rightarrow left/right derivatives differ \to not differentiable. The function can still be continuous.$
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Sample 5difficulty 3/5
$A function with a "V" shape at a point is:$
- A
  $Differentiable but not continuous$
- B
  $Neither$
- C
  $Both$
- D
  $Continuous but not differentiable there$
  check_circle
Why
$The function is continuous at the corner, but the derivative is undefined due to differing one-sided slopes.$
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