Practice question · Sort into groups
Sort each described point by whether the function is differentiable there.
Groups: Differentiable · Not differentiable
- A jump discontinuity
- A smooth curve with a single well-defined tangent
- A sharp corner where the slope jumps from -1 to +1
- A flat spot where the tangent is horizontal
- A point where the function is undefined
Hints
- The defining limit must exist and give one single number.
- A horizontal tangent is still a tangent.
Show the answer
Differentiable: A smooth curve with a single well-defined tangent, A flat spot where the tangent is horizontal
Not differentiable: A sharp corner where the slope jumps from -1 to +1, A jump discontinuity, A point where the function is undefined
Why
A flat spot is perfectly differentiable, with derivative zero, zero slope is not the same as no slope. Corners give two competing slopes, and jumps or gaps break continuity, which differentiability requires.
Practise Derivatives and Rates of Change
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