The definition of the derivative practice question 13
Let . Which statement about at is correct?
Continuity and differentiability are separate questions here. Which limit decides each one at ?
Compute the difference quotient for and for separately.
If the two one-sided limits differ, the limit that defines does not exist.
Theory UBC Math 100 §3.3CLP-1 §2.2Key results
Worked solution
Strategy
The derivative at 2 is the limit of |h|/h as h → 0. That quotient is +1 on one side and −1 on the other, so the limit does not exist. Continuity is a separate, weaker property that does hold.
Working
f(2) = 0 and f(2 + h) = |h|.
The one-sided limits are 1 and −1.
They disagree, so f′(2) does not exist: the graph has a corner. But lim f(x) = 0 = f(2), so f is continuous at 2.
Answer
The answer is continuous at 2 but not differentiable, with one-sided derivatives −1 and 1.
Check
f(1.99) = f(2.01) = 0.01: the chords from (2, 0) have slopes −1 and 1, yet both values close in on f(2) = 0.
Where this goes wrong
Assigning f′(2) = 0 because 2 is where the minimum is. A minimum at a smooth point has zero derivative; at a corner there is no derivative at all.
The idea behind it
The definition of the derivative lesson, a figure to move, in a few steps
Leans on
One-sided limits, from Limits
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Let . Which statement about at is correct?Saved in this browser only.