Cogito
AP Calculus AB · Chapter 2 · Lesson 2
Differentiability and Local Linearity
Where a derivative fails to exist.
12 problems · about 21 minutes · FUN-2.A
What this lesson teaches
The student identifies points of non-differentiability and relates differentiability to continuity.
- A derivative fails at corners, cusps, vertical tangents and breaks.
- Differentiable implies continuous, but not the reverse.
- Differentiable means locally linear.
Warm Up
Straightforward practice. Get the method working first.
5 problemsy = |x − 5|. At which x is the derivative undefined?
Answer 5
Why 5.
What does local linearity mean?
Answer Zooming in far enough makes the curve look like its tangent.
Why The curve is indistinguishable from its tangent up close.
y = |x|. Slope just to the right of zero?
Answer 1
Why The function is x there.
y = |x|. Slope just to the left of zero?
Answer -1
Why The function is −x there.
Continuous at x = 2. Must it be differentiable? 1 yes, 0 no.
Answer 0
Why A corner is a counterexample.
Build It Up
The same ideas with more to keep track of.
3 problemsy = |x + 7|. At which x is it not differentiable?
Answer -7
Why Where the inside is zero.
A graph jumps at x = 1. Differentiable there? 1 yes, 0 no.
Answer 0
Why Not even continuous.
At a vertical tangent, is the derivative defined? 1 yes, 0 no.
Answer 0
Why A vertical line has no slope.
Stretch Yourself
Mixed problems. Work out what kind of question it is before you start.
4 problemsWhich situations block differentiability?
Answer A corner; A vertical tangent; A jump discontinuity
Why A horizontal tangent has a perfectly good slope of zero.
y = x². Differentiable at zero? 1 yes, 0 no.
Answer 1
Why A smooth parabola.
The Corner: y = |x − 3|. At which x is it not differentiable?
Answer 3
Why 3.
The Implication: A function is differentiable at x = 5. Must it be continuous there? 1 yes, 0 no.
Answer 1
Why Yes.