Cogito
Calculus · Chapter 3 · Lesson 3
Differentiability
Where a function has no slope at all.
12 problems · about 21 minutes · FUN-2.A
What this lesson teaches
The student identifies points where a function is not differentiable and relates differentiability to continuity.
- A derivative fails to exist at corners, cusps, vertical tangents and breaks.
- Every differentiable function is continuous.
- Continuity does not guarantee differentiability.
Warm Up
Straightforward practice. Get the method working first.
5 problemsy = |x − 2|. At which x is the derivative undefined?
Answer 2
Why x = 2.
Which statement is true?
Answer Differentiable implies continuous, but not the reverse.
Why Differentiability is the stronger property.
For y = |x|, the slope just right of zero?
Answer 1
Why The function is x there.
A function is continuous at x = 2. Must it be differentiable there? 1 or 0.
Answer 0
Why A corner is a counterexample.
y = |x − 4|. At which x is it not differentiable?
Answer 4
Why Where the inside is zero.
Build It Up
The same ideas with more to keep track of.
3 problemsA graph jumps at x = 1. Is it differentiable there? 1 or 0.
Answer 0
Why It is not even continuous.
y = x². Is it differentiable at x = 0? 1 or 0.
Answer 1
Why A smooth parabola.
At a vertical tangent, is the derivative defined? 1 or 0.
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
Why A horizontal tangent has a perfectly good slope of zero.
y = |x + 7|. At which x is it not differentiable?
Answer -7
Why Where the inside is zero.
The Corner: For y = |x|, what is the slope just to the left of zero?
Answer -1
Why −1.
The Implication: A function is differentiable at x = 3. Must it be continuous there? 1 for yes, 0 for no.
Answer 1
Why Yes.