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Cogito

Calculus · Chapter 3 · Lesson 3

Practice Worksheet

Differentiability

Where a function has no slope at all.

12 problems · about 21 minutes · FUN-2.A

NameDate

Figure — use these to answer the problems

Look at the point of the V. Approaching from the left and the right gives two different slopes.
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y = 1|x + 0| + 0
a, vertical stretch1
b, horizontal shift0
Figure 1

Warm Up

Straightforward practice. Get the method working first.

5 problems

  1. y = |x − 2|. At which x is the derivative undefined?

    Answer

  2. Which statement is true?

    • Differentiable implies continuous, but not the reverse.
    • Continuous implies differentiable.

  3. For y = |x|, the slope just right of zero?

    Answer

  4. A function is continuous at x = 2. Must it be differentiable there? 1 or 0.

    Answer

  5. y = |x − 4|. At which x is it not differentiable?

    Answer

Build It Up

The same ideas with more to keep track of.

3 problems

  1. A graph jumps at x = 1. Is it differentiable there? 1 or 0.

    Answer

  2. y = x². Is it differentiable at x = 0? 1 or 0.

    Answer

  3. At a vertical tangent, is the derivative defined? 1 or 0.

    Answer

Stretch Yourself

Mixed problems. Work out what kind of question it is before you start.

4 problems

  1. Which situations block differentiability?

    • A corner
    • A vertical tangent
    • A jump
    • A horizontal tangent
    • Tick every box that applies.

  2. y = |x + 7|. At which x is it not differentiable?

    Answer

  3. The Corner

    For y = |x|, what is the slope just to the left of zero?

    Answer

  4. The Implication

    A function is differentiable at x = 3. Must it be continuous there? 1 for yes, 0 for no.

    Answer