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Cogito

AP Calculus AB · Chapter 2 · Lesson 2

Practice Worksheet

Differentiability and Local Linearity

Where a derivative fails to exist.

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 each side gives a different slope, so no derivative exists there.
-8-8-6-6-4-4-2-222446688
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 − 5|. At which x is the derivative undefined?

    Answer

  2. What does local linearity mean?

    • Zooming in far enough makes the curve look like its tangent.
    • The whole function is a straight line.

  3. y = |x|. Slope just to the right of zero?

    Answer

  4. y = |x|. Slope just to the left of zero?

    Answer

  5. Continuous at x = 2. Must it be differentiable? 1 yes, 0 no.

    Answer

Build It Up

The same ideas with more to keep track of.

3 problems

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

    Answer

  2. A graph jumps at x = 1. Differentiable there? 1 yes, 0 no.

    Answer

  3. At a vertical tangent, is the derivative defined? 1 yes, 0 no.

    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 discontinuity
    • A horizontal tangent
    • Tick every box that applies.

  2. y = x². Differentiable at zero? 1 yes, 0 no.

    Answer

  3. The Corner

    y = |x − 3|. At which x is it not differentiable?

    Answer

  4. The Implication

    A function is differentiable at x = 5. Must it be continuous there? 1 yes, 0 no.

    Answer