Non-Differentiable Points Gallery
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About This MicroSim
This interactive gallery presents the three types of non-differentiable points, helping you classify and distinguish between them. Each type shows why the derivative fails to exist, even though the function itself may be continuous.
The Three Types
| Type | Function | Example | Why No Derivative |
|---|---|---|---|
| Corner | \(f(x) = \|x\|\) | Absolute value at 0 | Left and right limits are different |
| Cusp | \(f(x) = x^{2/3}\) | Cusp at origin | Both limits are infinite, opposite signs |
| Vertical Tangent | \(f(x) = x^{1/3}\) | Cube root at origin | Both limits are \(+\infty\) (vertical line) |
The Key Question
For a derivative to exist at a point, the limit:
must exist as a single, finite number. This requires that the left-hand limit (approaching from negative \(h\)) and the right-hand limit (approaching from positive \(h\)) both exist and are equal.
Delta Moment
"See those two lines? The orange one approaches from the left, the green from the right. For me to have a well-defined tilt at a point, those lines need to settle on the SAME slope. When they don't agree, I'm confused about which way I'm actually tilted!"
How to Use
- Select a Type: Click the tabs to switch between Corner, Cusp, and Vertical Tangent
- Watch the Animation: Click "Watch h Approach 0" to see secant lines converge
- Adjust h Manually: Drag the slider to control how close the secant points are
- Zoom In: Use the + and - buttons to see detail near the origin
What to Observe
- Orange line: Secant from the LEFT (negative x values)
- Green line: Secant from the RIGHT (positive x values)
- Blue point: The non-differentiable point at the origin
- Info panel: Shows the current slopes from each side
Classification Guide
Corner (like \(|x|\) at 0): - Both one-sided derivatives exist and are finite - But they are DIFFERENT values (-1 and +1) - The function makes a "sharp turn"
Cusp (like \(x^{2/3}\) at 0): - Both one-sided derivatives approach infinity - One approaches \(-\infty\), the other \(+\infty\) - The curve comes to an infinitely sharp point
Vertical Tangent (like \(x^{1/3}\) at 0): - Both one-sided derivatives approach \(+\infty\) - They agree, but infinity isn't a valid slope - A tangent line would be vertical (undefined slope)
Lesson Plan
Learning Objectives
After using this MicroSim, students will be able to:
- Classify non-differentiable points into three categories: corner, cusp, or vertical tangent (Bloom Level 4: Analyze)
- Distinguish between cases where one-sided derivatives differ vs. where they are infinite
- Compare the behavior of secant lines from both sides at each type of point
- Predict which type of non-differentiability a given function exhibits
Prerequisite Knowledge
- Understanding of limits and one-sided limits
- Familiarity with the limit definition of the derivative
- Knowledge of basic function types (absolute value, fractional powers)
Suggested Activities
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Predict-Observe-Explain: Before switching to each type, predict what will happen to the secant lines. Then observe and explain any surprises.
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Slope Table: For each type, record the slopes at h = 1, 0.5, 0.1, 0.01. What patterns emerge?
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Classification Challenge: Given a graph of a function with a problematic point, classify the type without seeing the formula.
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Create Your Own: Can students think of other functions that exhibit each type? (Examples: \(f(x) = x^{4/3}\) at 0 for a cusp, \(f(x) = |x-2|\) at 2 for a corner)
Discussion Questions
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All three example functions are continuous at x = 0. Why doesn't continuity guarantee differentiability?
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For the cusp (\(x^{2/3}\)), both secant slopes go to infinity, but in opposite directions. Why does this mean no derivative exists?
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For the vertical tangent (\(x^{1/3}\)), both slopes agree as \(+\infty\). Why can't we just say the derivative is infinity?
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Which type would you expect at a "kink" in a piece of metal wire that's been bent?
Assessment Questions
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Classify the non-differentiable point of \(f(x) = |x - 3|\) at \(x = 3\). (Answer: Corner)
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The function \(g(x) = x^{4/5}\) has what type of point at \(x = 0\)? (Answer: Cusp, since \(g'(x) = \frac{4}{5}x^{-1/5}\) approaches \(\pm\infty\))
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If \(\lim_{h \to 0^-} \frac{f(a+h) - f(a)}{h} = 3\) and \(\lim_{h \to 0^+} \frac{f(a+h) - f(a)}{h} = 3\), does \(f'(a)\) exist? (Answer: Yes, both limits exist and are equal)
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Sketch a function that has a corner at \(x = -1\), is differentiable at \(x = 0\), and has a vertical tangent at \(x = 2\).
Embedding
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