Skip to content

One-Sided Derivatives

Run the One-Sided Derivatives MicroSim Fullscreen

Edit the One-Sided Derivatives MicroSim with the p5.js editor

Place the following line in your website to include this in your course.

1
<iframe src="https://dmccreary.github.io/calculus/sims/one-sided-derivatives/main.html" height="482px" scrolling="no"></iframe>

Description

This MicroSim helps students analyze one-sided derivatives to determine whether a function is differentiable at a given point. The visualization shows:

  • A function plotted on a coordinate plane - Select from smooth functions (like x^2 or sin(x)+1) or functions with corners/cusps (like |x| or max(x,0))
  • Point of interest at x = 0 - This is where we examine differentiability
  • Left secant line (blue) - Connects (a-h, f(a-h)) to (a, f(a)), representing the left-hand difference quotient
  • Right secant line (red) - Connects (a, f(a)) to (a+h, f(a+h)), representing the right-hand difference quotient
  • Slope values - Real-time display of left and right secant slopes

As students decrease the h-value using the slider, they can observe whether the two secant slopes converge to the same value (differentiable) or approach different limits (not differentiable).

Delta Moment

"When I roll toward a corner, my left wheel and right wheel suddenly disagree about which way is 'up.' That's what a non-differentiable point feels like - my two sides have different opinions about the slope!"

How to Use

  1. Select a function from the dropdown to explore different cases
  2. Adjust the h slider to control how close the secant points are to x = 0
  3. Observe the left (blue) and right (red) secant lines as h decreases
  4. Compare the slopes shown in the info panel
  5. Determine if the function is differentiable based on whether slopes converge

Key Observations

Function Type What Happens as h approaches 0 Differentiable?
Smooth (x^2, sin(x)) Both slopes approach the same value Yes
Corner (absolute value x) Left slope approaches -1, right approaches +1 No
Cusp (square root of absolute value x) Slopes diverge to infinity No

Lesson Plan

Learning Objective

Students will analyze one-sided derivatives to determine differentiability at a point (Bloom Level 4: Analyze)

Grade Level

High School AP Calculus (Grades 11-12)

Duration

15-20 minutes

Prior Knowledge Required

  • Understanding of limits
  • Concept of secant lines and slopes
  • Definition of the derivative as a limit of difference quotients

Activity Sequence

Part 1: Introduction (3 minutes)

  1. Review the definition of the derivative: \(\(f'(a) = \lim_{h \to 0} \frac{f(a+h) - f(a)}{h}\)\)

  2. Explain that for this limit to exist, both one-sided limits must exist and be equal

Part 2: Exploring Smooth Functions (5 minutes)

  1. Start with the "Smooth: x^2" function
  2. Set h = 1.0 and observe both secant lines
  3. Slowly decrease h and observe how both slopes converge to 0
  4. Discuss: "Why do both secant slopes approach the same value?"

Part 3: Discovering Non-Differentiability (7 minutes)

  1. Switch to "Corner: |x|" function
  2. With h = 1.0, note the difference between left and right slopes
  3. Decrease h and observe that slopes approach different values (-1 and +1)
  4. Ask students: "What does this tell us about the derivative at x = 0?"
  5. Explore the cusp function to see slopes diverging

Part 4: Synthesis (5 minutes)

  1. Have students predict which functions will be differentiable before testing
  2. Use the "max(x,0)" function as a test case
  3. Discuss real-world examples where non-differentiable points matter

Assessment Questions

  1. Explain why a corner causes non-differentiability in your own words
  2. If the left-hand derivative is 2 and the right-hand derivative is 5, is the function differentiable at that point?
  3. Sketch a function that is continuous but not differentiable at x = 2

Extensions

  • Have students create their own piecewise functions and predict differentiability
  • Connect to the concept of "sharp corners" in optimization problems
  • Discuss why physicists care about smoothness of position functions

References

Reminder: Please capture a screenshot for social media previews using ~/.local/bin/bk-capture-screenshot