Skip to content

Local Linearity

Run the Local Linearity MicroSim Fullscreen

Edit the Local Linearity 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/local-linearity/main.html" height="517px" scrolling="no"></iframe>

Description

This MicroSim demonstrates one of the most profound ideas in calculus: local linearity. At any point where a function is differentiable, if you zoom in far enough, the curve becomes indistinguishable from its tangent line.

Delta Moment

"Watch this! When I'm standing on this curve at a single point, and I look really, really closely at the ground beneath my wheels, it looks perfectly flat. The more I zoom in, the flatter it gets. That 'flatness' IS the tangent line!"

Why Local Linearity Matters

Local linearity is the reason calculus works:

  1. Derivatives exist because curves behave linearly at small scales
  2. Linear approximation works because near any point, f(x) ≈ f(a) + f'(a)(x-a)
  3. Differentials (dx and dy) are meaningful because tiny changes are approximately linear
  4. The fundamental theorem connects derivatives and integrals through this local linear behavior

How to Use This MicroSim

  1. Zoom slider: Drag to increase magnification from 1x to 1000x
  2. Point slider: Move the point of tangency along the curve
  3. Function buttons: Switch between x², sin(x), x³, and eˣ
  4. Show Tangent toggle: Hide/show the tangent line (dashed orange)

What to Observe

  • At low zoom (1-5x): The curve is clearly curved, distinct from the straight tangent line
  • At medium zoom (20-50x): The curve starts looking more linear
  • At high zoom (100-1000x): The curve and tangent line become nearly indistinguishable!

The error measurement in the info panel shows how the difference between the curve and tangent line shrinks as you zoom in.

Lesson Plan

Learning Objectives

After completing this activity, students will be able to:

  1. Explain why differentiable functions appear linear at small scales (Bloom Level 2)
  2. Demonstrate local linearity using different functions and zoom levels
  3. Interpret the relationship between zoom level and approximation error
  4. Connect local linearity to the concept of the derivative

Guided Exploration

Activity 1: Discovering Local Linearity (5 minutes)

  1. Start with f(x) = x² at x = 1
  2. Note the visible curvature at zoom level 1x
  3. Slowly increase zoom to 100x
  4. Predict-then-observe: Before each zoom increase, have students predict if the curve will still look curved

Discussion prompt: "At what zoom level did you start having trouble seeing the difference between the curve and the tangent?"

Activity 2: Comparing Functions (5 minutes)

  1. Try each function at the same point (x = 1)
  2. Zoom to 50x for each function
  3. Compare: Which function "straightens out" fastest? Which takes longer?

Key insight: Functions with smaller second derivatives straighten out faster!

Activity 3: Exploring Edge Cases (5 minutes)

  1. For sin(x), try x = 0 (where slope = 1) versus x = π/2 (where slope = 0)
  2. For x³, try x = 0 (inflection point)
  3. Question: Does local linearity work at inflection points?

Assessment Questions

  1. Conceptual: Why does the curve eventually look like its tangent line when you zoom in enough?

  2. Applied: If you're at zoom level 100x and the error is 0.001, what would you estimate the error to be at zoom level 1000x?

  3. Synthesis: How does local linearity explain why sin(x) ≈ x for small values of x?

Common Misconceptions to Address

  • Misconception: "The curve actually becomes straight at small scales"

  • Reality: The curve is always curved; we just can't visually distinguish the curvature at high zoom

  • Misconception: "Local linearity only works for polynomials"

  • Reality: It works for ALL differentiable functions (demonstrate with sin and eˣ)

Extension Activities

  1. Numerical exploration: At x = 1 for f(x) = x², calculate:
  2. f(1.001) using the actual function
  3. f(1.001) using linear approximation: f(1) + f'(1)(0.001)

  4. Compare the results and discuss the tiny error

  5. Connection to limits: Relate the zoom slider to the limit process in the derivative definition

References