Skip to content

EVT Conditions Explorer

Run the EVT Conditions Explorer Fullscreen

Edit with the p5.js Editor

Embedding This MicroSim

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

1
<iframe src="https://dmccreary.github.io/calculus/sims/evt-conditions/main.html" height="522px" width="100%" scrolling="no"></iframe>

Description

This MicroSim helps students understand why the Extreme Value Theorem (EVT) requires BOTH conditions to be satisfied:

  1. Continuity on the interval
  2. Closed interval [a, b]

The visualization allows students to experiment with violating each condition and observe the consequences.

How to Use

  1. Select a scenario from the dropdown:
  2. Continuous on closed: The standard case where EVT applies
  3. Jump discontinuity: A function with a sudden jump at x = 1

  4. Removable discontinuity: A function with a "hole" at its maximum

  5. Adjust the interval using the left (a) and right (b) endpoint sliders

  6. Toggle endpoint types using the buttons:

  7. Click "[a, (closed)" to toggle between [a (closed) and (a (open)

  8. Click "b] (closed)" to toggle between b] (closed) and b) (open)

  9. Find Extrema button highlights where the maximum and minimum occur (or shows why they may not exist)

What to Observe

  • When both conditions are met: Green checkmarks appear, and the global maximum and minimum are clearly marked
  • When continuity fails: See how the function's behavior at a discontinuity prevents a guaranteed extremum
  • When the interval is open: Watch the animation showing how values can get arbitrarily close to an extremum without reaching it

Lesson Plan

Learning Objectives

By the end of this activity, students will be able to:

  1. State the two conditions required for the Extreme Value Theorem to apply
  2. Predict whether EVT applies given a function and interval description
  3. Explain why each condition is necessary using counterexamples
  4. Evaluate real-world scenarios to determine if EVT guarantees extrema

Prerequisites

  • Understanding of function continuity
  • Knowledge of interval notation (open vs. closed)
  • Basic understanding of maximum and minimum values

Suggested Activities

Activity 1: Exploration (10 minutes)

Have students start with "Continuous on closed" scenario and verify that EVT applies. Then systematically explore what happens when each condition is violated.

Activity 2: Prediction Challenge (15 minutes)

Before clicking "Find Extrema," students predict: - Will a maximum exist? Why or why not? - Will a minimum exist? Why or why not? Then verify their predictions.

Activity 3: Real-World Connection (10 minutes)

Discuss scenarios where: - Temperature on a given day (closed interval, continuous) - Stock price at a specific moment (may have "jumps") - Distance traveled during a trip (endpoint considerations)

Assessment Questions

  1. A continuous function is defined on (0, 5]. Does EVT guarantee a maximum exists? Explain.

  2. Consider f(x) = 1/x on [1, 10]. Does EVT apply? What are the extrema?

  3. If f(x) has a removable discontinuity at its maximum point on [a, b], does the maximum actually exist as a function value? Why does this violate EVT?

Delta Moment

"I'm rolling along this curve, getting higher and higher... but wait, there's a HOLE right at the peak! I can get arbitrarily close to the top, but I'll never actually reach it. That's what happens when continuity fails!"

References

  1. Extreme Value Theorem - Wikipedia - Comprehensive mathematical treatment of EVT and its proof

  2. Khan Academy - Extreme Value Theorem - Video explanation with examples

  3. Paul's Online Math Notes - Finding Absolute Extrema - Detailed procedure for finding extrema on closed intervals