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Three Connected Graphs Explorer

Run Three Graphs Explorer Fullscreen

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You can include this MicroSim on your website using the following iframe:

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<iframe src="https://dmccreary.github.io/calculus/sims/three-graphs-explorer/main.html" width="100%" height="662px" scrolling="no"></iframe>

Description

This MicroSim displays three vertically stacked coordinate planes showing a function f(x), its first derivative f'(x), and its second derivative f''(x). A synchronized vertical cursor line appears on all three graphs, allowing students to examine how values at a single x-coordinate relate across all three functions.

Key Features

Visual Elements:

  • Three panels showing f(x), f'(x), and f''(x) with color-coded curves (blue, green, purple)
  • Synchronized vertical cursor line that spans all three graphs
  • Green shading indicates where f is increasing (f' > 0)
  • Red shading indicates where f is decreasing (f' < 0)
  • Blue dots mark local extrema (maxima and minima) on f(x), corresponding to zeros of f'(x)
  • Purple diamond markers show inflection points on f(x), corresponding to zeros of f''(x) where sign changes occur
  • Orange tangent line on f(x) showing the current slope

Interactive Controls:

  • Draggable x-position slider to move the cursor across all graphs
  • Click and drag directly on any graph panel to explore
  • Function selector with four pre-built examples:
  • Cubic: f(x) = x^3 - 3x
  • Quartic: f(x) = x^4 - 4x^2
  • Sine: f(x) = sin(x)
  • Rational: f(x) = x/(x^2+1)
  • Checkboxes to toggle visibility of each graph
  • "Animate Cursor" button to automatically sweep through x-values

Information Panel:

  • Displays current x-value and corresponding values of f(x), f'(x), and f''(x)
  • Provides contextual insights such as:
  • "f' = 0: Horizontal tangent!" when at a critical point
  • "f'' = 0: Inflection point!" when at an inflection point
  • "f' > 0: f increasing" or "f' < 0: f decreasing" for general points

How to Use

  1. Select a function using the buttons at the bottom to explore different curve shapes
  2. Drag the slider or click on any graph panel to move the vertical cursor
  3. Watch the information panel to see how f, f', and f'' values change together
  4. Look for patterns: When f'(x) crosses zero, f(x) has a horizontal tangent (potential max/min). When f''(x) crosses zero with a sign change, f(x) has an inflection point.
  5. Use the checkboxes to hide/show specific graphs and focus on particular relationships
  6. Click "Animate Cursor" to watch the cursor sweep through values automatically

Lesson Plan

Learning Objectives

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

  1. Examine how zeros of f'(x) correspond to horizontal tangents and potential extrema on f(x)
  2. Analyze the relationship between the sign of f'(x) and whether f(x) is increasing or decreasing
  3. Connect zeros of f''(x) (with sign changes) to inflection points on f(x)
  4. Interpret concavity by observing the sign of f''(x) and the behavior of f'(x)

Target Audience

  • AP Calculus AB/BC students
  • College Calculus I students
  • Prerequisites: Understanding of derivatives as rates of change and slopes of tangent lines

Suggested Activities

Activity 1: Discovering Critical Points (10 minutes)

  1. Select the Cubic function f(x) = x^3 - 3x
  2. Slowly drag the cursor from left to right
  3. Find the two points where f'(x) = 0 (the green curve crosses the x-axis)
  4. At these points, what do you notice about the tangent line on f(x)?
  5. Which is a local maximum? Which is a local minimum? How does f''(x) help you determine this?

Activity 2: Inflection Point Investigation (10 minutes)

  1. Keep the Cubic function selected
  2. Find the point where f''(x) = 0 (the purple curve crosses the x-axis)
  3. At this x-value, what happens to the concavity of f(x)?
  4. Observe the f'(x) graph at this point. What is special about f'(x) here?
  5. Repeat with the Quartic function - how many inflection points does it have?

Activity 3: Comparing Functions (15 minutes)

  1. Work through each of the four functions
  2. Create a table recording:
  3. Number of critical points
  4. Number of inflection points
  5. Intervals where f is increasing
  6. Intervals where f is concave up
  7. Discuss: What determines the number of critical points and inflection points a function can have?

Assessment Questions

  1. When f'(x) > 0, what can you conclude about f(x)?
  2. If f''(x) changes from positive to negative at x = a, what happens to f(x) at that point?
  3. Can f(x) have a local maximum where f''(x) > 0? Explain.
  4. The rational function f(x) = x/(x^2+1) has exactly one critical point. Why doesn't it have a local minimum or maximum in the traditional sense?

References

  1. Wikipedia: Derivative - Comprehensive overview of derivatives and their geometric interpretation as slopes of tangent lines.

  2. Wikipedia: Second Derivative - Explains the second derivative, concavity, and inflection points.

  3. Wikipedia: Critical Point (Mathematics) - Definition and significance of critical points in calculus.

  4. Wikipedia: Inflection Point - Detailed explanation of inflection points and their relationship to the second derivative.

  5. p5.js Reference - Documentation for the p5.js library used to create this interactive visualization.