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Curve Analysis Dashboard

Run Curve Analysis Dashboard Fullscreen

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Description

The Curve Analysis Dashboard is a comprehensive interactive tool that synthesizes all derivative-based analysis techniques to fully describe a function's behavior. This MicroSim brings together concepts from an entire calculus unit into one unified visualization.

What This MicroSim Shows

The dashboard displays four coordinated panels:

  1. Main Graph (Top): The original function f(x) with:
  2. Color-coded shading showing increasing (green) and decreasing (red) regions
  3. Curve style indicating concavity: solid line for concave up, dashed for concave down
  4. Critical points marked with circles (filled for local max/min, open for neither)

  5. Inflection points marked with purple diamonds

  6. First Derivative Panel (Left): Graph of f'(x) with:

  7. Zero crossings highlighted (corresponding to critical points)
  8. Sign chart showing + and - regions

  9. Synchronized vertical line showing current x position

  10. Second Derivative Panel (Right): Graph of f''(x) with:

  11. Zero crossings highlighted (corresponding to inflection points)
  12. Sign chart showing concavity regions

  13. Synchronized vertical line

  14. Summary Table (Bottom): Dynamic analysis showing:

  15. Current values of f(x), f'(x), and f''(x)
  16. Behavior at current point (increasing/decreasing, concave up/down)
  17. List of all critical and inflection points with classifications

Interactive Features

  • Function Selection: Choose from three preset functions of varying complexity
  • Toggle Checkboxes: Show/hide critical points, inflection points, concavity shading, and derivative graphs
  • Step-Through Mode: Reveals analysis features one at a time, guiding students through the complete analysis process
  • Interactive Exploration: Drag on the main graph or use the slider to explore how f, f', and f'' relate at any x-value
  • Hover Information: Move your mouse over the main graph to see detailed values at any point

Color Coding Legend

Visual Element Meaning
Green shading Function is increasing (f'(x) > 0)
Red shading Function is decreasing (f'(x) < 0)
Solid blue curve Concave up (f''(x) > 0)
Dashed orange curve Concave down (f''(x) < 0)
Green filled circle Local minimum
Red filled circle Local maximum
Open circle Critical point that's neither max nor min
Purple diamond Inflection point

Delta Moment

"This is where everything clicks! My tilt (the derivative) AND how my tilt is changing (the second derivative) together tell the complete story of every curve I explore. It's like having a full GPS readout for mathematical terrain!"

Lesson Plan

Learning Objectives

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

  1. Identify critical points by finding where f'(x) = 0 or is undefined
  2. Classify critical points as local maxima, minima, or neither using the second derivative test
  3. Locate inflection points where f''(x) = 0 and concavity changes
  4. Describe intervals of increase/decrease based on the sign of f'(x)
  5. Describe intervals of concavity based on the sign of f''(x)
  6. Synthesize all information into a complete curve sketch

Prerequisites

  • Understanding of derivatives and their interpretation as slope
  • Ability to find critical points (setting f'(x) = 0)
  • Understanding of concavity and inflection points
  • Second derivative test for classifying extrema

Suggested Activities

Activity 1: Step-Through Discovery (15 minutes)

  1. Select the function f(x) = x^4 - 4x^3 + 10
  2. Turn on Step Mode
  3. Progress through each step, discussing what each reveals
  4. At each step, have students predict what they'll see before clicking Next

Activity 2: Prediction Practice (20 minutes)

  1. Hide all overlays (uncheck all boxes)
  2. For a selected function, have students:
  3. Predict where critical points occur
  4. Predict where inflection points occur
  5. Sketch the sign charts for f'(x) and f''(x)
  6. Reveal each feature to check predictions

Activity 3: Comparative Analysis (15 minutes)

  1. Compare the three preset functions
  2. For each, answer:
  3. How many critical points? What type?
  4. How many inflection points?
  5. What's the relationship between the number of zeros of f'(x) and critical points?

Activity 4: The Big Picture Connection (10 minutes)

  1. Focus on one function
  2. Move the x slider slowly across the domain
  3. Observe how:
  4. f'(x) crossing zero corresponds to horizontal tangents on f(x)
  5. f''(x) crossing zero corresponds to curve shape changes
  6. The three graphs "tell the same story" in different ways

Assessment Questions

  1. At a local maximum, what must be true about f'(x) and f''(x)?
  2. If f''(x) > 0 on an interval, what does this tell us about f(x) on that interval?
  3. Can a function have an inflection point where f''(x) does not equal zero? Explain.
  4. How do you distinguish between a local maximum and a global maximum using this dashboard?

Extensions

  • Challenge: Sketch a function given only its sign charts for f'(x) and f''(x)
  • Research: Investigate the difference between inflection points where f''(x) = 0 versus where f''(x) is undefined
  • Connection: Relate concavity to the economic concept of "diminishing returns"

Technical Notes

  • Canvas dimensions: 800 x 680 pixels (responsive width, minimum 600px)
  • Control region: 100 pixels
  • Library: p5.js 1.11.10
  • All controls are canvas-based (no DOM elements) for iframe compatibility

References

  1. Stewart Calculus - Applications of Differentiation - Standard reference for curve sketching techniques
  2. Khan Academy - Analyzing Functions - Video lessons on using derivatives to analyze function behavior
  3. Paul's Online Math Notes - Shape of a Graph - Comprehensive guide to curve analysis