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Derivative from Graph

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About This MicroSim

This interactive exercise helps you develop your graphical intuition for derivatives. Instead of calculating derivatives algebraically, you will:

  1. See a curve with a marked point P
  2. Draw a tangent line by clicking and dragging through the point
  3. Check your answer to see the actual tangent line and compare slopes
  4. Get feedback on how close your estimate was

Delta Moment

"See that point P on the curve? That's me! I need to know my tilt right NOW at this exact spot. Help me figure out my slope by drawing a line that just kisses the curve at this point."

How to Use

  1. Observe the curve and the marked point P
  2. Click and drag through or near point P to draw your estimated tangent line
  3. Click "Check Answer" to reveal the actual tangent line (green dashed)
  4. Review your accuracy in the results panel showing percent error
  5. Click "New Function" to try a different curve
  6. Adjust difficulty to challenge yourself:
  7. Easy: Parabolas (quadratic functions)
  8. Medium: Cubic functions
  9. Hard: Trigonometric functions

Iframe Embedding

Copy this iframe to embed the MicroSim in your website:

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<iframe src="https://dmccreary.github.io/calculus/sims/derivative-from-graph/main.html" height="482px" width="100%" scrolling="no"></iframe>

Learning Objectives

After using this MicroSim, students will be able to:

  1. Estimate the slope of a tangent line visually at any point on a curve
  2. Recognize that the derivative at a point equals the slope of the tangent line
  3. Distinguish between tangent lines with positive, negative, and zero slopes
  4. Connect the graphical representation of a derivative to its numerical value
  5. Improve accuracy through practice and immediate feedback

Lesson Plan

Grade Level

High School (Grades 11-12) or Early College (Calculus I)

Duration

15-20 minutes for basic exploration; 30+ minutes for comprehensive practice

Prerequisites

  • Understanding of slope (rise over run)
  • Basic familiarity with the concept of a tangent line
  • Introduction to the derivative as instantaneous rate of change

Warm-Up Questions (5 minutes)

  1. What is the slope of a horizontal line?
  2. If you're skiing downhill, is your slope positive or negative?
  3. At the very top of a hill, what do you think the slope is?

Guided Exploration (10 minutes)

  1. Start with Easy mode
  2. Try 3-4 parabolas
  3. Notice: Where is the tangent line flat (slope = 0)?

  4. Notice: How does the slope change as you move along the curve?

  5. Key Discovery Questions

  6. At what point on y = x^2 is the slope equal to 0?
  7. Is the slope positive or negative when x > 0? When x < 0?

  8. How does the steepness change as you move away from the vertex?

  9. Progress to Medium and Hard

  10. Cubic functions have more interesting slope changes
  11. Trig functions show periodic slope patterns

Practice Activity (10-15 minutes)

Challenge: Try to get within 10% error on 5 consecutive problems at each difficulty level.

Discussion Prompts: - Which types of functions are easiest to estimate? Why? - At what kinds of points is estimation most challenging? - How does practicing with visual estimation help you understand derivatives?

Assessment Ideas

  1. Quick Check: Without using the MicroSim, sketch a tangent line on a given curve at a marked point
  2. Exit Ticket: Given a graph of f(x), estimate f'(2) by drawing and measuring
  3. Extension: If f(x) = x^3, predict where the tangent line will be steepest

Mathematical Background

The derivative of a function f at a point x = a is defined as:

\[f'(a) = \lim_{h \to 0} \frac{f(a+h) - f(a)}{h}\]

Geometrically, this limit represents the slope of the tangent line to the curve at the point (a, f(a)).

Functions Used in This MicroSim

Difficulty Function Derivative
Easy f(x) = x^2 f'(x) = 2x
Easy f(x) = -x^2 + 4 f'(x) = -2x
Medium f(x) = x^3/3 f'(x) = x^2
Medium f(x) = x^3 - x f'(x) = 3x^2 - 1
Hard f(x) = 2sin(x) f'(x) = 2cos(x)
Hard f(x) = cos(x) + x f'(x) = -sin(x) + 1

Tips for Accuracy

  1. Position carefully: Your line should pass through point P
  2. Use the grid: Count grid squares to estimate rise/run
  3. Think about curvature: The tangent barely touches the curve
  4. Check sign: Make sure your line slopes in the right direction
  5. Zero slopes: Horizontal tangents occur at peaks and valleys

References

  1. Tangent Line - Wikipedia - Comprehensive overview of tangent lines in mathematics
  2. Derivative - Khan Academy - Video introduction to derivatives
  3. Visual Calculus - David Tall - Research on visual approaches to teaching calculus
  4. AP Calculus AB Course Description - College Board - Official AP Calculus curriculum guidelines