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Critical Point Finder

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About This MicroSim

This interactive tool helps you find critical points of functions by identifying where the derivative equals zero or does not exist. The dual-graph display shows both f(x) and f'(x) simultaneously, making it easy to visualize the relationship between a function and its derivative.

Delta Moment

"Critical points are where I'm perfectly level - no tilt at all! Or sometimes they're at sharp corners where I can't even tell which way I'm facing. Either way, these are special places worth investigating!"

How to Use

  1. Select a function type from the dropdown menu:
  2. Polynomial: Standard polynomial functions with smooth derivatives
  3. Absolute Value: Functions with corners where f'(x) doesn't exist
  4. Rational: Functions with potential discontinuities

  5. Piecewise: Functions defined differently on different intervals

  6. Click "Find Critical Pts" to begin the step-by-step solution

  7. Click "Next Step" to reveal each step of the algebraic solution

  8. Toggle "Show f'(x)" to show or hide the derivative graph

  9. Adjust the Zoom slider to zoom in or out on the x-axis

  10. Use Prev/Next to cycle through different functions of the same type

Key Concepts

Critical points occur where: - f'(x) = 0: The function has a horizontal tangent line (potential local max/min) - f'(x) does not exist (DNE): The function has a corner, cusp, or vertical tangent

On the graphs: - Green vertical dashed lines mark critical points on both graphs - Green dots on f(x) show the actual critical points - Orange/red markers on f'(x) show where f' = 0 or DNE

Iframe Embedding

Copy this iframe to embed the MicroSim in your website:

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<iframe src="https://dmccreary.github.io/calculus/sims/critical-point-finder/main.html" height="622px" width="100%" scrolling="no"></iframe>

Learning Objectives

After using this MicroSim, students will be able to:

  1. Solve for critical points by setting f'(x) = 0 and solving algebraically
  2. Identify critical points where f'(x) does not exist (corners, cusps)
  3. Verify critical points graphically by observing horizontal tangents on f(x)
  4. Connect x-intercepts of f'(x) to critical points of f(x)
  5. Distinguish between different types of critical points (f'=0 vs. f' DNE)

Lesson Plan

Grade Level

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

Duration

20-30 minutes for guided exploration; 45 minutes for comprehensive practice

Prerequisites

  • Understanding of derivatives and derivative rules (power rule, quotient rule)
  • Familiarity with graphing functions and their derivatives
  • Basic equation-solving skills (factoring, quadratic formula)

Warm-Up Questions (5 minutes)

  1. What does it mean for a function to have a horizontal tangent line?
  2. At what points on a parabola y = x^2 is the tangent line horizontal?
  3. Can you think of a function that has a "corner" where you cannot draw a tangent?

Guided Exploration (15 minutes)

Part 1: Polynomial Functions

  1. Start with "Polynomial" type and the function f(x) = x^3 - 3x
  2. Before clicking "Find Critical Pts", predict:
  3. How many critical points might this cubic have?
  4. Where do you think they are approximately?
  5. Click through the solution steps and verify your predictions
  6. Notice how the critical points on f(x) correspond to x-intercepts on f'(x)

Part 2: Functions with Corners

  1. Switch to "Absolute Value" type
  2. Observe the corner at x = 0 for f(x) = |x| - 2
  3. Notice that f'(x) jumps from -1 to +1 at x = 0
  4. Key insight: f'(0) does not exist because left and right derivatives differ

Part 3: Connection Between Graphs

Toggle the derivative graph on and off to answer: - What happens on the f(x) graph at points where f'(x) crosses the x-axis? - What happens on the f(x) graph at points where f'(x) is discontinuous?

Practice Activity (10-15 minutes)

Challenge 1: For each function type, predict the critical points before revealing the solution.

Challenge 2: Given only the f'(x) graph, can you sketch what f(x) might look like?

Discussion Questions: - Why are critical points called "critical"? What makes them special? - If f'(c) = 0, does that mean f(c) is definitely a local maximum or minimum? - What's the difference between a critical point where f' = 0 vs. where f' DNE?

Assessment Ideas

  1. Quick Check: Given f(x) = x^4 - 8x^2, find all critical points without using the MicroSim
  2. Application: Sketch a function that has exactly three critical points: two where f' = 0 and one where f' DNE
  3. Extension: If a polynomial has degree n, what's the maximum number of critical points it can have?

Mathematical Background

Definition of Critical Point

A number c in the domain of f is a critical number if either:

\[f'(c) = 0 \quad \text{or} \quad f'(c) \text{ does not exist}\]

The point (c, f(c)) is called a critical point of f.

Finding Critical Points Algebraically

Method 1: Where f'(x) = 0 1. Find the derivative f'(x) 2. Set f'(x) = 0 3. Solve for x 4. Verify each solution is in the domain of f

Method 2: Where f'(x) DNE 1. Look for points where f is defined but f' is not 2. Common causes: - Corners (absolute value functions) - Vertical tangents (cube root functions) - Discontinuities in f' (piecewise functions)

Why Critical Points Matter

By the Extreme Value Theorem and Fermat's Theorem, if f has a local maximum or minimum at c and f'(c) exists, then f'(c) = 0. This means:

  • Local extrema can only occur at critical points (or endpoints)
  • Critical points are the "candidates" for optimization problems
  • Understanding critical points is essential for curve sketching

Functions in This MicroSim

Type Example Critical Points
Polynomial f(x) = x^3 - 3x x = -1, x = 1 (f' = 0)
Polynomial f(x) = x^4 - 4x^2 x = 0, plus or minus sqrt(2) (f' = 0)
Absolute Value f(x) = abs(x) - 2 x = 0 (f' DNE)
Rational f(x) = x + 1/x x = -1, x = 1 (f' = 0)
Piecewise f(x) = x^2 if x < 1, 2x-1 if x >= 1 x = 0 (f' = 0)

Tips for Success

  1. Always find f' first before looking for critical points
  2. Factor completely when setting f'(x) = 0
  3. Check the domain - critical points must be in the domain of f
  4. Use the graph as verification - critical points should match where tangent is horizontal
  5. Don't forget DNE points - check for corners, cusps, and vertical tangents

References

  1. Critical Point - Wikipedia - Mathematical definition and theory
  2. Critical Numbers - Khan Academy - Video explanation with examples
  3. Fermat's Theorem - MathWorld - The theorem connecting extrema to critical points
  4. AP Calculus AB: Applications of Derivatives - Official College Board curriculum guidelines