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Growth Rate Comparison

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About This MicroSim

This MicroSim visualizes one of calculus's most important insights: not all infinities grow at the same rate. When comparing functions as x approaches infinity, some functions completely dominate others.

The dominance hierarchy is:

  1. Logarithmic functions (slowest growth) - ln(x), log10(x)
  2. Polynomial functions (medium growth) - sqrt(x), x, x^2, x^3
  3. Exponential functions (fastest growth) - 2^x, e^x, 10^x

This hierarchy has profound implications for evaluating limits of the form infinity/infinity or determining which term dominates in complex expressions.

Delta Moment

"At x = 10, these functions look like they're in a close race. But watch what happens as we zoom out to x = 1000... the exponential just LEAVES everyone in the dust! It's not even close!"

How to Use

  1. Select Functions: Use the checkboxes to enable/disable different functions for comparison
  2. Adjust X-Range: Drag the slider to change the viewing window (10 to 1000)
  3. Race to Infinity: Click "Race!" to watch the functions grow in real-time
  4. Toggle Y-Scale: Switch between Linear and Logarithmic y-axis to see different scales
  5. Ratio Mode: Compare two functions directly by viewing their ratio

Key Observations

Stage 1 (x = 1 to 10): Functions appear similar in magnitude - At x = 10: ln(10) = 2.3, x^2 = 100, e^x = 22,026

Stage 2 (x = 10 to 100): Polynomial dominates logarithmic - Polynomial powers start separating from each other

Stage 3 (x = 100 to 1000): Exponential completely dominates - e^1000 is astronomically larger than any polynomial

Embedding

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

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

Lesson Plan

Learning Objective

Students will compare growth rates of logarithmic, polynomial, and exponential functions to predict limit behavior.

Bloom's Taxonomy Level

Evaluate (L5) - Students judge the relative growth rates and predict which function dominates in limit calculations.

Prerequisites

  • Understanding of basic function families (log, polynomial, exponential)
  • Familiarity with limits and infinity
  • Knowledge of indeterminate forms (infinity/infinity)

Warm-Up Activity (5 minutes)

Ask students to predict: - Which is bigger at x = 10: x^2 or 2^x? - Which is bigger at x = 100: x^2 or 2^x? - At what point does 2^x "pass" x^2?

Guided Exploration (15 minutes)

  1. Stage 1: Small Scale
  2. Set x-range to 10
  3. Enable ln(x), x^2, and e^x
  4. Observe that they appear somewhat comparable

  5. Click "Race!" and note the starting values

  6. Stage 2: Medium Scale

  7. Increase x-range to 100
  8. Watch the polynomial pull ahead of logarithmic

  9. Note that exponential is starting to accelerate

  10. Stage 3: Large Scale

  11. Increase x-range to 1000
  12. Switch to logarithmic y-scale (necessary to even see all functions!)

  13. Observe the exponential completely dominating

  14. Ratio Analysis

  15. Enable Ratio Mode
  16. Compare ln(x)/x - ratio approaches 0 (x dominates)
  17. Compare x/e^x - ratio approaches 0 (e^x dominates)
  18. Compare x^2/x^3 - ratio approaches 0 (higher power dominates)

Key Questions for Discussion

  1. If you're evaluating lim(x->infinity) of ln(x)/x^2, what's the answer? Why?
  2. In L'Hopital's rule for infinity/infinity forms, why do exponentials always "win"?
  3. How does this help us evaluate limits like lim(x->infinity) (x^100)/(2^x)?

Independent Practice

Have students predict the limit of these ratios, then verify with the MicroSim:

  1. lim(x->infinity) sqrt(x)/x
  2. lim(x->infinity) x^2/e^x
  3. lim(x->infinity) ln(x)/sqrt(x)
  4. lim(x->infinity) 10^x/e^x

Assessment

Students demonstrate understanding by: - Correctly predicting which function dominates in various comparisons - Explaining why "infinity/infinity" is indeterminate (depends on which infinity!) - Using growth rate knowledge to evaluate limits without L'Hopital's rule - Creating their own examples of each dominance relationship

Extension: The Dominance Ladder

Challenge students to arrange these functions from slowest to fastest growth: - ln(ln(x)) - ln(x) - sqrt(x) - x - x*ln(x) - x^2 - x^3 - e^x - x! - x^x

Mathematical Background

The formal statement of dominance is:

\[\lim_{x \to \infty} \frac{\ln(x)}{x^a} = 0 \quad \text{for any } a > 0\]
\[\lim_{x \to \infty} \frac{x^n}{e^x} = 0 \quad \text{for any } n\]

These facts mean: - Any positive power of x eventually beats any logarithm - Any exponential eventually beats any polynomial

References