Exponential growth is a constant rate of growth applied to a continuously expanding base that results in a very rapid increase in number.

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1 Exploring Exponential Growth with NetLogo Software: Net-Logo Teachers: Stephen Chiong and Jackie Da Ros Audience: High School Math/Science Students Allocated time - 80 min 1. Preparation, Grouping, and Introduction to the problem - 15 minutes 2. Activity - 55 minutes 3. Conclusion - 10 minutes Setup 1. 2 members per group. 2. Each student is provided with a computer. Students may choose to work together on one computer. Topic: Exponential Growth Exponential growth is a constant rate of growth applied to a continuously expanding base that results in a very rapid increase in number. For example, the number of E. coli doubles every 15 minutes. Scenario #1 If you started with 1 bacteria on your plate, how many would there be after 1 hour? Create a two-column chart with time in one column and number of bacteria in the other. Use your calculator to determine how many bacteria would be present after 1 hour. Time in Minutes Number of Bacteria

2 Work with a partner to determine an equation to describe this relationship. (Remember this is an exponential function.) How did you arrive at your answer? Write your formula or equation. Identify what each variable represents in your formula. Points to ponder: 1. Why is this an example of exponential growth? 2. What is the constant rate of growth in this example? 3. What is the continually expanding base in this example? Scenario #2 If you started with 10 bacteria on your plate, how many would there be after 2 hours? Write your name and your estimate on a piece of paper and hand it to the front. Create a two-column chart with time in one column and number of bacteria in the other. Use your calculator to determine how many bacteria would be present after 2 hours. Time in minutes Number of bacteria (1 hr) (2 hr) 2

Adjust the equation to reflect this difference. c.

3 Points to ponder: 1. How different (similar) is your result to your estimate? Explain why. 2. With your partner, compare the numbers for the first hour to those in scenario #1. a. How are they different? b. Adjust the equation to reflect this difference. c. Rewrite the equation with the base as 1 + % increase (as a decimal) Getting Started with NetLogo 1. Open the Models Library from the File menu (Ctrl + M) 1. Choose "Wolf Sheep Predation" from the Biology folder and press "Open". 2. Default Setup 3

4 3. Click on the info tab. Read HOW TO USE IT 4

5 Part A 1. Click the code tab 1 2. Disable the death feature 1. Look for to go 2. Look for ask sheep 3. Place ; (semicolon) 3. Disable the random feature in sheep reproduction 1. Place ; (semicolon) 5

6 4. Click on Interface tab 1 5. Set the following values Grass off grass-regrowth-time 30 Initial-number-sheep slider 11 sheep-gain-from-food 4 sheep-reproduce 4% Initial-number-wolves slider 0 wolf-gain-from-food 20 wolf-reproduce 5 6. Click setup 7. Click Go 8. Click Go to stop simulation when ticks (on top of view window) is slightly > Observe the graph produced a. Describe the slope of the graph. b. Determine the equation for this graph. c. Test your 11 ticks, you should get approximately sheep 10. Change the sheep-reproduce to 20% a. Click setup b. Click Go c. Click Go to stop simulation when ticks (on top of view window) is slightly > 10 d. Observe the graph produced i. Compare this graph to the previous graph? ii. How many sheep at 11 ticks? iii. What does this mean? 6

7 Part B 1. Click on code tab 1 2. Enable the random function but keep the energy divide disabled 2 Semicolons removed 3. Click on Interface tab 1 7

8 4. Set the following values: Grass off grass-regrowth-time 30 Initial-number-sheep slider 11 sheep-gain-from-food 4 sheep-reproduce 4% Initial-number-wolves slider 0 wolf-gain-from-food 20 wolf-reproduce 5 5. Click setup 6. Click Go 7. Click Go to stop simulation when ticks (on top of view window) is slightly > Observe the graph produced a. Describe the slope of the graph. b. In natural situations of exponential growth or decay, we use the formula where e, known as the natural exponential, is the base. (e is a mathematical constant, similar to π used in geometry, and has an approximate value of 2.718). Working with your partner, determine the equation for this graph. c. Test your approx. 100 ticks, 70 ticks, and 50 ticks. Hint: Place the pointer on the curve (it will change to cross-hairs) to determine the coordinates of points on the curve. d. Use any point on the curve to solve for r. Compare it to the number you used in the calculations above. Hint: ln e x = x Click on the link for more information on e and natural logarithms. Click on the page to advance through the slide. e. Compare your results from c and d with those of other groups. 8

9 f. Considering that this curve represents a natural system, provide a possible explanation for any discrepancies between the numbers you calculated and the numbers presented in the program. Part C 1. Set the following values: Grass on grass-regrowth-time 30 Initial-number-sheep slider 11 sheep-gain-from-food 4 sheep-reproduce 4% Initial-number-wolves slider 0 wolf-gain-from-food 20 wolf-reproduce 5% 2. Click setup 3. Click Go 4. Click Go to stop simulation when ticks 250 Point to ponder: 1. How does this change the original graph? Explain why. Part D 1. Set the following values: Grass on grass-regrowth-time 30 Initial-number-sheep slider 100 sheep-gain-from-food 4 sheep-reproduce 4% Initial-number-wolves slider 40 wolf-gain-from-food 20 wolf-reproduce 5% 1. Click setup 2. Click Go 3. Click Go to stop simulation when ticks 500 Points to ponder: 1. How does this graph compare to the other two? Explain what is happening. 9

10 2. How does an exponential function in a natural system differ from the mathematical equation? What did you learn? 1. List the distinguishing characteristics of an exponential growth curve. 2. Give the formula for exponential growth, including a description of each variable. a. How does the equation differ from other functions you have seen with exponents? b. In a natural system, how and why does the curve obtained for exponential growth vary from the formula describing it? Application 1. Find another simulation that demonstrates exponential growth. a. Describe how to determine the equation for the exponential growth curve. b. Write the equation. c. Describe how the other factors in the system affect the graph. References Homework depot. (2008). Natural logarithms. Retrieved from Wilensky, U. (1997). NetLogo Wolf Sheep Predation model. Center for Connected Learning and Computer-Based Modeling, Northwestern University, Evanston, IL. Wilensky, U. (1999). NetLogo. Center for Connected Learning and Computer-Based Modeling, Northwestern University, Evanston, IL. 10

CHAPTER 5: Exponential and Logarithmic Functions

MAT 171 Precalculus Algebra Dr. Claude Moore Cape Fear Community College CHAPTER 5: Exponential and Logarithmic Functions 5.1 Inverse Functions 5.2 Exponential Functions and Graphs 5.3 Logarithmic Functions