Concept Category 2 Logarithmic and Exponential Functions

Transcription

1 Concept Category 2 Logarithmic and Exponential Functions LT 2D: I can determine and create the most appropriate mathematical model to describe relationships between two quantities and use the model to analyze the relationships between the two quantities. I can explain why the model is an accurate representation of the rate of change in the data over time.

2 Task In 1996 the population of Russia was 148 million and the population of Nigeria was 104 million. If the population of Russia and Nigeria grow continuously at annual rates of of 0.62% and 3.0%, respectively, when will Nigeria have a greater population than Russia?

3 A. Definition Exponential Growth Model Growth Rate 01 N = N r n Time Length of time the rate is applied Quantity after t time has passed Initial Quantity Population Principal Exponential Constant Number of times the rate is applied per period

4 B. Visual N = N + (1 + r) 1 r < 0 r > 0 What are key features of the growth and decay models?

5 A. Definition Continuous Exponential Growth Model Continuous Growth Rate N = N + e 51 Time Years Months Quantity after t time has passed Initial Quantity Population Principal Exponential Constant

6 B. Visual N = N + e 51 r < Make 0 a prediction: r What > 0 will the graph of continuous exponential growth or decay look like? What are the key features?

7 B. Visual N = N + e 51 r < 0 r > 0

8 B. Visual Domain Range x-intercept y-intercept Asymptote End Behavior Continuity

9 C. Process Some bullfrogs were introduced into a small pond. The graph shows the bullfrog population for the next few years. Assume the population grows at a continuous exponential rate. Create a continuous exponential function model to find the projected bullfrog population after 15 years.

10 C. Process Estimate how long it takes the population to reach 75,000.

11 Small Spiral 1. Summarize what you know about exponential growth 2. Write out a process that can be applied to creating a model for exponential growth (How and Why) Include questions to help you when problem solving

12 What is a lingering question or curiosity?

13 Task In 1996 the population of Russia was 148 million and the population of Nigeria was 104 million. If the population of Russia and Nigeria grow continuously at annual rates of of 0.62% and 3.0%, respectively, when will Nigeria have a greater population than Russia?

14 Concept Category 2 Logarithmic and Exponential Functions LT 2D: I can determine and create the most appropriate mathematical model to describe relationships between two quantities and use the model to analyze the relationships between the two quantities. I can explain why the model is an accurate representation of the rate of change in the data over time.

15 How tall will Chloe be as an adult? Date Height (inches) 7/16/ /13/ /17/ /18/ /18/ /16/ /17/ /22/ /30/

16 A. Logistic Growth Logistic Growth models real-life quantities whose growth levels off because the rate of growth begins to decrease. y = c 1 + ae ;rx Carrying Capacity

17 A. Logistic Growth Logistic Growth models real-life quantities whose growth levels off because the rate of growth begins to decrease. y = c 1 + ae ;rx Carrying Capacity On your desk sketch a graph whose growth rate changes from increasing to decreasing

18 B. Visual Why does the graph continue What questions to increase if the do you have? growth rate is decreasing?

19 Point of Maximum Growth In biology class, you observed the biomass of a yeast population over a period of time. The table gives you the yeast mass y (in grams) after t hours Where does the point of maximum growth occur?

20 Point of Maximum Growth In biology class, you observed the biomass of a yeast population over a period of time. The population of the yeast mass y (in grams) after t hours is modeled by the equation y = e ;.AB1 How can the point of maximum growth when given the algebraic representation?

21 Small Spiral 1.Summarize what you know about logistic growth functions 2. Write out the key features of the logistic growth function and the process for finding the key features

22 What is a lingering question or curiosity?