Section 4.6 Modeling with Exponential and Logarithmic Functions

1. Solve Literal Equations for a Specified Variable 2. Create Models for Exponential Growth and Decay 3. Apply Logistic Growth Models 4. Create Exponential and Logarithmic Models Using Regression

Example 1: Solve for D

Example 2: Solve for x

1. Solve Literal Equations for a Specified Variable 2. Create Models for Exponential Growth and Decay 3. Apply Logistic Growth Models 4. Create Exponential and Logarithmic Models Using Regression

Create Models for Exponential Growth and Decay Let y be a variable changing exponentially with respect to t, and let y0 represent the initial value of y when t = 0. For a positive constant k For a negative constant k is a model for exponential growth. is a model for exponential decay.

Example 3: Suppose that at 19 years old you win $100,000 playing the lottery. If you would like to have $1,000,000 when you retire at age 67, determine the average rate of return needed under continuous compounding.

Example 4: According to a study published in 2012, Burmese pythons are becoming the new top predator in the Florida Everglades. Not many animals in the Everglades can stand against a python that can grow to 23 feet and weigh over 200 pounds. a) Twenty pythons were recorded as captured or killed in 1995. By 2009, that number had increased to 378. Write a function of the form to represent the number of pythons captured or killed t years after 1995.

Example 4 continued:

Example 4 continued: b) Use the model from part (a) to predict the number of pythons that will be captured or killed in 2017.

Example 4 continued: c) If the ratio of pythons living in the wild to pythons captured or killed is approximately 100:1, how many pythons were estimated to be in the Everglades in 2012?

Example 5: A sample collected from cave paintings on an archeological site in France shows that only 2% of the carbon-14 still remains. How old is the sample? Round to the nearest year. Use the model for radiocarbon dating: where Q0 is the original quantity of carbon-14.

1. Solve Literal Equations for a Specified Variable 2. Create Models for Exponential Growth and Decay 3. Apply Logistic Growth Models 4. Create Exponential and Logarithmic Models Using Regression

Apply Logistic Growth Models A logistic growth model is a function written in the form where a, b, and c are positive constants.

Example 6: The population of Los Angeles P(t) (in millions) can be approximated by the logistic growth function where t is the number of years since the year 1900.

Example 6 continued: Evaluate P(0) and interpret its meaning in the context of this problem.

Example 6 continued: Use this function to predict the population of Los Angeles on January 1, 2016.

1. Solve Literal Equations for a Specified Variable 2. Create Models for Exponential Growth and Decay 3. Apply Logistic Growth Models 4. Create Exponential and Logarithmic Models Using Regression

Example 7: Use a graphing utility to find an exponential model that best fits the data. x y 1 6.25 2 16.99 3 46.20 4 125.48 5 341.35 6 927.89

Example 8: Use a graphing utility to find a logarithmic model that best fits the data. x y 2 3.75 5 5.40 8 6.24 11 6.82 14 7.25