Precalculus Exponential, Logistic, and Logarithmic Functions Chapter 3 Section 1 Exponential and Logistic Functions

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1 Precalculus Exponential, Logistic, and Logarithmic Functions Chapter 3 Section 1 Exponential and Logistic Functions Essential Question: How and why are exponential and logistic functions used to model real-world growth in populations? Student Objectives: The student will learn how to construct a logistic function. The student will learn how to construct an exponential function. The student will identify the differences between exponential growth, continuous growth, and logistic growth. Terms: Base Continuous growth function Decay factor Exponential decay function Exponential function Exponential growth function Growth factor Limit to growth Logistic decay function Logistic growth function Maximum sustainable population - 1 -

2 Graphing Calculator Skills: The student is expected to graph an exponential growth function with an appropriate window. The student is expected to graph an exponential decay function with an appropriate window. The student is expected to graph a logistic growth function with an appropriate window. The student is expected to graph a logistic decay function with an appropriate window. Theorems and Definitions Exponential Functions Let a and b be real number constants. An exponential function in x is a function that can be written in the form f ( x) = ab x, where a is nonzero, b is positive and b 1. The constant a is the initial value of f x where and is the base. x = 0, b Exponential Growth and Decay For any exponential function f x and any real number, then = ab x x f ( x +1) = b f ( x). If a > 0 and b > 1, the function f x is increasing and is an exponential growth function. The base b is called the growth factor. If a > 0 and b < 1, the function f x is decreasing and is an exponential decay function. The base b is called the decay factor. Exponential Growth vs Continuous Growth An exponential growth function has the formula: growth function has the formula: f ( x) = ab x. = ae kx. f x A continuous The only difference between the two functions is the growth factor b e k

3 The Natural Base e The definition for the value of e is: e = lim 1+ 1 x x x. Continuous Growth and Decay The continuous growth function f ( x) = ae kx where k is an appropriate chosen real number constant. If a > 0 and k > 0, the function f x is increasing and is a continuous growth function. If a > 0 and k < 0, the function f x is decreasing and is a continuous decay function. Logistic Growth/Decay Functions Let A and M be positive constants and k be a negative constant. The logistic growth function can be written on the form: f ( x) = M 1+ Ae kt. The value of M is the limit to growth or the maximum sustainable population. If k < 0, the function f x is decreasing and is a logistic decay function. ** NOTE ** The vast majority of logistic exercises are logistic growth. Logistic decay is introduced here, but is rarely used throughout the course

4 Sample Questions: 1. What are the basic shapes of: (a) Exponential and/or continuous growth function; (b) Exponential and/or continuous decay function; (c) Logistic growth function; and (d) Logistic decay function. (a) (b) (c) (d) - 4 -

5 2. Hayden purchased 20 rabbits to raise on his farm. However, Hayden did not realize the that the rabbits basically double in population size every 4 months. How many rabbits did Hayden have after 3 years? 3. A small European city had a population of 58,000 people in The population had continuously decreased to 23,750 by Following this model, what is the expected population of the city by the year 2020? - 5 -

6 4. In 1990, several wolves were introduced into Yellowstone National in order to maintain the rodent population. Scientist predicted that the conditions of the Yellowstone National Park could support a maximum population of 450 wolves. In 2000, the wolf population had increased to 80. In 2012, the wolf population had increased to 185. (a) Determine the equation for the wolf population at any given time. (b) How many wolves were initially introduced into Yellowstone National Park? (c) What is the expected wolf population in the year 2020? (d) How long will it take for the wolf population to reach its maximum sustainable population? Homework: Pages Exercises: #13, 23, 35, 39, 41, 55, 57, 59, 61, and 63. Exercises: #14, 20, 38, 40, 44, 56, 58, 60, 62, and