1. 1 Exponential and Logarithmic Models 3/1/2011 DO NOW: I have a 12 inch balloon. It is doubling in size every minute. How big will it be in 7 minutes? How many minutes until it is a mile long? Psst. A mile is 5,280 feet SWBAT: Use exponential growth and decay functions to solve real-life problems Use logistic growth functions to solve real-life problems Use logarithmic functions to solve real-life problems

2. 2 Common Mathematical Models Many business applications and natural phenomena can be modeled by exponential and logarithmic functions. In this section we will explore logistic models and maybe even solve some real-life applications SWBAT: Use exponential growth and decay functions to solve real-life problems Use logistic growth functions to solve real-life problems Use logarithmic functions to solve real-life problems

3. 3 Applications of Logistic Growth Models  Logistic growth functions are used to model situations where initially there is an increasing rate of growth followed by a decreasing rate of growth.  Some common examples include the spread of a disease within a population and the growth of certain populations. In these cases, the functions have an upper bound which is equal to the maximum population capacity. This upper bound and the line  y = 0 are the horizontal asymptotes of the graphs of logistic functions. SWBAT: Use exponential growth and decay functions to solve real-life problems Use logistic growth functions to solve real-life problems Use logarithmic functions to solve real-life problems

4. 4 Graph of Logistic Model Domain: (- ,  ) Range: (0, c) Horizontal Asymptotes: y = 0, y = c SWBAT: Use exponential growth and decay functions to solve real-life problems Use logistic growth functions to solve real-life problems Use logarithmic functions to solve real-life problems Horizontal asymptote y=c provides a limit to growth. Original amount at x=0

5. 5 A Logistic Growth Model Example: Fruit flies are placed in a half-pint milk bottle with a banana (for food) and yeast plants (for food and to provide stimulus to lay eggs). Suppose that the fruit fly population P after t days is given by a) What is the carrying capacity of the half-pint bottle? (That is, what is the upper limit of the population?) b) How many fruit flies were initially placed in the half-pint bottle? c) When will the population be 180? SWBAT: Use exponential growth and decay functions to solve real-life problems Use logistic growth functions to solve real-life problems Use logarithmic functions to solve real-life problems

6. 6 A Logistic Growth Model (continued) Solution: a) We can use the TABLE feature on the graphing calculator table to find P(t) as t→ . P approaches 230 as t gets increasingly larger. Notice, the LIMITING CAPACITY IS THE NUMERATOR OF THIS FUNCTION. It is also the upper horizontal asymptote of the graph of this function. ( y = 0 is the lower asymptote ). So, the carrying capacity of the bottle is _______ fruit flies. y = 230 y = 0

7. 7 A Logistic Growth Model (continued) Solution: b) To find the initial number of fruit flies in the bottle, we need to find P(0). Algebraically:Graphically: Initially, there were ________ fruit flies in the half-pint bottle.

8. 8 A Logistic Growth Model (continued) Solution: c) To find when the population will reach 180, set P(t)=180. Algebraically:Graphically: It will take approx. _______ days for the pop. to reach 180 fruit flies.

9. 9 Applications of Logarithmic Models Many relations between variables are best modeled by a logarithmic function. Some common examples include: the relation between an earthquake’s magnitude and intensity on the Richter scale, the relation between atmospheric pressure and height, the relation between sound level (in decibels) and intensity, as well as many economic models.

10. 10 POINT OF INFLECTION: is a point on a curve at which the curvature changes sign. The curve changes from being concave upwards (positive curvature) to concave downwards (negative curvature), or vice versa. If one imagines driving a vehicle along a winding road, inflection is the point at which the steering-wheel is momentarily "straight" when being turned from left to right or vice versa. With Logistic models, it occurscurvecurvaturesignconcave where the graph ‘levels off’; it’s where the rate of increase is at it’s peak before it ‘slows down’ The End