8.1 Exponential Functions (inked)

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The document discusses exponential functions and provides examples of using them to model growth and decay scenarios. It asks how many mice there will be in a barn in 2 years if the number of mice quadruples every 4 months, beginning with 2 mice.

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Suppose two mice live in a barn. If the number of mice quadruples every 4 months, how many mice will there be in 2 years?

8.1 Exploring Exponential Functions

Exponential Functions ,[object Object],[object Object],[object Object],[object Object]

Example 1 . Graph y = 2 ∙ 3 x x y

[object Object],[object Object],[object Object],[object Object],[object Object]

Example 3. In 2000, the annual rate of increase in US population was 1.24%. ,[object Object],[object Object],[object Object],[object Object],http://www.census.gov/population/www/popclockus.html

[object Object],[object Object]

The exponential decay graph shows the expected depreciation for a car over four years. Estimate the value of the car after six years. Example 5 .

[object Object],[object Object]

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Understanding exponential growth is of critical importance in sustainability, resource conservation, and economics. This work contains a collection of practice problems and realistic case studies developed for the teaching of sustainability science and conservation, with an emphasis on learning and applying the concepts of exponential growth. The exercises are designed to foster quantitative competence (numeracy) as well as critical thinking and systems thinking. Students learn to work with tools such as spreadsheet software and online databases and practice the application of basic but powerful quantitative analyses techniques. The case studies are based on recent, high quality data and explore questions of high relevance for the study and application of sustainability science. This work is related to the Growth in a finite world presentation (http://www.slideshare.net/amenning/growth-in-a-finite-world-sustainability-and-the-exponential-function).

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1. If the profit from the sale of x units of a product is P = 105x − 300 − x2, what level(s) of production will yield a profit of $1050? (Enter your answers as a comma-separated list.) x = _________ units 2. The total costs for a company are given by C(x) = 5400 + 80x + x2 and the total revenues are given by R(x) = 230x. Find the break-even points. (Enter your answers as a comma-separated list.) x= __________ units 3. If total costs are C(x) = 900 + 800x and total revenues are R(x) = 900x − x2, find the break-even points. (Enter your answers as a comma-separated list.) x= _____________ 4. For the years since 2001, the percent p of high school seniors who have tried marijuana can be considered as a function of time t according to p = f(t) = 0.17t2 − 2.61t + 52.64 where t is the number of years past 2000.† In what year after 2000 is the percent predicted to reach 75%, if this function remains valid? _______________ 5. Using data from 2002 and with projections to 2024, total annual expenditures for national health care (in billions of dollars) can be described by E = 4.61x2 + 43.4x + 1620 where x is the number of years past 2000.† If the pattern indicated by the model remains valid, in what year does the model predict these expenditures will reach $15,315 billion? __________________ 6. The monthly profit from the sale of a product is given by P = 32x − 0.2x2 − 150 dollars. (a) What level of production maximizes profit? ___________ units (b) What is the maximum possible profit? $_____________ 7. Consider the following equation. y = 9 + 6x − x2 (a) Find the vertex of the graph of the equation. (x, y) = (__________) (b) Determine what value of x gives the optimal value of the function. x=_____________ (c) Determine the optimal (maximum or minimum) value of the function. y=______________ 8. Consider the following equation. f(x) = 6x − x2 (a) Find the vertex of the graph of the equation. (x, y) = (__________) (b) Determine what value of x gives the optimal value of the function. x=_____________ (c) Determine the optimal (maximum or minimum) value of the function. f(x)= _____________ 9. Find the maximum revenue for the revenue function R(x) = 358x − 0.7x2. (Round your answer to the nearest cent.) R = $______________ 10. The profit function for a certain commodity is P(x) = 150x − x2 − 1000. Find the level of production that yields maximum profit, and find the maximum profit. x= _________ units P=$ _________ 11. If, in a monopoly market, the demand for a product is p = 2000 − x and the revenue is R = px, where x is the number of units sold, what price will maximize revenue? $________________ 12. If the supply function for a commodity is p = q2 + 6q + 16 and the demand function is p = −3q2 + 4q + 436, find the equilibrium quantity and equilibrium price. equilibrium quantity_______________ equilibrium price $_______________ 13. If the supply and demand functions for a commodity are given by p ...

1. If the profit from the sale of x units of a product is P =

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1. If the profit from the sale of x units of a product is P =

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PAGE 1 Name:______________________________ HOW DO POPULATIONS GROW? Student Guide Thomas Austin was an Englishman who migrated to southern Australia to farm the land. His property, Barwon Park was located near Winchelsea, Victoria. In October of 1859, homesick for his homeland and the hunting he used to enjoy, Thomas enlisted his nephew, William Austin who still resided in England, to send two dozen wild English rabbits, which Thomas then released onto his land. Thomas dismissed the act as benign, not realizing the drastic consequences of his actions. Due to the well-known prolific nature of rabbits, and the suitability of the Australian climate, within 6 years, this population of 24 rabbits had increased to 22 million. By the 1930’s, Australia’s rabbit populations were estimated to exceed 750 million! How did the populations grow so large, so quickly? And what might the consequences be on the local ecosystem? Procedure 1. Select a partner to work with and obtain 10 pennies. The pennies represent 10 individual rabbits in a population. Place the pennies in a container and shake them up. Pour them out onto a table. Each penny that lands with a tail showing represents a rabbit that gets to produce an offspring that is added to the original population of 10. [So chances are that approximately five individuals got to reproduce and your new population contains about 15 individuals (i.e., about 15 pennies)]. Now remove 10% of your population representing individuals that have died. Round down if the number is not an integer. 2. Repeat this procedure several times until the rabbit population exceeds 100 individuals. After each episode of births and deaths (i.e., after each “generation”), record the population size (i.e., the total number of pennies) in the chart below. Also record the Idealized population size (given that exactly half of your individuals reproduced each generation with NO ONE dying—go ahead and report the idealized numbers in decimals, but only keep two decimals. I’ve gotten it started for you.) Flip/Generation Number Experimental Population Size Idealized Population Size (no death) 0 10 10 1 15 2 22.5 3 33.75 4 5 6 7 8 9 10 3. Using the graph on the next page, plot population size (on the y axis) versus flip/generation number (on the x axis). Use the data on your graph to determine the slope during each time interval (REMEMBER: slope is the change in y over the change in x, or rise over run; i.e., slope = (y/(x). Record the slopes below. Interval Slope Between generations 0 and 1 Between generations 1 and 2 Between generations 2 and 3 Between generations 3 and 4 Between generations 4 and 5 Between generations 5 and 6 Between generations 6 and 7 Between generations 7 and 8 .

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8.1 Exponential Functions (inked)

1. Suppose two mice live in a barn. If the number of mice quadruples every 4 months, how many mice will there be in 2 years?

2.

3. 8.1 Exploring Exponential Functions

4.

5. Example 1 . Graph y = 2 ∙ 3 x x y

6.

7.

8.

9. The exponential decay graph shows the expected depreciation for a car over four years. Estimate the value of the car after six years. Example 5 .

10.