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Warm Up

Warm Up. Exponential Growth with paper. Take a sheet of paper of the ordinary variety and fold it into half. Fold it a second time, and a third time. It's about as thick as your finger nail. Continue folding if you can. At 7 folds it is as thick as a notebook.

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Warm Up

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  1. Warm Up

  2. Exponential Growth with paper • Take a sheet of paper of the ordinary variety and fold it into half. • Fold it a second time, and a third time. • It's about as thick as your finger nail. • Continue folding if you can. • At 7 folds it is as thick as a notebook. • If you would have been able to fold it 10 times, it would be as thick as the width of your hand. http://www.intsoft.com/powers2.html

  3. Exponential Growth with paper • Unfortunately, it isn't possible to do so more than about 7 times. • At seventeen folds it would be taller than your average house. • Three more folds and that sheet of paper is a quarter way up the Sears tower. • Ten more folds and it has crossed the outer limits of the atmosphere. http://www.intsoft.com/powers2.html

  4. Exponential Growth with paper • Another twenty and it has reached the sun from the earth. • At sixty folds it has the diameter of the solar system. • At 100 folds it has the radius of the universe. • "Preposterous!", you exclaim. That is what I thought till I started calculating the thickness myself. http://www.intsoft.com/powers2.html

  5. Natural Exponential Applications exponential functions

  6. The Natural Base e f(x) = ex is the natural exponential function. e is a number e  2.71828

  7. 1) 2) 3) 4) The Natural Base e Ex. Use a calculator to evaluate each expression = 0.1353 = 0.3679 = 2.7183 = 7.3891

  8. Compound interest

  9. Compound Interest DO NOT WRITE THIS DOWN. JUST LISTEN. • When you put money into a savings account. It will earn interest.

  10. Compound Interest DO NOT WRITE THIS DOWN. JUST LISTEN. • Say you put $100 in an account that earns 5%. • The first time they “compound” the interest, how much money do you have? • Now, say they compound it again…

  11. Compound Interest DO NOT WRITE THIS DOWN. JUST LISTEN. • Back in the early 1970s, the interest rates that banks could pay on regular savings accounts were strictly regulated. • Typically, a bank could pay no more than 5% on such accounts. After a time, banking deregulation led to this limit being revoked and interest rates rose higher, at least until the 1990s.

  12. Compound Interest DO NOT WRITE THIS DOWN. JUST LISTEN. • Since the amount of interest that they could pay depositors was regulated by the government, one way that banks competed in the 1970s was by offering favorable compounding intervals. • If one bank compounded monthly while another compounded daily, the second bank would enjoy a competitive advantage even though both were paying 5% annual interest.

  13. Compound Interest DO NOT WRITE THIS DOWN. JUST LISTEN. • Of course, there was no reason to stop at daily compounding. A bank could, if it chose, compound every hour or every minute or every second. • And in fact, many banks took it to the limit and began compounding continuously.

  14. Notes Compound Interest A: Final Amount of Investment P: Principle (Initial value of investment) r: Annual Interest Rate n: Number of times interest is compounded t: Time in years

  15. Notes Compound Interest Number of compoundings

  16. A total of $ 12, 000 is invested at an annual interest rate of 9%. Find the balance after 5 years if it is compounded Ex. 8 a) quarterly b) monthly = $18726.11 = $18788.17

  17. Compound Interest For continuous compounding: A = balance P = principle t = time in years r = annual interest rate

  18. A total of $ 12, 000 is invested at an annual interest rate of 9%. Find the balance after 5 years if it is compounded Ex. 9 continuously. = $18819.75

  19. Exponential Growth • A quantity grows exponentially when its increase is proportional to what is already there.

  20. Exponential Growth DO NOT WRITE THIS DOWN. JUST LISTEN. • Consider a country (United States of McEachern) with 100 people, growing at 7% per year. • In 10 years, the population will double to 200 people, in another 10 years it will double again to 400 people, and ten years after that it will double again to 800 people.

  21. Population Growth DO NOT WRITE THIS DOWN. JUST LISTEN. • The U.S. is projected to double its population this century, practically within the lifetimes of today's children.

  22. Exponential Growth DO NOT WRITE THIS DOWN. JUST LISTEN. • A different example… • A snowball rolling downhill grows exponentially with time since when it is twice as big it gathers snow twice as fast.

  23. Doubling rice grains DO NOT WRITE THIS DOWN. JUST LISTEN. • A story…

  24. Doubling rice grains • Once upon a time, long ago there was a king who ruled a prosperous land. Poverty was unknown there and every person was gainfully employed. Hence the sight of a beggar making his way along Main Street caused quite a stir in the capital of the land. The king demanded to see this strange man. When brought to him the beggar revealed that he indeed did not have any possessions nor any money for the purchase of food. The king magnanimously offered all-you-can-eat meals for the rest of the week and clean clothes so that the beggar could continue his journey to the next land. Surprisingly, the beggar declined the royal offer and asked for a modest favor. The king demanded to know what the wish was. The beggar humbly requested a grain of rice for the first day, two on the second, four on the third day and so on - doubling the previous days contribution. • The king looked through the window at the overflowing granaries and almost accepted it when his grand advisor, remembering something that he had learnt in Elements of Numbers (Math 201 at the local University) advised his highness that he should reconsider. To calculate the implication of the wish he pulled out a dusty abacus to perform exponential calculations. He fumbled with it for a while but could not express the magnitude of the numbers involved because he ran out of beads. The king getting impatient with his advisor on such a simple wish from a poor man, officially granted the beggar the wish. Little did he know that he had sounded the death knell of his reign. • The next day the beggar came to claim his grain of rice. The townsfolk laughed at the beggar and said that he should have taken the king's kind offer for a full meal instead of the measly grain of rice. On the second day he was back for the two grains. A week later, he brought a teaspoon for the 128 grains that was due to him. In two weeks it was a non-negligible amount of half a kilo. At the end of the month it had grown to a whopping 35 tons. A few days later the king had to declare bankruptcy. That is how long it was needed to bring down the kingdom.

  25. Notes Exponential Growth y: Final amount a: Initial amount b: growth factor x: Length of growth (days, minutes, etc.)

  26. Discussion Sally starts a chain email that requires each recipient to pass it on to five new people after 24 hours else they be cursed with a lifetime of bad-hair days. Sally sends the email to three of her friends. If everyone who gets the email obeys and sends it on to five people, how many people will receive the email on the 15th day after it started? Initial amount: 3 Growth Factor: 5 Length of Growth: 15 Final amount: y

  27. Notes If the growth is given as a rate in percent, it must be converted to the growth factor by first changing it to a decimal, then adding 1. Example: Growth rate: 32% Growth factor: 1.32 Growth Rate b = 1 + r

  28. Discussion Two grey squirrels stow away on a freight ship heading for Greenland. When they arrive, they take to their new environment and begin to thrive. Their population grows at a rate of 45% per month. How many squirrels will there be after one and a half years? Initial amount: 2 Growth rate: 45% Growth factor: 1.45 Length of Growth: 18 months Final amount: y

  29. Ex. You inherited some land in 1960 worth $30,000. The value of the property increases 5% per year. What it be in 2016? a= initial amount, r= interest rate x = time in years $344,021

  30. Exponential Decay • A quantity decays exponentially when its decrease is proportional to what is already there.

  31. The Exponential Decay of Coca-Cola DO NOT WRITE THIS DOWN. JUST LISTEN. • If you don't understand exponential decay -- just pour yourself a coke over ice and watch what happens. • The rate at which the foam on a coke shrinks models exponential decay.

  32. The Exponential Decay of Coca-Cola DO NOT WRITE THIS DOWN. JUST LISTEN. • To understand exponential decay, think of the coke foam as a collection of bubbles. • In one time period, half of the bubbles will pop, and the height of the foam will drop by half. • In a second time period, half of the remaining half will pop, leaving only a quarter of the original foam.

  33. The Exponential Decay of Coca-Cola DO NOT WRITE THIS DOWN. JUST LISTEN. • In a third time period, another eighth of the original bubbles (half of the remaining quarter) will pop, and the height will be an eighth of where it started. • As bubbles pop, there are fewer left to pop, which is why the decay slows.

  34. Notes Exponential Decay y: Final amount a: Initial amount b: decay factor x: Length of decay (days, minutes, etc.)

  35. Notes If the decay is given as a rate in percent, it must be converted to the decay factor by changing it to a decimal and subtracting it from 1. Example: Decay Rate: 12% Decay Factor: 0.88 Decay Rate b = 1 – r

  36. Discussion You purchase a brand new 2007 BMW for $45,000.00. Each year the value of a car depreciates about 15%. How much is your BMW worth in the year 2020? Initial amount: 45,000 Decay Rate: 15% Decay Factor: 0.85 Length of Decay: 13 Final amount: y

  37. Ex. • You bought a new car for $22,000. The car depreciates 12.5% per year. • What is the value of the car after 3 years? • When will the car have a value of $8000? a= initial amount, r= interest rate t = time in years $14738 7.65 years

  38. Homework • Worksheet

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