2. E XPONENTIAL G ROWTH 4 th Year Level Cristina P. David ERNESTO RONDON HIGH SCHOOL

3. Target Group: 4 th Year Level Duration : 50 minutes Prerequisite Skill: Exponential Function 2

4. O BJECTIVES : At the end of the lesson, the students shall be able to:  Define exponential growth  Differentiate between simple and compound interest  Compute for the final amount of compound interest and population growth. 3

5. H OW DO BACTERIA MULTIPLY ? www.wadsworth.org 4

6. D EMONSTRATING E XPONENTIAL G ROWTH fym.la.asu.edu www.ultimatedisney... www.sd4history.com 5

7. ANOTHER EXAMPLE OF EXPONENTIAL GROWTH 6

8. M ORE EXAMPLES OF EXPONENTIAL GROWTH multi-level networking social networking compound interest 7

9. I N Y OUR O WN W ORDS, D EFINE EXPONENTIAL GROWTH 1. 2. 3. 4. 5. 8

10. R ECALL E XPONENTIAL F UNCTION Definition: If b>0 and b ≠ 1, and k is a constant, then y = kb x, where x is a real number, is called an exponential function with base b. Definition: If b>0 and b ≠ 1, and k is a constant, then y = kb x, where x is a real number, is called an exponential function with base b. 9

11. S AMPLE P ROBLEM The number of bacteria in a culture is given by the function y= 50(2 t ) where t is the number of hours after the start of observation. Find the number of bacteria in the culture : a.) at the start of observation b.) 3 hours later image.spreadshirt... 10

12. S OLUTION : Given y = 50(2 t ) a. At the start, t = 0. Putting t = 0 into the function, y = 50(2 0 ) = 50(1) = 50 hence the number of bacteria at the start is 50. b. When t = 3 y = 50 (2 3 ) = 50 (8) = 400 hence the number of bacteria after 3 hours is 400. 11

13. A NOTHER F ORM OF E XPONENTIAL F UNCTION The Exponential Growth is in the form y = P( 1 + r) t where P = original number or amount r = rate of change t = unit of time y = total number or amount after t years ( Party or Pretty) The Exponential Growth is in the form y = P( 1 + r) t where P = original number or amount r = rate of change t = unit of time y = total number or amount after t years ( Party or Pretty) 12

14. E XAMPLE N O. 2 A certain city has a population of 200,000 and a growth rate of 2.5%.What will be the expected population after 3 years ? telegraph.co.uk 13

15. S OLUTION : Given: P = 200,000 r = 2.5 % or.025 t = 3 yrs. Find : y Using the formula y = P( 1 + r) t and substituting the given : y = 200,000 ( 1.025) 3 = 200,000 ( 1.076891) = 215,378 Therefore after 3 years, the population will be 215,378. 14

16. C OMPOUND I NTEREST Do watch the ffg. videos and take note how compound interest is computed compared to simple interest. Compound Interest VS S imple Interest www.nycon.org/joinus/index.asp www.mahalo.com/answers/money/h 15

17. T ASKS : Sammy deposited $2000 in a bank that pays 5% per annum. Complete the table below and make a graph showing the two kinds of interest. YearSimple InterestCompound Interest PrincipalInterestFinal Amount PrincipalInterestFinal Amount 1 2 3 4 5 16

18. G RAPH YEAR FINAL AMOUNTFINAL AMOUNT 17

19. YearSimple InterestCompound Interest PrincipalInterestFinal Amount PrincipalInterestFinal Amount 12000100210020001002100 22000200220021001052205 3200030023002205110.252315.25 4200040024002315.25115.762431.01 5200050025002431.01121.552552.56 10 15 20 25 18

20. G RAPH FINAL AMOUNTFINAL AMOUNT YEAR 19

21. Q UESTIONS 1. Describe the changes in the simple interest and the compound interest as the period of time increases. 2. After how many years will the balance be at least double the original principal if the interest is calculated using : a. simple interest b. compound interest 20

22. M ORE Q UESTIONS … 3. If the interest rate increases, what can you observe from the balance –year graph? 4. If the amount will be compounded semi- annually, what can you observe on the same graph? Can we still use the same formula in no.2? 21

23. E XPLAIN IN YOUR OWN WORDS WHAT THE N ARRATOR MEANS BY H IS M ORAL S TORY Earn compound interest. Don’t pay it. Earn c ompound interest. Don’t pay it. 22

24. F ORMULA FOR C OMPOUND I NTEREST Compound Interest Formula A = P(1 + r/n) nt where A = the final amount P = principal r = rate of interest n = no. of compounding in a year t = time or no.of years (Apparent or A parent) 23

25. N UMBER OF C OMPOUNDING IN A Y EAR A certain principal or deposit can be compounded annually, semi-annually, quarterly, monthly, bi-monthly or even daily. For a given kind of compounding in a year, we use the ff. for the value of n : n = 1 (annually or yearly) n = 2 ( semi-annually or bi-annually) n = 4 ( quarterly) n = 12 ( monthly) 24

26. E XAMPLE N O.3 Mr. Sy deposited an amount of Php 500,000 at Banco De Oro which gives 3% interest compounded annually. How much will be his money after 5 years? Compute for the final amount if the bank will compound his money: a. semi-annually b. quarterly c. monthly cdn4.wn.com 25

27. S OLUTION a. Given : P = 500,000 r = 3% or.03 n = 2 t = 5 Find : A To solve for A : Use the Apparent formula A = P(1+r/n) nt = 500,000(1+.03/2) 2(5) = 500,000(1.015) 10 = 500,000(1.160541) = 580,270.41 26

28. b. Using the same given facts in (a) except for n. Since compounding is done quarterly, n = 4. To solve for A: (Use the same apparent formula) A = 500,000(1+.03/4) 4(5) = 500,000(1.0075) 20 = 500,000(1.161184) = 5 80,592.07 c.Using n =12 since interest is compounded monthly A = 500,000(1+.03/12) 12(5) = 500,000(1.0025) 60 = 5 80,808.39 27

29. A NY Q UESTION ? Based on the given example, which scenario or method of compounding will yield a higher interest? 28

30. E VALUATION Three banks made the following offers to their customers. If you are to deposit your money, which of the following banks do you think will give you the best deal? P 3% interest compounded annually 5 years 29

31. 2% interest P n =4 3 years 2.5 % interest compo- unded mont hly 2 years cdn4.wn.com 30

32. A SSIGNMENT 1. Visit at least 3 banks nearest to your community. (in a mall for example) and ask an officer regarding the rate of interest and the method of compounding. 2. Using a specific number of years and amount to be invested or deposited, compare the final amount and decide which bank offers the great deal. 3. Ask your parent if they have a credit card or have experienced obtaining a loan from a lending institution and ask them the rate of interest and how compounding is done. 31 FIN