1. BMayer@ChabotCollege.edu MTH55_Lec-54_sec_8-5a_PolyNom_InEqual.ppt 1 Bruce Mayer, PE Chabot College Mathematics Bruce Mayer, PE Licensed Electrical & Mechanical Engineer BMayer@ChabotCollege.edu Chabot Mathematics §9.1b The Base e

2. BMayer@ChabotCollege.edu MTH55_Lec-54_sec_8-5a_PolyNom_InEqual.ppt 2 Bruce Mayer, PE Chabot College Mathematics Review §  Any QUESTIONS About §9.1 → Exponential Functions, base a  Any QUESTIONS About HomeWork §9.1 → HW-43 9.1 MTH 55

3. BMayer@ChabotCollege.edu MTH55_Lec-54_sec_8-5a_PolyNom_InEqual.ppt 3 Bruce Mayer, PE Chabot College Mathematics Compound Interest  Terms  INTEREST ≡ A fee charged for borrowing a lender’s money is called the interest, denoted by I  PRINCIPAL ≡ The original amount of money borrowed is called the principal, or initial amount, denoted by P Then Total AMOUNT, A, that accululates in an interest bearing account if the sum of the Interest & Principal → A = P + I

4. BMayer@ChabotCollege.edu MTH55_Lec-54_sec_8-5a_PolyNom_InEqual.ppt 4 Bruce Mayer, PE Chabot College Mathematics Compound Interest  Terms  TIME: Suppose P dollars is borrowed. The borrower agrees to pay back the initial P dollars, plus the interest amount, within a specified period. This period is called the time (or time-period) of the loan and is denoted by t.  SIMPLE INTEREST ≡ The amount of interest computed only on the principal is called simple interest.

5. BMayer@ChabotCollege.edu MTH55_Lec-54_sec_8-5a_PolyNom_InEqual.ppt 5 Bruce Mayer, PE Chabot College Mathematics Compound Interest  Terms  INTEREST RATE: The interest rate is the percent charged for the use of the principal for the given period. The interest rate is expressed as a decimal and denoted by r.  Unless stated otherwise, it is assumed the time-base for the rate is one year; that is, r is an annual interest rate.

6. BMayer@ChabotCollege.edu MTH55_Lec-54_sec_8-5a_PolyNom_InEqual.ppt 6 Bruce Mayer, PE Chabot College Mathematics Simple Interest Formula  The simple interest amount, I, on a principal P at a rate r (expressed as a decimal) per year for t years is

7. BMayer@ChabotCollege.edu MTH55_Lec-54_sec_8-5a_PolyNom_InEqual.ppt 7 Bruce Mayer, PE Chabot College Mathematics Example  Calc Simple Interest  Rosarita deposited $8000 in a bank for 5 years at a simple interest rate of 6% a)How much interest will she receive? b)How much money will she receive at the end of five years?  SOLUTION a) Use the simple interest formula with: P = 8000, r = 0.06, and t = 5

8. BMayer@ChabotCollege.edu MTH55_Lec-54_sec_8-5a_PolyNom_InEqual.ppt 8 Bruce Mayer, PE Chabot College Mathematics Example  Calc Simple Interest  SOLUTION a) Use Formula  SOLUTION b) The total amount, A, due her in five years is the sum of the original principal and the interest earned

9. BMayer@ChabotCollege.edu MTH55_Lec-54_sec_8-5a_PolyNom_InEqual.ppt 9 Bruce Mayer, PE Chabot College Mathematics Compound Interest Formula  A = $-Amount after t years  P = Principal (original $-amount)  r = annual interest rate (expressed as a decimal)  n = number of times interest is compounded each year  t = number of years

10. BMayer@ChabotCollege.edu MTH55_Lec-54_sec_8-5a_PolyNom_InEqual.ppt 10 Bruce Mayer, PE Chabot College Mathematics Compare Compounding Periods  One hundred dollars is deposited in a bank that pays 5% annual interest. Find the future-value amount, A, after one year if the interest is compounded: a)Annually. b)SemiAnnually. c)Quarterly. d)Monthly. e)Daily.

11. BMayer@ChabotCollege.edu MTH55_Lec-54_sec_8-5a_PolyNom_InEqual.ppt 11 Bruce Mayer, PE Chabot College Mathematics Compare Compounding Periods  SOLUTION In each of the computations that follow, P = 100 and r = 0.05 and t = 1. Only n, the number of times interest is compounded each year, is changing. Since t = 1, nt = n∙1 = n. a)Annual Amount:

12. BMayer@ChabotCollege.edu MTH55_Lec-54_sec_8-5a_PolyNom_InEqual.ppt 12 Bruce Mayer, PE Chabot College Mathematics Compare Compounding Periods b)Semi Annual Amount: c)Quarterly Amount:

13. BMayer@ChabotCollege.edu MTH55_Lec-54_sec_8-5a_PolyNom_InEqual.ppt 13 Bruce Mayer, PE Chabot College Mathematics Compare Compounding Periods d)Monthly Amount: e)Daily Amount:

14. BMayer@ChabotCollege.edu MTH55_Lec-54_sec_8-5a_PolyNom_InEqual.ppt 14 Bruce Mayer, PE Chabot College Mathematics The Value of the Natural Base e  The number e, an irrational number, is sometimes called the Euler constant.  Mathematically speaking, e is the fixed number that the expression approaches as h gets larger and larger  The value of e to 15 places: e = 2.718281828459045

15. BMayer@ChabotCollege.edu MTH55_Lec-54_sec_8-5a_PolyNom_InEqual.ppt 15 Bruce Mayer, PE Chabot College Mathematics Continuous Compound Interest  The formula for Interest Compounded Continuously; e.g., a trillion times a sec.  A = $-Amount after t years  P = Principal (original $-amount)  r = annual interest rate (expressed as a decimal)  t = number of years

16. BMayer@ChabotCollege.edu MTH55_Lec-54_sec_8-5a_PolyNom_InEqual.ppt 16 Bruce Mayer, PE Chabot College Mathematics Example  Continuous Interest  Find the amount when a principal of $8300 is invested at a 7.5% annual rate of interest compounded continuously for eight years and three months.  SOLUTION: Convert 8-yrs & 3-months to 8.25 years. P = $8300 and r = 0.075 then use Formula

17. BMayer@ChabotCollege.edu MTH55_Lec-54_sec_8-5a_PolyNom_InEqual.ppt 17 Bruce Mayer, PE Chabot College Mathematics Compare Continuous Compounding  Italy's Banca Monte dei Paschi di Siena (MPS), the world's oldest bank founded in 1472 and is today one the top five banks in Italy  If in 1795 Thomas Jefferson Placed a Deposit of $450k the MPS bank at an interest rate of 6%, then find the value $-Amount for the this Account Today, 213 years Later

18. BMayer@ChabotCollege.edu MTH55_Lec-54_sec_8-5a_PolyNom_InEqual.ppt 18 Bruce Mayer, PE Chabot College Mathematics Compare Continuous Compounding  SIMPLE Interest  YEARLY Compounding

19. BMayer@ChabotCollege.edu MTH55_Lec-54_sec_8-5a_PolyNom_InEqual.ppt 19 Bruce Mayer, PE Chabot College Mathematics

20. BMayer@ChabotCollege.edu MTH55_Lec-54_sec_8-5a_PolyNom_InEqual.ppt 20 Bruce Mayer, PE Chabot College Mathematics The NATURAL Exponential Fcn  The exponential function  with base e is so prevalent in the sciences that it is often referred to as THE exponential function or the NATURAL exponential function.

21. BMayer@ChabotCollege.edu MTH55_Lec-54_sec_8-5a_PolyNom_InEqual.ppt 21 Bruce Mayer, PE Chabot College Mathematics Compare 2 x, e x, 3 x  Several Exponentials Graphically

22. BMayer@ChabotCollege.edu MTH55_Lec-54_sec_8-5a_PolyNom_InEqual.ppt 22 Bruce Mayer, PE Chabot College Mathematics Example  Xlate e x, Graphs  Use translation to sketch the graph of  SOLUTION  Move e x graph: 1 Unit RIGHT 2 Units UP

23. BMayer@ChabotCollege.edu MTH55_Lec-54_sec_8-5a_PolyNom_InEqual.ppt 23 Bruce Mayer, PE Chabot College Mathematics Example  Graph Exponential  Graph f(x) = 2 − e −3x  SOLUTION Make T-Table, Connect-Dots 22 1.951 1 −18.09 −401.43 y = f(x) 0 −1−1 −2−2 x

24. BMayer@ChabotCollege.edu MTH55_Lec-54_sec_8-5a_PolyNom_InEqual.ppt 24 Bruce Mayer, PE Chabot College Mathematics Exponential Growth or Decay  Math Model for “Natural” Growth/Decay:  A(t) = amount at time t  A 0 = A(0), the initial amount  k = relative rate of Growth (k > 0) Decay (k < 0)  t = time

25. BMayer@ChabotCollege.edu MTH55_Lec-54_sec_8-5a_PolyNom_InEqual.ppt 25 Bruce Mayer, PE Chabot College Mathematics Exponential Growth  An exponential GROWTH model is a function of the form  where A 0 is the population at time 0, A(t) is the population at time t, and k is the exponential growth rate The doubling time is the amount of time needed for the population to double in size A0A0 A(t)A(t) t 2A02A0 Doubling time

26. BMayer@ChabotCollege.edu MTH55_Lec-54_sec_8-5a_PolyNom_InEqual.ppt 26 Bruce Mayer, PE Chabot College Mathematics Exponential Decay  An exponential DECAY model is a function of the form  where A 0 is the population at time 0, A(t) is the population at time t, and k is the exponential decay rate The half-life is the amount of time needed for half of the quantity to decay A0A0 A(t) t ½A 0 Half-life

27. BMayer@ChabotCollege.edu MTH55_Lec-54_sec_8-5a_PolyNom_InEqual.ppt 27 Bruce Mayer, PE Chabot College Mathematics Example  Exponential Growth  In the year 2000, the human population of the world was approximately 6 billion and the annual rate of growth was about 2.1 percent.  Using the model on the previous slide, estimate the population of the world in the years a)2030 b)1990

28. BMayer@ChabotCollege.edu MTH55_Lec-54_sec_8-5a_PolyNom_InEqual.ppt 28 Bruce Mayer, PE Chabot College Mathematics Example  Exponential Growth  SOLUTION a) Use year 2000 as t = 0 Thus for 2030 t = 30  The model predicts there will be 11.26 billion people in the world in the year 2030

29. BMayer@ChabotCollege.edu MTH55_Lec-54_sec_8-5a_PolyNom_InEqual.ppt 29 Bruce Mayer, PE Chabot College Mathematics Example  Exponential Growth  SOLUTION b) Use year 2000 as t = 0 Thus for 1990 t = −10  The model postdicted that the world had 4.86 billion people in 1990 (actual was 5.28 billion).

30. BMayer@ChabotCollege.edu MTH55_Lec-54_sec_8-5a_PolyNom_InEqual.ppt 30 Bruce Mayer, PE Chabot College Mathematics WhiteBoard Work  Problems From §9.1 Exercise Set 40, 58, 63  Calculating e

31. BMayer@ChabotCollege.edu MTH55_Lec-54_sec_8-5a_PolyNom_InEqual.ppt 31 Bruce Mayer, PE Chabot College Mathematics All Done for Today World Population

32. BMayer@ChabotCollege.edu MTH55_Lec-54_sec_8-5a_PolyNom_InEqual.ppt 32 Bruce Mayer, PE Chabot College Mathematics Bruce Mayer, PE Licensed Electrical & Mechanical Engineer BMayer@ChabotCollege.edu Chabot Mathematics Appendix –