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Bruce Mayer, PE Licensed Electrical & Mechanical Engineer BMayer@ChabotCollege

Chabot Mathematics. §9.1b The Base e. Bruce Mayer, PE Licensed Electrical & Mechanical Engineer BMayer@ChabotCollege.edu. MTH 55. 9.1. Review §. Any QUESTIONS About §9.1 → Exponential Functions, base a Any QUESTIONS About HomeWork §9.1 → HW-42. Compound Interest  Terms.

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Bruce Mayer, PE Licensed Electrical & Mechanical Engineer BMayer@ChabotCollege

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  1. Chabot Mathematics §9.1bThe Base e Bruce Mayer, PE Licensed Electrical & Mechanical EngineerBMayer@ChabotCollege.edu

  2. MTH 55 9.1 Review § • Any QUESTIONS About • §9.1 → Exponential Functions, base a • Any QUESTIONS About HomeWork • §9.1 → HW-42

  3. 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. 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. 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 thus an annual interest rate.

  6. 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. Example  Calc Simple Interest • Rosarita deposited $8000 in a bank for 5 years at a simple interest rate of 6% • How much interest will she receive? • 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. 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. 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. 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: • Annually. • SemiAnnually. • Quarterly. • Monthly. • Daily.

  11. Compare Compounding Periods • SOLUTIONIn 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. • AnnualAmount:

  12. Compare Compounding Periods • Semi Annual Amount: • Quarterly Amount:

  13. Compare Compounding Periods • Monthly Amount: • Daily Amount:

  14. 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 e as h gets larger & larger • The value of e to 15 places: e = 2.718281828459045

  15. 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. 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 to8.25years. P =$8300and r =0.075 thenuse Formula

  17. 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 1797 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. Compare Continuous Compounding • SIMPLE Interest • YEARLY Compounding

  19. 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.

  20. Compare 2x, ex, 3x • SeveralExponentialsGraphically

  21. Example  Xlate ex, Graphs • Use translation to sketch the graph of • SOLUTION Move ex graph: • 1 Unit RIGHT • 2 Units UP

  22. x y = f(x) −2 −401.43 −1 −18.09 0 1 1 1.95 2 2 Example  Graph Exponential • Graph f(x) = 2 − e−3x • SOLUTIONMake T-Table,Connect-Dots

  23. Exponential Growth or Decay • Math Model for “Natural” Growth/Decay: • A(t) = amount at time t • A0 = A(0), the initial amount • k = relative rate of • Growth (k> 0); i.e., k is POSITIVE • Decay (k< 0); i.e., k is NEGATIVE • t = time

  24. Exponential Growth • An exponential GROWTH model is a function of the form A(t) 2A0 A0 • where A0 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 t Doubling time

  25. Exponential Decay • An exponential DECAY model is a function of the form A(t) A0 • where A0 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 ½A0 t Half-life

  26. 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 • 2030 • 1990

  27. Example  Exponential Growth • SOLUTION a) Use year 2000 as t = 0 Thusfor 2030 t = 30 • The model predicts there will be 11.26 billion people in the world in the year 2030

  28. Example  Exponential Growth • SOLUTION b) Use year 2000 as t = 0 Thusfor 1990 t = −10 • The model postdicted that the world had 4.86 billion people in 1990 (actual was 5.28 billion).

  29. WhiteBoard Work • Problems From §9.1 Exercise Set • 40, 58, 63 • Calculatinge

  30. All Done for Today WorldPopulation

  31. Chabot Mathematics Appendix Bruce Mayer, PE Licensed Electrical & Mechanical EngineerBMayer@ChabotCollege.edu –

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