advertisement

Chabot Mathematics §9.1b The Base e Bruce Mayer, PE Licensed Electrical & Mechanical Engineer BMayer@ChabotCollege.edu Chabot College Mathematics 1 Bruce Mayer, PE BMayer@ChabotCollege.edu • MTH55_Lec-54_sec_8-5a_PolyNom_InEqual.ppt Review § 9.1 MTH 55 Any QUESTIONS About • §9.1 → Exponential Functions, base a Any QUESTIONS About HomeWork • §9.1 → HW-42 Chabot College Mathematics 2 Bruce Mayer, PE BMayer@ChabotCollege.edu • MTH55_Lec-54_sec_8-5a_PolyNom_InEqual.ppt 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 Chabot College Mathematics 3 Bruce Mayer, PE BMayer@ChabotCollege.edu • MTH55_Lec-54_sec_8-5a_PolyNom_InEqual.ppt 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. Chabot College Mathematics 4 Bruce Mayer, PE BMayer@ChabotCollege.edu • MTH55_Lec-54_sec_8-5a_PolyNom_InEqual.ppt 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. Chabot College Mathematics 5 Bruce Mayer, PE BMayer@ChabotCollege.edu • MTH55_Lec-54_sec_8-5a_PolyNom_InEqual.ppt 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 I Prt. Chabot College Mathematics 6 Bruce Mayer, PE BMayer@ChabotCollege.edu • MTH55_Lec-54_sec_8-5a_PolyNom_InEqual.ppt 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 Chabot College Mathematics 7 Bruce Mayer, PE BMayer@ChabotCollege.edu • MTH55_Lec-54_sec_8-5a_PolyNom_InEqual.ppt Example Calc Simple Interest SOLUTION a) Use Formula I Prt I $8000 0.06 5 I $2400 SOLUTION b) The total amount, A, A P I due her in five years A $8000 $2400 is the sum of the A $10, 400 original principal and the interest earned Chabot College Mathematics 8 Bruce Mayer, PE BMayer@ChabotCollege.edu • MTH55_Lec-54_sec_8-5a_PolyNom_InEqual.ppt Compound Interest Formula r A P 1 n nt 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 Chabot College Mathematics 9 Bruce Mayer, PE BMayer@ChabotCollege.edu • MTH55_Lec-54_sec_8-5a_PolyNom_InEqual.ppt 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. Chabot College Mathematics 10 Bruce Mayer, PE BMayer@ChabotCollege.edu • MTH55_Lec-54_sec_8-5a_PolyNom_InEqual.ppt 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. n r a) Annual Amount: A P 1 n A 100 1 0.05 $105.00 Chabot College Mathematics 11 Bruce Mayer, PE BMayer@ChabotCollege.edu • MTH55_Lec-54_sec_8-5a_PolyNom_InEqual.ppt Compare Compounding Periods b) Semi Annual Amount: r A P 1 n n 2 0.05 A 100 1 $105.06 2 c) Quarterly A P 1 r Amount: 4 4 4 0.05 A 100 1 $105.09 4 Chabot College Mathematics 12 Bruce Mayer, PE BMayer@ChabotCollege.edu • MTH55_Lec-54_sec_8-5a_PolyNom_InEqual.ppt Compare Compounding Periods d) Monthly Amount: r A P 1 12 12 0.05 A 100 1 12 e) Daily Amount: r A P 1 365 13 $105.12 365 0.05 A 100 1 365 Chabot College Mathematics 12 365 $105.13 Bruce Mayer, PE BMayer@ChabotCollege.edu • MTH55_Lec-54_sec_8-5a_PolyNom_InEqual.ppt 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 h number that 1 the expression 1 h approaches e as h gets larger & larger The value of e to 15 places: e = 2.718281828459045 Chabot College Mathematics 14 Bruce Mayer, PE BMayer@ChabotCollege.edu • MTH55_Lec-54_sec_8-5a_PolyNom_InEqual.ppt Continuous Compound Interest The formula for Interest Compounded Continuously; e.g., a trillion times a sec. rt A Pe A = $-Amount after t years P = Principal (original $-amount) r = annual interest rate (expressed as a decimal) t = number of years Chabot College Mathematics 15 Bruce Mayer, PE BMayer@ChabotCollege.edu • MTH55_Lec-54_sec_8-5a_PolyNom_InEqual.ppt 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 A Pert to 8.25 years. 0.075 8.25 A $8300e P = $8300 and r = 0.075 then A $15, 409.83 use Formula Chabot College Mathematics 16 Bruce Mayer, PE BMayer@ChabotCollege.edu • MTH55_Lec-54_sec_8-5a_PolyNom_InEqual.ppt 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 Chabot College Mathematics 17 Bruce Mayer, PE BMayer@ChabotCollege.edu • MTH55_Lec-54_sec_8-5a_PolyNom_InEqual.ppt Compare Continuous Compounding SIMPLE Interest A P Prt P 1 rt A $450, 000 1 0.06 213 A $6.201 million. YEARLY Compounding A P 1 r $450, 000 1 0.06 t 213 A $1.105 1011 A $110.5 million. Chabot College Mathematics 18 Bruce Mayer, PE BMayer@ChabotCollege.edu • MTH55_Lec-54_sec_8-5a_PolyNom_InEqual.ppt Simple Yearly 0.11 Quarterly 0.01 145.30 Continuous Interest Compounding Account $Value for $450k invested at 6% Interest for 213 Years 159.80 0 10 20 30 40 50 60 70 80 90 Account Value ($B) Chabot College Mathematics 19 100 110 120 130 140 150 160 170 M55_Sec9_1_Compare_Compounding_0810.xls Bruce Mayer, PE BMayer@ChabotCollege.edu • MTH55_Lec-54_sec_8-5a_PolyNom_InEqual.ppt The NATURAL Exponential Fcn The exponential function f x e x with base e is so prevalent in the sciences that it is often referred to as THE exponential function or the NATURAL exponential function. Chabot College Mathematics 20 Bruce Mayer, PE BMayer@ChabotCollege.edu • MTH55_Lec-54_sec_8-5a_PolyNom_InEqual.ppt Compare 2x, ex, 3x Several Exponentials Graphically Chabot College Mathematics 21 Bruce Mayer, PE BMayer@ChabotCollege.edu • MTH55_Lec-54_sec_8-5a_PolyNom_InEqual.ppt Example Xlate ex, Graphs Use translation to sketch the graph of g x e x 1 2 SOLUTION Move ex graph: • 1 Unit RIGHT • 2 Units UP Chabot College Mathematics 22 Bruce Mayer, PE BMayer@ChabotCollege.edu • MTH55_Lec-54_sec_8-5a_PolyNom_InEqual.ppt Example Graph Exponential Graph f(x) = 2 − e−3x SOLUTION Make T-Table, Connect-Dots x y = f(x) −2 −401.43 −1 −18.09 0 1 1 1.95 2 2 Chabot College Mathematics 23 Bruce Mayer, PE BMayer@ChabotCollege.edu • MTH55_Lec-54_sec_8-5a_PolyNom_InEqual.ppt Exponential Growth or Decay Math Model for “Natural” Growth/Decay: A t A0 e kt 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 Chabot College Mathematics 24 Bruce Mayer, PE BMayer@ChabotCollege.edu • MTH55_Lec-54_sec_8-5a_PolyNom_InEqual.ppt Exponential Growth An exponential GROWTH model is a function of the form At A0e kt k 0 A(t) 2A0 A0 A A0e kt where A0 is the Doubling population at time 0, A(t) time is the population at time t, and k is the exponential growth rate t • The doubling time is the amount of time needed for the population to double in size Chabot College Mathematics 25 Bruce Mayer, PE BMayer@ChabotCollege.edu • MTH55_Lec-54_sec_8-5a_PolyNom_InEqual.ppt Exponential Decay An exponential DECAY model is a function of the form At A0e kt k 0 A(t) A A0e kt A0 where A0 is the ½A0 population at time 0, A(t) Half-life is the population at time t, and k is the exponential decay rate t • The half-life is the amount of time needed for half of the quantity to decay Chabot College Mathematics 26 Bruce Mayer, PE BMayer@ChabotCollege.edu • MTH55_Lec-54_sec_8-5a_PolyNom_InEqual.ppt 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 Chabot College Mathematics 27 Bruce Mayer, PE BMayer@ChabotCollege.edu • MTH55_Lec-54_sec_8-5a_PolyNom_InEqual.ppt Example Exponential Growth SOLUTION a) Use year 2000 as t = 0 Thus for 2030 t = 30 A0 6 k 0.021 t 30 A t 6e 0.02130 A t 11.265663 The model predicts there will be 11.26 billion people in the world in the year 2030 Chabot College Mathematics 28 Bruce Mayer, PE BMayer@ChabotCollege.edu • MTH55_Lec-54_sec_8-5a_PolyNom_InEqual.ppt Example Exponential Growth A0 6 SOLUTION b) Use year 2000 k 0.021 as t = 0 Thus t 10 for 1990 t = −10 0.02110 A t 6e A t 4.8635055 The model postdicted that the world had 4.86 billion people in 1990 (actual was 5.28 billion). Chabot College Mathematics 29 Bruce Mayer, PE BMayer@ChabotCollege.edu • MTH55_Lec-54_sec_8-5a_PolyNom_InEqual.ppt WhiteBoard Work Problems From §9.1 Exercise Set • 40, 58, 63 Calculating e Chabot College Mathematics 30 Bruce Mayer, PE BMayer@ChabotCollege.edu • MTH55_Lec-54_sec_8-5a_PolyNom_InEqual.ppt All Done for Today World Population Chabot College Mathematics 31 Bruce Mayer, PE BMayer@ChabotCollege.edu • MTH55_Lec-54_sec_8-5a_PolyNom_InEqual.ppt Chabot Mathematics Appendix r s r s r s 2 2 Bruce Mayer, PE Licensed Electrical & Mechanical Engineer BMayer@ChabotCollege.edu – Chabot College Mathematics 32 Bruce Mayer, PE BMayer@ChabotCollege.edu • MTH55_Lec-54_sec_8-5a_PolyNom_InEqual.ppt