1. BMayer@ChabotCollege.edu MTH15_Lec-18_sec_4-1_Exponential_Fcns.pptx 1 Bruce Mayer, PE Chabot College Mathematics Bruce Mayer, PE Licensed Electrical & Mechanical Engineer BMayer@ChabotCollege.edu Chabot Mathematics §4.1 ax Functions

2. BMayer@ChabotCollege.edu MTH15_Lec-18_sec_4-1_Exponential_Fcns.pptx 2 Bruce Mayer, PE Chabot College Mathematics Review §  Any QUESTIONS About §3.5 → Applied Optimization  Any QUESTIONS About HomeWork §3.5 → HW-17 3.5

3. BMayer@ChabotCollege.edu MTH15_Lec-18_sec_4-1_Exponential_Fcns.pptx 3 Bruce Mayer, PE Chabot College Mathematics §4.1 Learning Goals  Define exponential functions  Explore properties of the natural exponential function  Examine investments involving continuous compounding of interest

4. BMayer@ChabotCollege.edu MTH15_Lec-18_sec_4-1_Exponential_Fcns.pptx 4 Bruce Mayer, PE Chabot College Mathematics Exponential Function  A function, f(x), of the form  is called an EXPONENTIAL function with BASE a.  The domain of the exponential function is (−∞, ∞); i.e., ALL Real Numbers

5. BMayer@ChabotCollege.edu MTH15_Lec-18_sec_4-1_Exponential_Fcns.pptx 5 Bruce Mayer, PE Chabot College Mathematics Recall Rules of Exponents  Let a, b, x, and y be real numbers with a > 0 and b > 0. Then Product Rule Quotient Rule Product to a Power Rule Power to a Power Rule Zero Power Rule Negative Power Rule Equal Powers Rule

6. BMayer@ChabotCollege.edu MTH15_Lec-18_sec_4-1_Exponential_Fcns.pptx 6 Bruce Mayer, PE Chabot College Mathematics Evaluate Exponential Functions  Example   Solution   Example   Solution 

7. BMayer@ChabotCollege.edu MTH15_Lec-18_sec_4-1_Exponential_Fcns.pptx 7 Bruce Mayer, PE Chabot College Mathematics Evaluate Exponential Functions  Example   Solution 

8. BMayer@ChabotCollege.edu MTH15_Lec-18_sec_4-1_Exponential_Fcns.pptx 8 Bruce Mayer, PE Chabot College Mathematics Solve Exponential Equation  Solve the following for x  Using the Transitive Property  Need to state 2187 in terms of a Base-3 to a power  Using the Equal Powers Rule

9. BMayer@ChabotCollege.edu MTH15_Lec-18_sec_4-1_Exponential_Fcns.pptx 9 Bruce Mayer, PE Chabot College Mathematics Example  Graph y = f(x) =3 x  Graph the exponential fcn:  Make T-Table, & Connect Dots xy 0 1 –1 2 –2 3 1 3 1/3 9 1/9 27

10. BMayer@ChabotCollege.edu MTH15_Lec-18_sec_4-1_Exponential_Fcns.pptx 10 Bruce Mayer, PE Chabot College Mathematics Example  Graph Exponential  Graph the exponential fcn:  Make T-Table, & Connect Dots xy 0 1 –1 2 –2 –3 1 1/3 3 1/9 9 27 This fcn is a REFLECTION of y = 3 x

11. BMayer@ChabotCollege.edu MTH15_Lec-18_sec_4-1_Exponential_Fcns.pptx 11 Bruce Mayer, PE Chabot College Mathematics Example  Graph Exponential  Graph the exponential fcn:  Construct SideWays T-Table x −3−3 −2−2 −1−1 0123 y = (1/2) x 84211/21/41/8  Plot Points and Connect Dots with Smooth Curve

12. BMayer@ChabotCollege.edu MTH15_Lec-18_sec_4-1_Exponential_Fcns.pptx 12 Bruce Mayer, PE Chabot College Mathematics Example  Graph Exponential  As x increases in the positive direction, y decreases towards 0

13. BMayer@ChabotCollege.edu MTH15_Lec-18_sec_4-1_Exponential_Fcns.pptx 13 Bruce Mayer, PE Chabot College Mathematics Exponential Fcn Properties  Let f(x) = a x, a > 0, a ≠ 1. Then A.The domain of f(x) = a x is (−∞, ∞). B.The range of f(x) = a x is (0, ∞); thus, the entire graph lies above the x-axis. C.For a > 1 (e.g., a = 7) i.f is an INcreasing function; thus, the graph is RISING as we move from left to right ii.As x→∞, y = a x increases indefinitely and VERY rapidly

14. BMayer@ChabotCollege.edu MTH15_Lec-18_sec_4-1_Exponential_Fcns.pptx 14 Bruce Mayer, PE Chabot College Mathematics Exponential Fcn Properties  Let f(x) = a x, a > 1, a ≠ 1. Then iii.As x→−∞, the values of y = a x get closer and closer to 0. D.For 0 < a < 1 (e.g., a = 1/5 = 0.2) i.f is a DEcreasing function; thus, the graph is falling as we scan from left to right. ii.As x→−∞, y = a x increases indefinitely and VERY rapidly iii.As x→ ∞, the values of y = a x get closer and closer to 0

15. BMayer@ChabotCollege.edu MTH15_Lec-18_sec_4-1_Exponential_Fcns.pptx 15 Bruce Mayer, PE Chabot College Mathematics Exponential Fcn Properties  Let f(x) = a x, a > 0, a ≠ 1. Then E.Each exponential function f is one-to-one; i.e., each value of x has exactly ONE target. Thus: i. –The Basis of the Equal Powers Rule ii.f has an inverse

16. BMayer@ChabotCollege.edu MTH15_Lec-18_sec_4-1_Exponential_Fcns.pptx 16 Bruce Mayer, PE Chabot College Mathematics Exponential Fcn Properties  Let f(x) = a x, a > 0, a ≠ 1. Then F.The graph f(x) = a x has no x-intercepts In other words, the graph of f(x) = a x never crosses the x-axis. Put another way, there is no value of x that will cause f(x) = a x to equal 0 G.The x-axis is a horizontal asymptote for every exponential function of the form f(x) = a x.

17. BMayer@ChabotCollege.edu MTH15_Lec-18_sec_4-1_Exponential_Fcns.pptx 17 Bruce Mayer, PE Chabot College Mathematics ExponentialFcn ≠ PowerFcn  The POWER Function is the Variable (x) Raised to a Constant Power; e.g.: Note that PolyNomials are simply SUMS of Power Functions:  The EXPONENTIAL Function is a Constant Raised to a Variable Power (x); e.g.:

18. BMayer@ChabotCollege.edu MTH15_Lec-18_sec_4-1_Exponential_Fcns.pptx 18 Bruce Mayer, PE Chabot College Mathematics ExponentialFcn ≠ PowerFcn The Exponential is NEVER Negative

19. BMayer@ChabotCollege.edu MTH15_Lec-18_sec_4-1_Exponential_Fcns.pptx 19 Bruce Mayer, PE Chabot College Mathematics Example  Bacterial Growth  A technician to the Great French MicroBiologist Louis Pasteur noticed that a certain culture of bacteria in milk doubled every hour.  Assume that the bacteria count B(t) is modeled by the equation Where t is time in hours

20. BMayer@ChabotCollege.edu MTH15_Lec-18_sec_4-1_Exponential_Fcns.pptx 20 Bruce Mayer, PE Chabot College Mathematics Example  Bacterial Growth  Given Bacterial Growth Equation  Find: a)the initial number of bacteria, b)the number of bacteria after 10 hours; and c)the time when the number of bacteria will be 32,000.

21. BMayer@ChabotCollege.edu MTH15_Lec-18_sec_4-1_Exponential_Fcns.pptx 21 Bruce Mayer, PE Chabot College Mathematics Example  Bacterial Growth a)INITIALLY time, t, is ZERO → Sub t = 0 into Growth Eqn: b)At Ten Hours Sub t = 10 into Eqn:

22. BMayer@ChabotCollege.edu MTH15_Lec-18_sec_4-1_Exponential_Fcns.pptx 22 Bruce Mayer, PE Chabot College Mathematics Example  Bacterial Growth c)Find t when B(t) = 32,000  Thus 4 hours after the starting time, the number of bacteria will be 32k

23. BMayer@ChabotCollege.edu MTH15_Lec-18_sec_4-1_Exponential_Fcns.pptx 23 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 e as n gets larger & larger  The value of e to 15 places: e = 2.718 281 828 459 045

24. BMayer@ChabotCollege.edu MTH15_Lec-18_sec_4-1_Exponential_Fcns.pptx 24 Bruce Mayer, PE Chabot College Mathematics The “Natural” base e  The most “common” base for people is 10; e.g., 7.3x10 5  However, analysis of physical; i.e., Natural, phenomena leads to base e  Check the Definition Graphically  0.495% less than the actual e-Value

25. BMayer@ChabotCollege.edu MTH15_Lec-18_sec_4-1_Exponential_Fcns.pptx 25 Bruce Mayer, PE Chabot College Mathematics MATLAB Code MATLAB Code % Bruce Mayer, PE % MTH-15 16Jul13 % XYfcnGraph6x6BlueGreenBkGndTemplate1306.m % % The Limits xmin = 0; xmax = 100; ymin = 1; ymax = 2.75; % The FUNCTION x = linspace(xmin,xmax,1000); y = (1 +1./x).^x; % % The ZERO Lines zxh = [xmin xmax]; zyh = [0 0]; zxv = [0 0]; zyv = [ymin ymax]; % % the 6x6 Plot axes; set(gca,'FontSize',12); whitebg([0.8 1 1]); % Chg Plot BackGround to Blue-Green subplot(1, 2, 1) plot(x,y, 'LineWidth', 4),axis([xmin xmax ymin ymax]),... grid, xlabel('\fontsize{14}n'), ylabel('\fontsize{14}y = f(n) = (1 + 1/n)^n'),... title(['\fontsize{16}MTH15 e Value',]),... annotation('textbox',[.75.05.0.1], 'FitBoxToText', 'on', 'EdgeColor', 'none', 'String', 'BMayer 16Jul13','FontSize',7) hold on plot([xmin, xmax], [2.7182818, 2.7182818], '--m', 'LineWidth', 2) set(gca,'XTick',[xmin:10:xmax]); set(gca,'YTick',[ymin:.25:ymax]) hold off % % xmin1 = 0; xmax1 = 10; ymin1 = 1; ymax1 = 2.75; % The FUNCTION n = linspace(xmin,xmax,1000); z = (1 +1./n).^n; % % The ZERO Lines % % the 6x6 Plot axes; set(gca,'FontSize',12); whitebg([0.8 1 1]); % Chg Plot BackGround to Blue-Green subplot(1, 2, 2) plot(n,z, 'LineWidth', 4),axis([xmin1 xmax1 ymin1 ymax1]),... grid on, xlabel('\fontsize{14}n'),... title(['\fontsize{16}MTH15 e Value',]),... annotation('textbox',[.75.05.0.1], 'FitBoxToText', 'on', 'EdgeColor', 'none', 'String', 'BMayer 16Jul13','FontSize',7) hold on plot([xmin1, xmax1], [2.7182818, 2.7182818], '--m', 'LineWidth', 2) set(gca,'XTick',[xmin1:1:xmax1]); set(gca,'YTick',[ymin1:.25:ymax1])

26. BMayer@ChabotCollege.edu MTH15_Lec-18_sec_4-1_Exponential_Fcns.pptx 26 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.

27. BMayer@ChabotCollege.edu MTH15_Lec-18_sec_4-1_Exponential_Fcns.pptx 27 Bruce Mayer, PE Chabot College Mathematics Compare 2x, ex, 3x  Several Exponential Functions Graphically Note that EVERY Exponetial intercepts the y-Axis at x = 1

28. BMayer@ChabotCollege.edu MTH15_Lec-18_sec_4-1_Exponential_Fcns.pptx 28 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

29. BMayer@ChabotCollege.edu MTH15_Lec-18_sec_4-1_Exponential_Fcns.pptx 29 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, or time-zero, amount  k = relative rate of Growth (k > 0); i.e., k is POSITIVE Decay (k < 0); i.e., k is NEGATIVE  t = time

30. BMayer@ChabotCollege.edu MTH15_Lec-18_sec_4-1_Exponential_Fcns.pptx 30 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

31. BMayer@ChabotCollege.edu MTH15_Lec-18_sec_4-1_Exponential_Fcns.pptx 31 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

32. BMayer@ChabotCollege.edu MTH15_Lec-18_sec_4-1_Exponential_Fcns.pptx 32 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

33. BMayer@ChabotCollege.edu MTH15_Lec-18_sec_4-1_Exponential_Fcns.pptx 33 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

34. BMayer@ChabotCollege.edu MTH15_Lec-18_sec_4-1_Exponential_Fcns.pptx 34 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).

35. BMayer@ChabotCollege.edu MTH15_Lec-18_sec_4-1_Exponential_Fcns.pptx 35 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

36. BMayer@ChabotCollege.edu MTH15_Lec-18_sec_4-1_Exponential_Fcns.pptx 36 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.

37. BMayer@ChabotCollege.edu MTH15_Lec-18_sec_4-1_Exponential_Fcns.pptx 37 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 thus an annual interest rate.

38. BMayer@ChabotCollege.edu MTH15_Lec-18_sec_4-1_Exponential_Fcns.pptx 38 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

39. BMayer@ChabotCollege.edu MTH15_Lec-18_sec_4-1_Exponential_Fcns.pptx 39 Bruce Mayer, PE Chabot College Mathematics Example  Calc Simple Interest  Rosarita deposited $8000 in a bank for 5 years at a simple interest rate of 6% a)How much interest-$’s 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

40. BMayer@ChabotCollege.edu MTH15_Lec-18_sec_4-1_Exponential_Fcns.pptx 40 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

41. BMayer@ChabotCollege.edu MTH15_Lec-18_sec_4-1_Exponential_Fcns.pptx 41 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

42. BMayer@ChabotCollege.edu MTH15_Lec-18_sec_4-1_Exponential_Fcns.pptx 42 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.

43. BMayer@ChabotCollege.edu MTH15_Lec-18_sec_4-1_Exponential_Fcns.pptx 43 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:

44. BMayer@ChabotCollege.edu MTH15_Lec-18_sec_4-1_Exponential_Fcns.pptx 44 Bruce Mayer, PE Chabot College Mathematics Compare Compounding Periods b)Semi Annual Amount: c)Quarterly Amount:

45. BMayer@ChabotCollege.edu MTH15_Lec-18_sec_4-1_Exponential_Fcns.pptx 45 Bruce Mayer, PE Chabot College Mathematics Compare Compounding Periods d)Monthly Amount: e)Daily Amount:

46. BMayer@ChabotCollege.edu MTH15_Lec-18_sec_4-1_Exponential_Fcns.pptx 46 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

47. BMayer@ChabotCollege.edu MTH15_Lec-18_sec_4-1_Exponential_Fcns.pptx 47 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

48. BMayer@ChabotCollege.edu MTH15_Lec-18_sec_4-1_Exponential_Fcns.pptx 48 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 1797 Thomas Jefferson Placed a Deposit of $450k in the MPS bank at an interest rate of 6%, then find the value $-Amount for the this Account in 2010; 213 years Later

49. BMayer@ChabotCollege.edu MTH15_Lec-18_sec_4-1_Exponential_Fcns.pptx 49 Bruce Mayer, PE Chabot College Mathematics Compare Continuous Compounding  SIMPLE Interest  YEARLY Compounding

50. BMayer@ChabotCollege.edu MTH15_Lec-18_sec_4-1_Exponential_Fcns.pptx 50 Bruce Mayer, PE Chabot College Mathematics Compare Continuous Compounding  Quarterly Compounding  Continuous Compounding

51. BMayer@ChabotCollege.edu MTH15_Lec-18_sec_4-1_Exponential_Fcns.pptx 51 Bruce Mayer, PE Chabot College Mathematics

52. BMayer@ChabotCollege.edu MTH15_Lec-18_sec_4-1_Exponential_Fcns.pptx 52 Bruce Mayer, PE Chabot College Mathematics Effective Interest rate → APR  To help people compare simple, MultiPeriod-compounded, and continuous-compounded Interest rates, ALL advertised interest rates are stated in the effective Annual Percentage Rate, or APR or r e  APR is the simple annual interest, r e, that produces the same Change in $-Value in ONE year

53. BMayer@ChabotCollege.edu MTH15_Lec-18_sec_4-1_Exponential_Fcns.pptx 53 Bruce Mayer, PE Chabot College Mathematics Effective Interest rate → APR  APR Defined For MultiPeriod Compounding at k times per year For Continuous Compounding –Where r is the stated, or nominal, interest rate  When Assessing a Loan or a Savings Instrument the Consumer should consider ONLY the APR for comparisons

54. BMayer@ChabotCollege.edu MTH15_Lec-18_sec_4-1_Exponential_Fcns.pptx 54 Bruce Mayer, PE Chabot College Mathematics WhiteBoard Work  Problems From §4.1 P71 → Beer-Lambert (absorption) Law See Also ENGR45

55. BMayer@ChabotCollege.edu MTH15_Lec-18_sec_4-1_Exponential_Fcns.pptx 55 Bruce Mayer, PE Chabot College Mathematics All Done for Today Very Good Car Loan Rate But what about the Purchase $-Price, and Loan Fees?

56. BMayer@ChabotCollege.edu MTH15_Lec-18_sec_4-1_Exponential_Fcns.pptx 56 Bruce Mayer, PE Chabot College Mathematics Bruce Mayer, PE Licensed Electrical & Mechanical Engineer BMayer@ChabotCollege.edu Chabot Mathematics Appendix –

57. BMayer@ChabotCollege.edu MTH15_Lec-18_sec_4-1_Exponential_Fcns.pptx 57 Bruce Mayer, PE Chabot College Mathematics ConCavity Sign Chart abc −−−−−−++++++−−−−−−++++++ x ConCavity Form d 2 f/dx 2 Sign Critical (Break) Points InflectionNO Inflection Inflection

58. BMayer@ChabotCollege.edu MTH15_Lec-18_sec_4-1_Exponential_Fcns.pptx 58 Bruce Mayer, PE Chabot College Mathematics

59. BMayer@ChabotCollege.edu MTH15_Lec-18_sec_4-1_Exponential_Fcns.pptx 59 Bruce Mayer, PE Chabot College Mathematics

60. BMayer@ChabotCollege.edu MTH15_Lec-18_sec_4-1_Exponential_Fcns.pptx 60 Bruce Mayer, PE Chabot College Mathematics