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

1. Chabot Mathematics §4.1 axFunctions Bruce Mayer, PE Licensed Electrical & Mechanical EngineerBMayer@ChabotCollege.edu

2. 3.5 Review § • Any QUESTIONS About • §3.5 → Applied Optimization • Any QUESTIONS About HomeWork • §3.5 → HW-17

3. §4.1 Learning Goals • Define exponential functions • Explore properties of the natural exponential function • Examine investments involving continuous compounding of interest

4. 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. Recall Rules of Exponents • Let a, b, x, and y be real numbers with a > 0 and b > 0. Then Product Rule Power to a Power Rule Quotient Rule Zero Power Rule Product to a Power Rule Negative Power Rule Equal Powers Rule

6. Evaluate Exponential Functions • Example  • Solution  • Example  • Solution 

7. Evaluate Exponential Functions • Example  • Solution 

8. 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. y 8 7 6 5 4 3 2 1 -5 -4 -3 -2 -1 1 2 3 4 5 -1 x -2 Example  Graph y = f(x) =3x • Graph the exponential fcn: • Make T-Table,& Connect Dots

10. y 8 7 6 5 4 3 2 1 -5 -4 -3 -2 -1 1 2 3 4 5 -1 x -2 Example  Graph Exponential • Graph the exponential fcn: • Make T-Table,& Connect Dots • This fcn is a REFLECTION of y = 3x

11. Example  Graph Exponential • Graph the exponential fcn: • Construct SideWays T-Table • Plot Points and Connect Dots with Smooth Curve

12. Example  Graph Exponential • As x increases in the positive direction, y decreases towards 0

13. Exponential Fcn Properties • Let f(x) = ax, a > 0, a ≠ 1. Then • The domain of f(x) = ax is (−∞, ∞). • The range of f(x) = ax is (0, ∞); thus, the entire graph lies above the x-axis. • For a > 1 (e.g., a = 7) • f is an INcreasing function; thus, the graph is RISING as we move from left to right • As x→∞, y = ax increases indefinitely and VERY rapidly

14. Exponential Fcn Properties • Let f(x) = ax, a > 1, a ≠ 1. Then • As x→−∞, the values of y = ax get closer and closer to 0. • For 0 < a < 1 (e.g., a = 1/5 = 0.2) • f is a DEcreasing function; thus, the graph is falling as we scan from left to right. • As x→−∞, y = ax increases indefinitely and VERY rapidly • As x→ ∞, the values of y = ax get closer and closer to 0

15. Exponential Fcn Properties • Let f(x) = ax, a > 0, a ≠ 1. Then • Each exponential function f is one-to-one; i.e., each value of x has exactly ONE target. Thus: • The Basis of the Equal Powers Rule • f has an inverse

16. Exponential Fcn Properties • Let f(x) = ax, a > 0, a ≠ 1. Then • The graph f(x) = ax has no x-intercepts • In other words, the graph of f(x) = axnever crosses the x-axis. Put another way, there is no value of x that will cause f(x) = ax to equal 0 • The x-axis is a horizontal asymptote for every exponential function of the form f(x) = ax.

17. 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. ExponentialFcn ≠ PowerFcn The Exponential is NEVER Negative

19. 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. Example  Bacterial Growth • Given Bacterial Growth Equation • Find: • the initial number of bacteria, • the number of bacteria after 10 hours; and • the time when the number of bacteria will be 32,000.

21. Example  Bacterial Growth • INITIALLY time, t, is ZERO → Sub t = 0 into Growth Eqn: At Ten Hours Sub t = 10 into Eqn:

22. Example  Bacterial Growth • Find t when B(t) = 32,000 • Thus 4 hours after the starting time, the number of bacteria will be 32k

23. 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 ngets larger & larger • The value of e to 15 places: e = 2.718 281 828 459 045

24. The “Natural” base e • The most “common” base for people is 10; e.g., 7.3x105 • 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. % 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 = [xminxmax]; zyh = [0 0]; zxv = [0 0]; zyv = [yminymax]; % % 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([xminxmaxyminymax]),... 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]) MATLAB Code

26. 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. Compare 2x, ex, 3x • SeveralExponentialFunctionsGraphically • Note that EVERY Exponetial intercepts the y-Axisat x = 1

28. 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

29. Exponential Growth or Decay • Math Model for “Natural” Growth/Decay: • A(t) = amount at time t • A0 = 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. 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

31. 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-lifeis the amount of time needed for half of the quantity to decay ½A0 t Half-life

32. 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

33. 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

34. 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).

35. 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. 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. 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. 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. Example  Calc Simple Interest • Rosarita deposited $8000 in a bank for 5 years at a simple interest rate of 6% • How much interest-$’s 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

40. 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. 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. 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.

43. 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:

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

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

46. 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. 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

48. Compare Continuous Compounding • Italy's Banca Monte deiPaschidi 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. Compare Continuous Compounding • SIMPLE Interest • YEARLY Compounding

50. Compare Continuous Compounding • Quarterly Compounding • Continuous Compounding