Week 2: Exponential Functions Goals: Introduce exponential functions Study the compounded interest and introduce the number e Suggested Textbook Readings: Chapter 4: 4.1, and Chapter 5: 5.1. Practice Problems: 4.1: 3, 7, 13, 15, 21, 25, 31, 41, 47
Week 2: Exponential Functions 2 Exponential Functions To prepare for this topic, please read section 4.1 in the textbook. Exponential functions provide better models in some applications including compounded interest, population growth and radioactive decay. To work effectively with exponential functions, we need to know the rules of exponents. Law of Exponents a m a n = a m+n a m a n = am n ( a b ) n = a n a 1 = a (a m ) n = a mn a 0 = 1 b n (ab) n = a n b n a 1 n = n a a n = 1 a n a m n = n a m = ( n a) m (For practice on the basic laws of exponents, try some questions in Problems 0.3 on page 14 (12th), or page 13-14 (13th).) Definition The function f defined by f(x) = b x where b > 0, b 1, and the exponent x is any real number, is called an exponential function with base b. [Textbook, Section 4.1] Example 1: The number of bacteria present in a culture after t minutes is given by N(t) = 4 t. 1. How many bacteria are present initially? Approximately how many bacteria are present after 3 seconds? 2. What multiplication results when the time t is increased 3 seconds? t 0 3 6 9 t t + 3 N(t)
Week 2: Exponential Functions 3 A useful characteristic feature of exponential function is that the value of the function y = b x is multiplied by b for every one unit increase of x Example 2: 1. Which (if any) of the functions in the follow table could be linear? Find formulas for those functions. 2. Which (if any) of these functions could be exponential? Find formulas for those functions. x 2 1 0 1 2 f(x) 9 3 1 g(x) 7 3 1 5 9 h(x) 1 25 1 5 1 3 1 9 1 5 25
Week 2: Exponential Functions 4 Graphs of Exponential Functions The domain of y = a x consists of all real numbers, and the range consists of all positive numbers. All exponential functions have graphs that pass through the point (0, 1), are concave up, and lie entirely above the x-axis. If 0 < a < 1, then the output of f decreases as the input increases and f models exponential decay. If 1 < a, then the output of f increases as the input increases and f models exponential growth. Draw graphs of some exponential functions f(x) = a x (for example, y = ( ) x 1, y = 2 x, y = 3 x, or y = 5 x ). 5 ( ) x 1, y = 2 y 16 14 12 10 8 6 4 2 1 2 3 4 x
Week 2: Exponential Functions 5 Remark: 1. Exponential functions are not defined for b 0 and b = 1 2. More generally, exponential functions are in the form y = Cb x, for C R, b > 0, b 1 Concept Question 1 (Math 121 December Exam 2007) Find the exponential function whose graph is given. (A) f(x) = 3 x+1 (B) f(x) = 3 (C) f(x) = 2 2x (D) f(x) = 3 ( ) x 1 2 ( ) x 1 ANSWER: 4
Week 2: Exponential Functions 6 Example 3: In the early stage of the H1N1 epidemic in 2009, the number of cases in Canada was increased exponentially. There were 719 cases reported on May 20 and 1530 cases reported on June 1 (12 days after). 1. Find an exponential function that models the number of people infected. Let x be the number of days after May 20, 2009. 2. Predicts the number of people infected on June 15, 2009, i.e., x = 27. 3. Assuming the exponential growth model from the part (1), find the number of people infected on July 15, 2010. Here is a table of H1N1 Flu cases reported from April 26, 2009 to July 15, 2009. Time Line April 26 May 1 May 15 May 20 June 1 June 15 June 29 July 15 t 0 5 19 34 46 60 74 90 Canada 6 51 496 719 1530 4049 7987 10156
Week 2: Exponential Functions 7 Compound Interest To prepare for this topic, please read sections 4.1 and 5.1 in the textbook. Exponential functions arise in finance and economics mainly through the idea of compounded interest. Example 4: Suppose that $100 is invested at the rate of 5% compounded annually. How much money will it be after ten years. In general, the compound amount S of the principal P at the end of n years at the rate of r compound annually is given by S = P(1 + r) n Example 5: Suppose that $1 is invested at an annual rate of 100%. How much money will it be after one year under the following compounding schemes. (a) Simple interest: interest is paid once at the end of the year. (b) Compounded semi-annually: This means that the 100% is broken into two pieces of 50%, with each piece awarded to the account every half year.
Week 2: Exponential Functions 8 (c) Compounded quarterly: 25% interest is awarded on the current balance every quarter year. (d) Compounded monthly: Compounded daily: Compounded hourly: Compounded every minute: Compounded every second: (e) Compounded continuously: interest is compounded n times a year and n increase without bound.
Week 2: Exponential Functions 9 When n increases without bound, ( 1 + 1 n) n = e 2.718281828459 The exponential function with base e is called the natural exponential function which is denoted by y = e x. [Textbook, Section 4.1] Example 6: Draw the graphs of y = 2 x, y = 3 x, and y = e x on the same set of axes. y 16 14 12 10 8 6 4 2 1 2 3 4 x
Week 2: Exponential Functions 10 Example 7: (Example 1, Section 5.1) Suppose that 500 amounted to 588.38 in a saving account after three years. If interest was compounded semiannually, find the annual rate (or the nominal rate) of interest, compounded semiannually, that was earned by the money. Example 8: (Example 9, Section 4.1) The projected population P of a city is given by P = 100000e 0.05t where t is the number of years after 1990. Predict the population for the year 2010.
Week 2: Exponential Functions 11 Radioactive Decay Radioactive elements are such that the amount of the element decreases with respect to time. We say that the element decays. Example 9: (Example 11, Section 4.1) A radioactive element decays such that after t days the number of milligrams present is given by N = 100e 0.062t a. How many milligrams are initially present? b. How many milligrams are present after 10 days? If N is the amount at time t, then N = N 0 e λt where N 0 > 0 and λ > 0. The positive constant N 0 represents the amount of the element present at t = 0 and is called initial amount. λ is called the decay constant.