Exponential Functions

Transcription

1 MathsStart (NOTE Feb 2013: This is the old version of MathsStart. New books will be created during 2013 and 2014) Topic 7 Eponential Functions e 20 1 e 10 (0, 1) MATHS LEARNING CENTRE Level 3, Hub Central, North Terrace Campus The Universit of Adelaide, SA, 00 TEL FAX mathslearning@adelaide.edu.au

2 This Topic... This topic introduces eponential functions, their graphs and applications. Eponential functions are used to model growth and deca in man areas of the phsical and natural sciences and economics. Further properties of eponential functions will be covered in Topic 8. Prerequisites You will need a scientific calculator. Contents Chapter 1 Powers. Chapter 2 Eponential Functions. Chapter 3 Growth and Deca. Appendices A. Answers Printed: 18/02/2013

3 1 Powers In the epression 2, 2 is called the base and is called the inde or power or eponent. We use the inde rules to simplif epressions involving powers. Rule 1: a n a m a n m (i) (2 2) (2 2 2) 2 (ii) a n Rule 2: a m an m (i) (ii) (iii) (iv) or Rule 3: (a n ) m a nm or or 1 (i) (2 2 ) 3 (2 2) (2 2) (2 2) If two powers with the same bases are multiplied, add the indices. If two powers with the same bases are divided, subtract the indices. An power with inde 0 is equal to 1. To find the power of a power, multipl the powers. (ii) (iii) (2 2 ) (2 2 )

4 Eponential Functions 2 Rule 3: (ab) n a n b n Rules 3 and 4 deal with powers to different bases (i) (2 3) 3 (2 3) (2 3) (2 3) (ii) (2 10 ) (10 ) a Rule 4: n an b b n Note that 2 1 = 2. Problems 1 1. Use the inde rules to simplif the following: (a) (a) b 3 b 4 (b) b 4 b 3 2w 2 (w) 2 (c) b 4 b 3 b 2 b 2. Write the following as simpl as possible without brackets: 3. Simplif: (a) (2n 3 ) 4 (b) (b) (c) ( 2 ) (c) (d) (d) (d) 3 (b 3 ) 4 10 b 4

5 2 Eponential Functions 2.1 Eponential Functions and their Graphs Eponential functions are functions like f () a, where the base a is a fied number and the inde is given different values. When a 1 the function increases rapidl as increases, and when a 1 the function decreases rapidl as increases. (a) = 2, (b) (1/2)

6 Eponential Functions 4 The graph of 2 combines the shapes of the two graphs above (0, 1) The graph of 2 or is a reflection of 2 across the -ais (0, 1)

7 Eponential Functions The graphs of eponential functions a have the following properties: all are above the -ais, as a 0 for ever value of, all have -intercept (0, 1), as a 0 1 for each value of a, graphs with a 1 increase rapidl as increases, graphs with a 1 decrease rapidl as increases, all have the -ais as a horizontal asmptote. The eample below compares eponential functions with different bases. If a b, then the graph of a is above (below) the graph of b when is positive (negative). Can ou see wh?

8 Eponential Functions Calculating Powers General powers are calculated using the power ke (or or or ^ ) on a scientific calculator. This last one is used because when tping in a computer program or an , ou would write 3 2 as 3^2. To calculate 3 2, use: 3 2 a=a To calculate 3 2, use: 3 a(-) 2 a=a Problems 2.2 Calculate the following, giving our answers to 4 significant figures. (a) (b) 2 (c) (d) (e)

9 3 Growth and Deca 3.1 The number e The most commonl used powers are powers to the bases 10, 2, and Powers to the base 10 are used in scientific notation and also in chemistr (when describing the ph levels of solutions). Powers to the base 2 are used in computing and (occasionall) growth and deca models. Powers to the base are used commonl in man areas of mathematics and its applications. As it is so commonl encountered, it is alwas represented b the letter e, and the function e is called the eponential function. The graph of e has a similar shape to the graphs in the previous section, and lies between the graphs of 2 and 3. Like those graphs, it cuts the -ais at (0, 1) and has the -ais as an asmptote. The graph of e is the reflection of e across the -ais. e 20 e 1 10 (0, 1) You can calculate powers to base e b using the eponential ke e (or Ep ) on a calculator. Can ou see it? It ma need to be done b using SHIFT or 2ndF followed b ln. 7

10 Growth and Deca 8 To calculate e 2, use: e 2 a=a To calculate e 2, use: e a(-)a 2 a=a To calculate e 2 for = 3.71, use: e a(a 2 a a 3.71 a)a a=a Problems Calculate the following, giving our answers to 4 significant figures. (a) e 0 (b) e 1 (c) e.89 (d) e 1.2 (e) 10e e The relative risk of a car accident after drinking alcohol is the probabilit of having an accident after drinking alcohol divided b the probabilit of having an accident without drinking alcohol. The relative risk after having n standard drinks in one hour is given b the function R(n) e 0.4n. (a) Sketch the graph of the function for values of n between 0 and 6. (You can have a part of a drink.) (b) What is the value of R(0)? What does it represent? (c) The legal limit for blood alcohol corresponds to having three drinks in an hour. Interpret the value R(3). (d) How man time more likel are ou to have an accident if ou have 6 drinks in an hour compared to having two drinks in an hour? 3.2 Growth and Deca Models A population that is growing at a constant rate will have P(t) = P(0) e rt members after time t, where P(0) is the initial population and r is the constant growth rate per unit time. The population of China was 80, 000, 000 in 1990 and was growing at the rate of 4% per ear. What would ou predict the population to be in 2002?

11 9 Eponential Functions Answer The initial population (in 1990) is P(0) = 80, 000, 000. The growth rate is 0.04 per ear. We need to find the population in 2002, ie. P(12). Rewrite 4% as a decimal number. P(12) P(0)e r 12 80,000,000 e ,370,000,000 (3 sf) A quantit which is decaing at a constant rate will have the amount Q(t) = Q(0) e rt left after time t, where Q(0) is the initial amount and r is the constant deca rate per unit time. The quantit in the model above could refer to a decaing population or a decaing chemical. It is traditional to choose our own letters for these functions in growth and deca models. For eample, P(t) could be used with population growth, B(t) with bacterial growth, and M(t) with decaing mass. One kilogram of a radioactive isotope of iodine decas at a rate of 8.7% per da. How much would be left after one week, after 30 das. Answer The initial mass is M(0) = 1. The deca rate is per da. We need to find M(7) and M(30) M (7) M (0)e r 7 1 e (3 sf) M (30) M (0)e r 30 1 e (3 sf) 9

12 Growth and Deca 10 Problems The population of the earth at the beginning of 1990 was billion and is growing at the rate of 2% per ear. What will the population be in 2040? 2. If three grams of a radioactive material decas at a constant rate of 0% per ear, how much will be left after ears. 3. A culture of bacteria initiall weighs 1 gm and is growing at a constant rate of 70% per hour. What will be its weight after hours?

13 A Appendi: Answers Section 1.1 1(a) b 7 (b) b (c) 1 (d) b (a) 16n 12 (b) 0w 4 (c) 2 (d) 3 3 3(a) 2 2 (b) (c) (d) Section 2.2 (a) 7.67 (b) (c) (d) (e) Section 3.1 1(a) 1 (b) (c) (d) (e) (b) R(0) = 1 => Relative risk is 1 (c) R(3) = 3.32 => Having 3 standard drinks increases he chance of an accident 3.32 times. (d) R(6)/R(2) = 4.9 => five times more likel to have an accident. Section billion gm gm 12