9.1 Exponential Growth

Transcription

1 9.1 Exponential Growth 1. Complete Activity 1 a. Complete the chart using the x formula y = Advanced Algebra Chapter 9 - Note Taking Guidelines Complete each Now try problem after studying the previous example x y Complete Activity 2 x a. Use the formula y = to estimate the population 1.5 hours before the experiment began. 3. The exponential function f(x) = ab x has a horizontal asymptote. What is the equation for the horizontal asymptote? 4. What is the y-intercept of the exponential function f(x) = ab x?

2 5. Study example 1 6. Now try the following problem: a. The Consumer Price Index (CPI) measures the costs of goods and services in the U.S. In 1980 the CPI was $100. Between 1980 and 1990, the CPI rose at an average rate of 4.7% per year. Let x = number of years after 1980, and let y = the CPI. Then the formula y = 100(1.047) x can be used to estimate the CPI. Graph the equation and trace the graph to estimate: i. The CPI in ii. The year when the CPI will be approximately $ In your own words, what is an exponential function? 8. What do we call the graph of an exponential function?

3 9. Definition of Exponential Function 10. What must be true about an exponential function in order for it to model exponential growth? 11. What two situations already studied this year are examples of exponential functions? 12. What information is needed to write an equation that models exponential growth?

4 13. Study example Now try the following problem: a. The world population in 1985 was 4.9 billion people. At the growth rate of that time, the population would double every 35 years. x = years after 1985 P(x) = population i. Write an exponential equation modeling this situation. Let x equal the number of years after 1985, and let p equal the population. ii. With this model, what would be an estimate for the world s population in the year 2000? iii. What was the population in 1980, assuming the same growth factor? Summarize what you learned in sections 9.1:

5 9.2 Exponential Decay 1. In your own words describe exponential decay 2. What is one situation which is modeled with exponential decay? 3. Study example 1 4. Now try the following problem: 1. Some used-car dealers use the general ruleof-thumb that the trade-in value of a car decreases by 30% each year. This means that the car retains 70% of its value. i. Erica has a car worth $6400. Write an equation that models the value of the car in x years. ii. How much will the car be worth in three years? iii. Erica has owned the car for two years. How much was it worth when she bought it? 5. What is meant by the term half-life?

6 6. Complete the activity on page 540 Number of Number of years half-life periods g g Study example 2 8. Now try the following problem: Amount of 90 Sr remaining b. c.

7 9. Exponential Growth Model 10. How do you determine if an Exponential Growth Model should be exponential growth vs. exponential decay? 11. Write out all the properties of the exponential functions of the form y = ab x, a > 0:

8 12. Study example Now try the following problem: x 2 f ( x) =. Prove that f(x) is an 3 a. Let x exponential decay function. Summarize what you learned in sections 9.2:

9 9.3 Continuous Growth or Decay 1. What are the origins of the number e? 2. What was the formula used earlier this year for compound interest? 3. Continuously Compounded Interest Formula: 4. Study example 1 5. Now try the following problem: a. If $850 is invested at an annual rate of 6% compounded continuously, how much will be in the account after 10 years? 6. Study example 2 7. Now try the following problem: a. It is said that in 1626 the Native Americans who sold Manhattan Island paid about $24 in beads. If that money had been put into an account earning 6% compounded continuously, how much would be in the account today?

10 8. What is the function notation often used to describe continuous change models? 9. Now try the following problem: a. A machine used in an industry depreciates so that its value after t years is given by V = Vo e t i. What is the annual rate of depreciation? ii. If after 3 years the machine is worth $12,000, what was its original value? 10. Now try the following problem: a. There were 500 bacteria growing by 10% each hour. How many bacteria were there after 8 hours? Summarize what you learned in sections 9.3:

11 9.4 Fitting Exponential Models to Data 1. What is the general form for an exponential function and what information would you need in order to write a specific equation in this form? 2. Study example 1 3. Now try the following problem: a. Use the date from Example 1: i. Use the two points (0, 100) and (50, 0.001) to obtain an equation for the data. ii. Use the two points (20, 1.15) and (30, 0.12) to obtain an equation for the data.

12 4. Explain how to determine whether an exponential model is appropriate for a situation. 5. Study example 2 6. Now try the following problem: a. The U.S. Bureau of the Census estimated that the population of Cuba would be 11,613,000 in the year 2000, and 12,795,000 in the year What annual growth rate is assumed between 2000 and 2020? 7. Study example 3 8. Now try the following problem: a. Find a model for the population between 1790 and 1860 for which the starting point is i. Use the equation to predict the population in 1860 Summarize what you learned in sections 9.4:

13 9.5 Common Logarithms 1. Complete Activity 1 on page 557. a. Using your calculator complete the table. x X y=10 x b. Using your calculator NEATLY sketch a graph of the table. c. Plot the points obtained by switching the x s and y s from the table in part a. on the same graph. d. What is the equation for the ordered pairs in part c.? e. Draw the line y=x on the graph above.

14 2. To solve the equation in d. above we need a new function called the logarithmic function. a. Definition of logarithm of x base 10: b. What is one of the most important things to keep in mind when working with logarithms? 3. What does the term Common Logarithm refer to? 4. What are two ways to write a common log function? 5. How do we read log 10 6

15 6. Study example 1 7. Now try the following problem: a. Evaluated the following. i. log 10 10,000 ii. log Study example 2 9. Now try the following problem: a. Evaluate without a calculator i. log 10 1,000,000 ii. log 0.1 iii. log Study example Now try the following problem: a. Between which two consecutive integers does log π lie?

16 12. Complete activity 2 on page 560 a. Use your calculators graph y = 10 x and y = log x in a window -2 to 12 for x and -2 to12 for y then do a zoom square. b. Using y = log x, when x = 2, what is y? c. Using y = 10 x, when x =0.301, what is y? 13. Write all the properties of y = 10 x and y = log x y = 10 x y = log x 14. Study example Now try the following problem: a. Solve for x: log x = 3.5

17 1) Solve log 10 w = 4 3) Solve log 10 (y - 2) = -1 2) Solve log 10 (2x) = 1 4) Solve 2log (2w+8) =6 5) Solve 5 + log (x - 6) - 7= 0

18 Summarize what you learned in sections 9.5:

19 9.6 Logarithmic Scales 1. What unit is used with sound intensity? 2. What unit is used with Relative Intensity? 3. What formula relates sound intensity to relative intensity? 4. Study example 1 5. Now try the following problem: a. Find the relative intensity in decibels of music with a sound intensity of 1 watt per square meter. 6. Study example 2 7. Now try the following problem: a. What is the intensity of a sound with a volume of 100 decibels?

20 8. How does increasing the relative intensity by n decibels change the sound intensity? 9. Study example Now try the following problem: a. The sound of heavy traffic (90 db) is how many times as intense as the sound of normal conversation (60 db)? 11. What is the difference between linear scales (db) and logarithmic scales (w/m 2 )?

21 12. What is the ph scale used for and what is the formula? 13. Study example Now try the following problem: a. Seawater has a ph of 8.5. i. Is it acidic or alkaline? ii. What is the concentration of hydrogen ions? iii. Rewrite your answer from ii in scientific notation.

22 Summarize what you learned in section 9.6

23 9.7 Logarithms to Bases Other Than Definition of logarithm of m to the base b: 2. What is the inverse of the logarithm function with base b, f(x) = log b x? 3. Write all the properties of logarithmic equations of the form y = log b x, where b>0 not equal to zero. i. ii. iii. iv. 4. Study example 1 5. Now try the following problem: a. Evaluate i. log 9 81 ii. log 9 27 iii. 1 log 9 9

24 6. What is one of the most important things to keep in mind when working with logarithms? 7. Study example 2 8. Now try the following problem: a. Solve for n: log 64 n = Study example Now try the following problem: log 9 = 2 a. Find x if x 3

25 1) Solve log 3 w = 4 3) Solve log (y - 2) = -1 2) Solve log 3 (2x) = 1 4) Solve 2log (2w+8) =6 5) Solve 5 + log 4 (x - 6) - 7= 0

26 Summarize what you learned in sections 9.7:

27 9.8 Properties of Logarithms 1. What properties which we have studied earlier are the basis for the properties of logarithms? 2. Theorem (Logarithm of 1) n 3. Theorem ( log of b ) b 4. Study example 1 5. Now try the following problem: a. Evaluate log Theorem (Product Property of Logarithms)

28 7. Study example 2 8. Now try the following problem: a. Find 5 log 45 log Theorem (Quotient Property of Logarithms) 10. Study example Now try the following problem: a. Use the Quotient Property of Logarithms to show that: 1 logb = log b N N 12. Theorem (Power Property of Logarithms)

29 13. Study example Now try the following problem: a. A person uses a calculator and discovers that log b. How can this value be used to find 8? log b 15. Now try the following problem: a. Solve for x: x 1 = log7 49 Summarize what you learned in sections 9.8:

30 1) Solve without using a calculator 2) Solve without using a calculator 3) Evaluate withour a calculator 4) Evaluate without a calculator 5) Rewrite as a single logarithm without a calculator

31 9.9 Natural Logarithms 1. What is a natural logarithm? 2. What is the notation for the natural logarithm? 3. Definition of natural logarithm: 4. How do we read ln x? 5. Complete Activity 1 on page 584: a. ln 1 b. ln 10 c. ln 2 d. ln 0.5

32 6. Now try the following problem: a. ln (xy) = x b. ln = y c. ln x n = 7. Complete activity 2 on page 584: a. Verify with a calculator that ln (2 6 ) = 6 ln 2 8. The natural logarithm function y = ln x is the inverse of what function?

33 9. Study example 10. Now try the following problem: a. Under certain geographic conditions, the wind velocity v at a height h cm above the h ground by v = k ln, where k is a h o positive constant that depends on air density, average wind velocity, and other factors, and where ho is a "roughness value " depending on the roughness of the vegetation on the ground. Suppose that ho = 0.7 cm, a value that applies to a lawn 3 cm high, and k = 300 cm/ sec. i. At what height above the ground is the wind velocity zero? ii. At what height is the wind velocity 1500 cm/sec? 11. Now try the following problem: a. Rewrite in logarithmic form i. e ii. e

34 12. Now try the following problem: a. Rewrite in exponential form i. ln ii. ln 1 = Now try the following problem: a. Evaluate lne 6 1) Give the general property used in the following equation 2) Summarize what you learned in sections 9.9:

35 9.10 Using Logarithms to Solving Exponential Equations 1. Write the four steps for solving equations of the form: b x = a 2. Study example 1 3. Now try the following problem: a. Solve 3 x = Which base for the logarithm should be used when solving equations of the form b x = a?

36 5. Study example 2 6. Now try the following problem: a. At what rate of interest compounded continuously would you have to invest your money so that it would quadruple in 25 years? 7. Study example 3 8. Now try the following problem: a. The number of milligrams of radium present at the end of t years is given by the t formula A = A o , where Ao is the initial amount of radium. What is the half-life of radium; that is, how long will it take for an initial amount to reduce to half its size?

37 9. Can you find the log of any base on your calculator? 10. Theorem (Change of Base Property) 11. Study example Now try the following problem: a. Approximate log to the nearest thousandth Summarize what you learned in sections 9.10:

38 Using Exponential and Logarithmic equations 1) Solve log x 343 = 3 2) Solve 2. ln 6 = ln x + ln 12 3) Solve 3 4 m+ 1 = 24 4) Solve 3 17 m+ 1 = 128