Lesson 8. Diana Pell. Monday, January 27

Transcription

1 Lesson 8 Diana Pell Monday, January 27 Section 5.2: Continued Richter scale is a logarithmic scale used to express the total amount of energy released by an earthquake. The Richter scale gives the magnitude R of an earthquake using the formula ( ) I R = log Exercise 1. Solve the inequalities. a) x 2 3x > 3 5x I 0 1

2 b) x 2 6x 7 2

3 Exercise 2. A model rocket is projected straight upward from ground level according to the equation h = 16t t, t 0 where h is the height in feet and t is the time in seconds. During what time interval will the height of the rocket exceed 320 feet? 3

4 Exercise 3. The future value of $3000 invested for 3 years at rate r, compounded annually, is given by S = 3000(1 + r) 3. What interest rate will give a future value of at least $3630? 4

5 Section 5.1: Exponential Functions Paramecia: is a single-cell animal which reproduces by splitting into two pieces. If we assume that population begins with 1 paramecium and doubles each hour, then Exponential Function If b is a positive real number, b 1, then the function f(x) = b x is an exponential function. Graph y = 3 x 1) x-intercept: 2) y intercept: 3) Domain: 4) Range: 5) Horizontal Asymptote: 5

6 Exercise 4. Suppose that inflation is predicted to average 4% per year for each year from 2012 to This means that an item that costs $10, 000 one year will cost $10, 000(1.04) the next year and $10, 000(1.04)(1.04) = $10, 000(1.04) 2 the following year. a) Write the function that gives the cost of a $10, 000 item t years after b) If an item costs $10, 000 in 2012, use the model to predict its cost in Exercise 5. Total personal income in the U.S. (in billions of dollars) for selected years from 1960 and projected to 2018 can be modeled by y = 492.4(1.07) x with x equal to the number of years after What does the model predict the total U.S. personal income to be in 2014? 6

7 Exercise 6. It pays to advertise, and it is frequently true that weekly sales will drop rapidly for many products after an advertising campaign ends. This decline in sales is called sales decay. Suppose that the decay in the sales of a product is given by S = x dollars where x is the number of weeks after the end of a sales campaign. a) What is the level of sales when the advertising campaign ends? b) What is the level of sales 1 week after the end of the campaign? c) Use a graph of the function to estimate the week in which sales equal $500. d) According to this model, will sales ever fall to zero? 7

8 The Number e Graph y = e x e The future value S of an investment of P dollars for t years at interest rate r, compounded continuously, is given by S = P e rt Exercise 7. If $10, 000 is invested for 15 years at 10%, compounded continuously, what is the future value of the investment? 8

9 Exercise 8. A breeder reactor converts stable uranium-238 into the isotope plutonium-239. The decay of this isotope is given by A(t) = 100e t where A(t) is the amount of the isotope at time t (in years) and 100 grams is the original amount. a) How many grams remain after 100 years? b) Graph this function for 0 t 50, 000 c) The half-life is the time it takes for half of the initial amount to decay; use graphical methods to estimate the half-life of this isotope. 9

10 Section 5.2: Logarithmic Functions; Properties of Logarithms Logarithmic Function For x > 0, b > 0, and b 1, the logarithmic function with base b is y = log b x which is defined by x = b y. This function is the inverse function of the exponential function y = b x. Exercise 9. Find the value of the logarithms. a) log

11 b) log c) log Exercise 10. Graph y = log 2 x 11

12 Note: Logarithms with base 10 are called common logarithms, and log 10 x is denoted as log x. Exercise 11. Find f(10, 000) if f(x) = log x. Natural Logarithms The logarithmic function with base e is y = log e x, defined by x = e y for all positive numbers x and denoted by Properties of Logarithms ln x = log e x 1. log a 1 = 0 2. log a a = 1 3. log a a x = x and a log a x = x (Inverse Properties) 4. If log a x = log a y, then x = y (One-to-One Property) Exercise 12. Simplify a) log 4 1 b) log

13 c) 6 log 6 20 More Properties of Logarithms 1. Product Property: log a (uv) = log a u + log a v 2. Quotient Property: log a u v = log a u log a v 3. Power Property: log a u n = n log a u Exercise 13. Expand each logarithmic expression. a) log 5 125yz b) ln[e 2 (e + 3)] ( ) x 8 c) log x 13

14 d) log 4 y 6 ( ) 1 e) ln x 5 Exercise 14. Condense each logarithmic expression. a) log 3 x + 4 log 3 y b) 3 log b 1 3 log c c) ln(5x) 3 ln z 14