ࡱ> eg\]^_`abcdi|oR sbjbjVV /<<N1vv4555ht5y8::::::p:v:3VVV8V8VV6`L]j^5|cZ$I0ycB+D0+jj?+V::ty+v : Activity 5.1: Going Shopping SOLs: None Objectives: Students will be able to: Define growth factor Determine growth factors from percent increases Apply growth factors to problems involving percent increases Define decay factor Determine decay factors from percent decreases Apply decay factors to problems involving percent decreases Vocabulary: Growth Factor a percentage increase in the original value of an item Decay Factor a percentage decrease in the original value of an item Key Concept: Growth factor problems are when the ratio of the new value to the original value is always the same and the values are increasing. This ratio is the growth factor. Growth factor = 1 + percentage growth Original value growth factor = new value Decay factor problems are when the ratio of the new value to the original value is always the same, but the values are decreasing. This ratio is the decay factor. Decay factor = 1 - percentage decay Original value decay factor = new value Activity: To earn revenue (income), many state and local governments require merchants to collect sales tax on the items they sell. In several localities, the sales tax is assessed at as much as 8% of the selling price and is passed on directly to the purchaser. Determine the total cost (including 8% sales tax) to the customer of the following items: a) Greeting card selling for $1.50 b) A DVD player selling for $300 Example 1: Determine the growth factor represented by the following percent increases 30% 75% 15% 5.5% Example 2: Determine the new values given the growth factor $400 and 50% increase 120,000 people and a 30% increase Example 3: Determine the growth factor for any quantity that increases 20%. Use the growth factor to determine this years budget if last years was $75,000 Example 4: Determine the decay factor represented by the following percent decreases 30% 75% 15% 5.5% Example 5: Determine the new values given the decay factor $400 and 20% decrease 120,000 people and a 12% decrease Example 6: Determine the original price of an answering machine if you got a 40% discount and paid $150. Determine the original price of a Nordic Track if you got a 25% discount and paid $1140. Concept Summary: The growth factor is when things increase: is the ratio new / original value formed by adding specified percent increase to 100% and then changing to decimal form original value ( growth factor = new value new value ( growth factor = original value The decay factor is when things decrease: is the ratio new / original value formed by subtracting specified percent decrease from 100% and then changing to decimal form original value ( decay factor = new value new value ( decay factor = original value Homework: pg 533 37; problems 1 3, 5, 10 12, 15 Activity 5.2: Take an Additional 20% Off SOLs: None Objectives: Students will be able to: Define consecutive growth and decay factors Determine a consecutive growth or decay factor from two or more consecutive percent change Apply consecutive growth or decay factors from to solve problems involving percent changes Vocabulary: Cumulative Factors the consecutive growth or decay factors from two or more consecutive percent changes Key Concept: You can form a single decay (or growth) factor that represents the cumulative effect of applying the consecutive factors; the single decay factor is the product of the three decay factors. For example: a 10% off coupon on top of 25% off all Holiday items yields (1 0.10)(1 0.25) = (0.9)(0.75) = (0.675) Activity: Your friend arrives at you house. Todays newspaper contains a 20% off coupon at Old Navy. tHe $100 jacket she had been eyeing all season was already reduced by 40%. She clipped the coupon, drove to the store, selected her jacket and walked up to the register. The cashier brought up a price of $48; your friend insisted that the price should have been only $40. The store manager arrived and re-entered the transaction, and again the registered displayed $48. Your friend left without purchasing the jacket and drove straight to your house to tell you her story. How do you think your friend calculated a price of $40? You grab a pencil and start your own calculation. First you determine the ticketed price that reflects the 40% reduction. At what price is Old Navy selling the jacket? Explain how you calculated this price. To what price does the 20%-off coupon apply? Apply the 20% discount to determine the final price of the jacket. If you applied the discounts in reverse order, that is, applying the 20% coupon, followed by a 40% reduction, would the final sales price change? Example 1: A stunning $2000 gold and diamond necklace you saw was far too expensive to even consider. However, over several weeks you tracked the following successive discounts: 20% off list; 30% off marked price; and an additional 40% off every item. Determine the selling price after each of the discounts is taken. Example 2: You purchased $1000 of a recommended stock last year and watched gleefully as it rose quickly by 30%. Unfortunately, the economy turned downward, and your stock recently fell 30% from last years high. Have you made or lost money on your investment? Concept Summary: Cumulative effect of a sequence of percent changes is the product of the associated growth or decay factors Cumulative effect of a sequence of percent changes is the same regardless of the order the changes are applied Homework: page 542; problems 1 6 Activity 5.3: Inflation SOLs: None Objectives: Students will be able to: Recognize an exponential function as a rule for applying a growth factor or a decay factor Graph exponential functions from numerical data Recognize exponential functions from equations Graph exponential functions using technology Vocabulary: Exponential Function when the independent variable appears as an exponent of the growth factor Exponential Growth when the independent variable appears as an exponent of the growth factor that is greater than 1 Exponential Decay when the independent variable appears as an exponent of the decay factor that is less than 1 Key Concepts: Classic exponential functions are in the form y = akx, where a is any constant, and k is called the factor. If k > 1, then it is a growth factor and if 0 < k < 1, then k is a decay factor. Inflation is a typical growth factor type of problem. The growth factor is 1 + inflation rate. Depreciation is a typical decay factor type of problem. The decay factor is 1 depreciation rate. Activity: Inflation means that a current dollar will buy less in the future. According to the US Consumer Price Index, the inflation rate for 2005 was 4%. This means that a one-pound loaf of white bread that cost a dollar in January 2005 cost $1.04 in January 2006. The change in price is usually expressed as an annual percentage rate, known as the inflation rate. At the current inflation rate of 4%, how much will a $20 pair of shoes cost next year? Assume that inflation is constant (4%) next year too; how much will the shoes cost in the year after? Assume that inflation remains at 5% per year for the next decade. Calculate the cost of a currently priced $8 pizza for each of the next ten years and graph it. Years From Now Pizzas Cost 08.0012345678910 Example 1: In the late 1970s and early 1980s inflation in the United States was a big problem. In 1980 the inflation rate was 14.3%. Gasoline was 50 cents a gallon. Assume inflation remains constant. What is mathematical model for the cost of gas? What is the cost of gas in 1990? What is the cost of gas in 2010? Example 2: You have just purchased a new car for $16,000. Much to your dismay, you have learned that you can expect the value of your car to depreciate by 15% per year (taken as soon as you drive it off the dealers lot). What is the decay factor? What is the model to represent the cars value? How much is the car worth after 6 years? When will it be worth about half of its original value? Concept Summary: Exponential function is a function in which the independent variable appears as an exponent of the growth factor or a decay factor Growth: factor greater than 1 Decay: factor between 0 and 1 Homework: pg 547-551; problems 1, 2, 4 Activity 5.4: The Summer Job SOLs: None Objectives: Students will be able to: Determine the growth or decay factor of an exponential function Identify the properties of the graph of an exponential function defined by y = bx where b > 0 and b `" 1 Graph exponentials functions using transformations Vocabulary: Exponential Function  when the independent variable appears as an exponent of a constant Key Concepts: x-intercept is called a zero of the function and is the solution to a quadratic equation Activity: Your brother will be attending college in the fall, majoring in mathematics. On July 1, he goes to your neighbors house looking for summer work to help pay for college expenses. Your neighbor is interested since he needs some odd jobs done. Your brother can start right away and will work all day July 1 for 2 cents. This gets your neighbors attention, but you wonder if there is a catch. Your brother says that he will work July 2 for 4 cents, July 3 for 8 cents, July 4 for 16 cents and so on for every day of the month of July. Complete the following table. Day in July Pay in Cents 1224384165678Do you notice a pattern in the output values? Use this pattern to determine his pay on the 9th of July Use the pattern to determine an equation that relates the day, n, to the amount of pay P(n) What was the average rate of change between day 3 and 4? What was the average rate of change between day 7 and 8? Is the function linear? Use the equation above to figure out: What does your brother make on the 20th of July? What would he make on the 31st of July? Growth Factor Examples: Identify the growth factor, if any, for the given function. h(x) = 1.08x g(x) = 0.8x f(t) = 8t s(t) = 10t Transformations: Describe in words how the graph of y = 2x is transformed by each equation. y = - 2x y = 5 2x y = 2x + 3 (What is the horizontal asymptote?) y = 2x+3 Exponential Growth: Graph the functions f(x) = 2x and g(x) = 10x What two things are the same for both functions? How do they differ? When x > 0? When x < 0? Decay Factor Examples: Identify the decay factor, if any, for the given function. h(x) = 0.98x g(x) = 1.01x f(t) = 0.8t s(t) = (5/8)t Exponential Decay: Given a function g(x) = (1/2)x, answer the following: Identify the x- and y-intercepts Complete the following table: x124610g(x)Does it have a horizontal asymptote? What is function always doing? (Hint: slope) Concept Summary: Exponential functions in form y = bx, where b > 0 and b `" 1 have the following characteristics Domain is all real numbers and Range is y > 0 Line y = 0 is a horizontal asymptote y-intercept is (0, 1) and there is no x-intercept Function is continuous If 0 < b < 1, then its a decay function; If b > 1, then its a growth function Homework: pg 562 9; problems 4, 7, 8 - 13 Activity 5.5: Cellular Phones SOLs: None Objectives: Students will be able to: Determine the growth and decay factor for an exponential function represented by a table of values or an equation Graph exponential functions defined by y = abx, where a `" 0, b > 0 and b `" 1 Identify the meaning of a in y = abx as it relates to a practical situation Determine the doubling and halving time Vocabulary: Zero-product principle  if a"b = 0, Key Concepts: Exponential Functions of the form y = a"bx, where b is > 0 and b `" 1 a is called the initial value, y-intercept (0, a) at x = 0 Exponential functions have successive ratios that are constant The constant ratio is a growth factor, if y-values are increasing (b > 1) The constant ratio is a decay factor, if y-values are decreasing ( 0 < b < 1) Doubling time set by growth factor Half-life is set by the decay factor Activity: During a meeting, you hear the familiar ring of a cell phone. Without hesitation, several of your friends reach into their jacket pockets, brief cases and purses to receive the anticipated call. Although sometimes annoying, cell phones have become part of our way of life. The following table shows the increase in the number of cell phone users in the late 1990s. YearCell Phones ( in millions)Rate of ChangeRatio between Years199644.248199755.312199869.14199986.4252000108.031 Is this a linear function? Why or why not? Is the rate of change (slope) the same? Is the ratio between consecutive years the same? Does the relationship in the table represent an exponential function? What is the growth factor? Set up an equation, N = a"bt, where N represents the number of cell phones in millions and t represents the number of years since 1996 What is the practical domain of the function N? Example 1: Identify the y-intercept, growth or decay factor, and whether the function is increasing or decreasing f(x) = 5(2)x g(x) = (0.8)x h(x) = (5/6)x f(t) = 3(4/3)x Example 2: An investment accounts balance, B(t), in dollars, is defined by B(t) = 5500(1.12)t, where t is the number of years. What was the initial investment? What is the interest rate on the account? When will the investment double in value? When will the investment quadruple in value? Example 3: Chocolate chip cookie freshness decays over time due to exposure to air. If the cookie freshness is defined by f(t) = (0.8)t, find the following information. What was the initial cookie freshness? What is the decay rate on the cookies? When will the cookies freshness be halved? Concept Summary: Functions defined by y = abx, where a is the initial value and b is the growth or decay factor are exponential functions Y-intercept is (0, a) Growth factor, b > 1, y-values are increasing Decay factor, 0 < b < 1, y-values are decreasing Doubling time is the time for the y-value to double Half-life is the time for the y-value to be halved Homework: pg 576 580; problems 2, 3, 6, 7 Activity 5.6: Population Growth SOLs: None Objectives: Students will be able to: Determine annual growth or decay rate of an exponential function represented by a table of values or an equation Graph an exponential function having equation y = a(1 + r)x Vocabulary: Growth rate percentage of growth, r Growth factor the growth rate plus 100 percent; (1 + r) Decay rate percentage of decay, r Decay factor 100 percent minus the decay rate; (1 - r) Key Concepts: Linear functions represent quantities that change at a constant rate (slope) Exponential functions represent quantities that change at a constant ratio, expressed as a percent. Activity: According to the 2000 US Census, the city of Charlotte, North Carolina, had a population of approximately 541,000. Assuming that the population increases at a constant rate of 3.2%, determine the population of Charlotte (in thousands) in 2001. Determine the population of Charlotte (in thousands) in 2002. Divide the population in 2001 by the population in 2000 and record this ratio. Divide the population in 2002 by the population in 2001 and record this ratio Are the ratios the same? What does this mean? Fill in the table below for Charlottes population: Years (since 2000) 012345Population 541 What is the growth rate? What is the growth factor? What is the population of Charlotte in 2006? How long will it take for Charlottes population to double? Exponential Growth Example: Determine the growth rate, r, given the following factors: b = 1.12 b = 1.07 b = 1.33 Determine the growth factor, b, given the following rates: r = 5.4% r = 25% Exponential Decay Example: Determine the decay rate, r, given the following factors: b = 0.82 b = 0.87 b = 0.93 Determine the decay factor, b, given the following rates: r = 6.4% r = 15% You are working at a waste-treatment facility. You are presently treating water contaminated with 18 micrograms of pollutant per liter. Your process is designed to remove 20% of the pollutant during each treatment. Your goal is to reduce the pollutant to less than 3 micrograms per liter. Complete the table: Treatments 0 1 2 3 4 5 Pollutant Concentration  What percent of the pollutant remains after treatment? What is the concentration after the first treatment? Write the equation for the concentration, C, of the pollutant as a function on the number of treatments, n. How many treatments are necessary to achieve the needed pollutant concentration? Concept Summary: Exponential functions are used to describe phenomena that grow or decay by a constant percentage rate over time Annual growth rate problems are modeled by P = P0(1 + r)t where P0 is the initial amount, r is the annual growth rate, and t is time in years (1 + r) represents the growth factor Annual decay rate problems are modeled by P = P0(1 - r)t where P0 is the initial amount, r is the annual decay rate, and t is time in years (1 - r) represents the decay factor Homework: pg 586 588; problems 1, 2, 5 Activity 5.7: Time is Money SOLs: None Objectives: Students will be able to: Distinguish between simple and compound interest Apply compound interest formula to determine the future value of a lump-sum investment earning compound interest Apply the continuous compounding formula A = Pert Vocabulary: Compound Interest interest earned is added to principal before the new interest is calculated Future Value the value of an amount in the future at a specific interest rate and compounding structure Effective Yield percentage by which the balance will grow in one year Continuous Compounding compounds interest each instant of time Key Concepts: Simple versus Compound Interest Simple interest means that you earn only interest on the principal over the time period invested. Compounded interest, the interest paid on most banks savings accounts, pays interest on the principal and the interest it has already earned. A = P(1 + r/n)nt where A is the current balance P is the principal (original deposit) r is the annual interest rate (in decimal form) n is the number of time per year interest is compounded t is the time in years the money has been invested Effective Rates Determine the growth factor (n is number of times compounded per year) b = ( 1 + r/n)n Subtract 1 from b and write the result as a decimal re = b 1 = (1 + r/n)n 1 Effective yield, re, will always be slightly greater than the interest rate. Continuous Compounding Compounded interest formula approaches A = P(1 + r/n)nt ( A = Pert as the number of times the interest is compounded approaches infinity (continuous compounding). Like a horizontal asymptote on a graph. Some banks use this method for compounding interest. Activity: Congratulations, you have inherited $20,000! Your grandparents suggest that you save half of the inheritance for a rainy day. Suppose the $10,000 is deposited in a bank at 6.5% annual simple interest. What is the interest earned after 1 year? What would be the interest in 10 years? Simple vs Compound Interest Example: Suppose you won (tax-free) a million dollars and deposited it in an account earning 5% interest simple interest. How much will you have after 10 years? If it was deposited in an account earning 5% compounded yearly, how much will you have after 10 years? Compounding Comparisons: Determine the effective yield associated with each of the growth factors (interest rate of 6.5% with different compounding schedules) in the following table. n1412365Growth Factor, b 1.0651.06661.066971.06715Effective Yield, re How Compounded  Does the number of times compounded make a difference? Effective Yield Example: Determine the effective yields for the following rates and compounding schedules: r = 4.5%, compounded monthly r = 2.5%, compounded quarterly r = 5.5%, compounded daily Compounding Example: Calculate the balance of your $10,000 investment in 10 years with an annual interest rate of 6.5% compounded continuously. What is the growth factor in this case? What is the effective yield? Retirement Example: Historically, investments in the stock market have yielded an average rate of 11.7% per year (over the long haul). Suppose on graduating high school a rich aunt deposits 10,000 in an account at an 11% annual interest rate that compounds continuously for your retirement. She claims that you will have over a million dollars by retirement time (age 65). Is she right? Concept Summary: Compound interest formula: A = P(1 + r/n)nt where A = current balance P is the principal (original deposit) r is the annual interest rate (in decimal form) n is the number of times per year of compounding t is the time in years money is invested Continuous compounding formula: A = Pert If the number of compounding periods is large, then the compound interest formula can be approximated by continuous compounding formula. Homework: pg 595 597; problems 1-3 Activity 5.8: Continuous Growth and Decay SOLs: None Objectives: Students will be able to: Discover the relationship between the equations of exponential functions defined by y = abt and the equations of continuous growth and decay exponential functions defined by y = aekt Solve problems involving continuous growth and decay models Graph base e exponential functions using transformations Vocabulary: Vary Directly if Key Concept: Continuous growth is modeled by the equation: y = aekt where a is the initial amount, k is the constant continuous growth rate and t is time Activity: The US Census Bureau reported that the US population on April 1, 2000 was 281,421,906. The US population on April 1, 2001 was 284,236,125. Assuming exponential growth, the US population y can be modeled by the equation y = abt, where t is the number of years since April 1, 2000 (when t = 0). What is the initial value, a? What is the annual growth factor, b? Assuming exponential growth, the US population y can be modeled by the equation y = abt. What is the annual growth rate? What is the equation for US population as a function of t? Use this to estimate the US population on 1 Apr 2011. Change the equation y = abt, to a continuous growth form of y = aekt. So bt = ekt and from algebra ekt = (ek)t How are b and ek related? Using our calculator, let Y1 = ex and Y2 = 1.01 and find their intersection (solution for b = ek)? Rewrite the US population function in continuous growth format. Example 1: A bacterial growth in a culture increases by 25% every hour. If 10000 are present when the experiment starts: Determine the constant, k, in continuous growth model Write the equation for the continuous model When will the sample double? Example 2: Tylenol (acetaminophen) is metabolized in your body and eliminated at a rate of 24% per hour. You take two Tylenol tablets (1000 milligrams) at 1200 noon. What is the initial value? Determine the decay factor, b. Find the constant continuous decay rate, k. Write the continuous decay function  Example 3: Compared to y = ex, describe the graphic relationship between its graph and the following graphs: y = - ex y = ex+2 y = ex + 2 y = 2ex y = e-x y = 1 2ex Concept Summary: Quantities that increase or decrease continuously at a constant rate can be modeled by y = aekt. Increasing: k > 0 k is continuous rate of increase Decreasing: k < 0 |k| is continuous rate of decrease The initial quantity at t=0, a, may be written in other forms such as y0, P0, etc Remember the general shapes of the graphs Homework: pg 604-09; problems 2, 3, 8 Activity 5.9: Bird Flu SOLs: None Objectives: Students will be able to: Determine the equation of an exponential function that best fits the given data Make predictions using an exponential regression equation Determine whether a linear or exponential model best fits the data Vocabulary: None new Activity: In 2005, the avian flu, also known as the bird flu, received international attention. Although there were very few documented case of the avian flu infecting humans worldwide, world health organizations including the Centers for Disease Control in Atlanta expressed concern that a mutant strain of the bird flu virus capable of infecting humans would develop and produce a worldwide pandemic. The infection rate (the number of people that any single infected person will infect) and the incubation period (the time between exposure and the development of symptoms) of this flu cannot be known precisely but they can be approximated by studying the infection rates and incubation periods of existing strains of the virus. A conservative infection rate would be 1.5 and a reasonable incubation period would be about 15 days (0.5 months). This means that the first infected person could be expected to infect 1.5 people in about 15 days. After 15 days that person cannot infect anyone else. This assumes that the spread of the virus is not checked by inoculation or vaccination. So after 15 days from the first infection there are 2.5 people infected. In the next 15 days the 1.5 newly infected people would infect 2.25 more people (1.5 ( 1.5). Now we have 4.75 people who have been infected (at the end of one month). Fill in the following table (Round each value to the nearest person): Months 00.511.522.533.544.55Newly Infected 11.52.253.385Total Infected 12.55813 Construct a scatterplot of the data L1: Months, L2: Total  Does the scatterplot indicate a linear relationship? Use your calculator to model the data with an exponential function. STAT CALC, option 0: ExpReg to determine an exponential model of best fit. Round all values to 3 decimal places. What is the practical domain of this function? How does the N-intercept compare with the table? Use the model to predict the total number infected after 1 year. When will 2 million people be infected? Exponential Growth Example: According to the US Department of Education, the number of college graduates increased significantly during the 20th century. The following table gives the number (in thousands) of college degrees awarded from 1990 to 2000: Year 19001910192019301940195019601970198019902000 Grads 30547312722343253087893510171180  What is the exponential regression equation? What is the base of the exponential model? What is the annual growth rate? Use the exponential model to determine the number of college graduates in 2010 (t = 110). When will the number of college grads equal 2 million? What is the doubling time for the exponential model? Exponential Decay Example: Students in US public schools have had much greater access to computers in recent years. The following table shows the number of students per computer in selected years: Year19831984198519871989199219951999Students / computer 305473127223432530878 What is the exponential regression equation? What is the base of the exponential model? What is the annual decay rate? Concept Summary: Quantities that increase or decrease continuously at a constant rate can be modeled by y = aekt. Increasing: k > 0 k is continuous rate of increase Decreasing: k < 0 |k| is continuous rate of decrease The initial quantity at t=0, a, may be written in other forms such as y0, P0, etc Remember the general shapes of the graphs Homework: pg 614 617; problems 1-3 Exponential and Logarithmic Properties Properties of Exponents The following examples are given to the base e (the natural number, 2.718281828) Multiplication Property: ea ( eb = ea+b Example: e3x ( e2x = e3x+2x = e5x Division Property: ea ( eb = ea- b Example: e3x ( e2x = e3x-2x = ex Exponent Property: (ea)b = ea( b Example: (ex)4 = e4x You must have common bases for the above properties to apply. Other reminders: any number (other than 0) raised to the zero power is 1 Example: (12,345)0 = 1 Negative powers are equivalent to reciprocals. Example: e-x = 1/ex Logarithms are made up of three parts: a base, b, an exponent, y, and a quantity, x: y = logb x (Note: b > 0 and x > 0) They have a corresponding exponential equation: by = x corresponds to y = logb x The logarithms on a calculator are the common log (to the base 10) and the natural log (to the base e) To calculate logarithms to other bases you must perform a change of base (to 10 or e) calculation: Log a ln a ln 10 Logb a = -------- = ------- Example: Log4 10 = -------- = 1.661 Log b ln b ln 4 Properties of Logarithms The following examples are given with log to the base e (also known as natural logs, abbreviated ln) Multiplication Property: ln ab = ln a + ln b Example: ln 14 = ln 2 + ln 7 Division Property: ln (a/b) = ln a ln b Example: ln 16/7 = ln 16 ln 7 Exponent Property: ln (a)b = b ln a Example: ln 8 = ln 23 = 3 ln 2 Other reminders: log (to any base) of 1 = 0 Logs of (1/x) are always equal to log x (regardless of the base; application of division property) Activity 5.10: The Diameter of Spheres SOLs: None Objectives: Students will be able to: Define logarithm Write an exponential statement in logarithmic form Write a logarithmic statement in exponential form Determine log and ln values using a calculator Vocabulary: Logarithms are inverses of exponential functions. The equation logb x = y is equivalent to by = x. Common logarithms are logarithms to the base 10 Natural logarithms are logarithms to the base e e is the natural number, apx 2.712818 Key Concepts: In general, a statement in logarithmic form is logb x = y, where b is the base of the logarithm, x is a power of b, and y is the exponent. The base b for a logarithm can be any positive number except 1. Common logs are to the base 10 and natural logs are to the base e. The equation logb x = y is equivalent to by = x. If b > 0 and b `" 1 then Logb bn = n Logb 1 = 0 Logb b = 1 Activity: Spheres are all around us (pardon the pun). You play sports with spheres like baseballs, basketballs, and golf balls. You live on a sphere, the Earth; which is orbited by a sphere, the moon; and both of them orbit another sphere, the sun. All spheres have properties in common. For example, from Geometry, the formula for volume is V = 4/3r3 and the formula for surface area is SA = 4r2 (where r represents the radius of the sphere). But not all spheres are the same size! The table below has the diameter (remember d = 2r) of several spheres mention before: SphereDiameter (in meters)SphereDiameter (in meters)Golf Ball0.043 Moon3,476,000 Baseball0.075 Earth12,756,000 Basketball0.239 Jupiter142,984,000  If we wanted to plot diameter as the x-variable and either volume or surface area as the y-variable, then we would need to put all these values on the horizontal axis somehow. If we plot the first three on the axis below  If we plot the last three on the axis below  If we scale the axis in exponential powers of 10 (each tick mark is ten times larger than the one before), then we can fit them all on the same scale. This is called a logarithmic scale.  Many times a logarithmic scale will be marked with just the exponents (and a note about it being logarithmic).  Richter scale (measuring earthquakes) The magnitude value is proportional to the logarithm of the amplitude of the strongest wave during an earthquake. A recording of 7, for example, indicates a disturbance with ground motion 10 times as large as a recording of 6. The energy released by an earthquake increases by a factor of 30 for every unit increase in the Richter scale. The table below gives the frequency of earthquakes and the effects of the earthquakes based on this scale Richter scale No. of earthquakes per yearTypical effects of this magnitude< 3.4800 000Detected only by seismometers3.5 - 4.230 000Just about noticeable indoors4.3 - 4.84 800Most people notice them, windows rattle.4.9 - 5.41400Everyone notices them, dishes may break, open doors swing.5.5 - 6.1500Slight damage to buildings, plaster cracks, bricks fall.6.2 6.9100Much damage to buildings: chimneys fall, houses move on foundations.7.0 - 7.315Serious damage: bridges twist, walls fracture, buildings may collapse.7.4 - 7.94Great damage, most buildings collapse.> 8.0One every 5 to 10 yearsTotal damage, surface waves seen, objects thrown in the air.Example 1: Evaluate each of the following Log 104 Log (1/100) Log 1000 Log4 64 Ln e Log3 (1/27) Example 2: Evaluate each of the following with your calculator Log 20 Ln 15 Log 0.02 Rewrite each logarithmic expression as exponential: 3 = log2 8 0 = log5 1 Concept Summary: Notation of logarithms is y = logb x where b is the base of the log, x is the resulting power of b and y is the exponent Common logs (base 10) y = log x Natural logs (base e) y = ln x y = logb x is equivalent to exponential by = x Special log properties: logb 1 = 0 logb b = 1 logb bn = n Homework: pg 632 633; problems 1- 5 Activity 5.11: Walking Speed of Pedestrians SOLs: None Objectives: Students will be able to: Determine the inverse of the exponential function Identify the properties of the graph of a logarithmic function Graph the natural logarithmic function using transformations Vocabulary: Logarithmic function defined by the inverse function of an exponential function to the base, b Inverse functions interchange the domain and range of the original function. The graph of an inverse function is the reflection of the original function about the line y = x. You determine the inverse function by switching x and y and solve the new equation for y. Vertical asymptote a vertical line that the function approaches but never reaches (usually a restricted value in the domain) Key Concept: The logarithmic function base b is defined by y = logb x where b is the base (b > 0 and b `" 1) x is the power of the base, b and y is the exponent needed on b to obtain x Properties of f(x) = log x (common log) Base is 10 Domain of x is set of all positive numbers (x > 0) Range of y is set of all real numbers f(x) is inverse of g(x) = 10x Properties of f(x) = ln x (natural log) Base is e (natural number H" 2.718281828) Domain of x is set of all positive numbers (x > 0) Range of y is set of all real numbers f(x) is inverse of g(x) = ex Activity: On a recent visit to Boston, you notice that people seem rushed as the move about the city. Upon returning to school, you mention this observation to your psychology teacher. The instructor refers you to a psychology study that investigates the relationship between the average walking speed of pedestrians and the population of the city. The study cites statistics presented in the graph.  Does the data appear to be linear? Does the data appear to be exponential? Natural Logarithm Fill in the table and graph y = ln x x0.010.10.250.515102050y = ln x Inverse Functions: Fill in the table below using your calculator xf(x) = 10xxg(x) = log x-20.01-10.1011102100 Find the inverse of the following functions y = 2x 3 y = x2 y = 5x Natural Logarithmic Transformations Compared to y = ln x, describe the graphic relationship between its graph and the following graphs: y = -ln x y = ln (x + 3) y = ln (-x) y = ln x + 3 y = 2 ln x y = 3 2 ln x Concept Summary: Properties of the function y = log x: Domain is x > 0 Range is all real numbers Is the inverse of the function x = 10y Graph of the function y = logb x or y = ln x: is increasing for all x > 0 x-intercept of (1,0) and no y-intercept x=0 (y-axis) is a vertical asymptote is continuous for x > 0 Inverse functions are determined by interchanging x and y variables Homework: pg 640-43; problems 1, 3-5 Activity 5.12: Walking Speed of Pedestrians, continued SOLs: None Objectives: Students will be able to: Compare the average rate of change of increasing logarithmic, linear, and exponential functions Determine the regression equation of a natural logarithmic function, y = a + b ln x, that best fits a set of data Vocabulary: None new Key Concept: Lets compare three of the major function families that we have studied this year: linear, exponential, and logarithmic. Graph the following: Y1 = x; Y2 = ex; and Y3 = ln x And compare them using the following table: Function Slope Grows Horizontal AsymptoteVertical Asymptote Domain Range y = xy = exy = ln x This illustrates the relationship between f(x) = bx, b > 1; g(x) = logb x, b > 1 and y = mx + b, m > 0 Activity: In Activity 5-11 we looked at a psychological study that investigated the relationship between the average walking speed of pedestrians and the population of the city. The table below summarizes the data: Population (in K) 50 100 200 500 1000 2000 Ave Walking Speed 3.7 4.3 4.9 5.7 6.3 6.9 Run a natural logarithmic regression on the data What is the model? y = a + b ln x What is the practical domain of the function? Predict the walking speed in Boston (pop = 589,121) Predict the walking speed in New York (pop = 8,008,278) If the average walking speed is 5.2, write an equation that can be used to find the population, P, of the city. How do we solve it graphically? How do we solve it algebraically? Concept Summary: As the input of a logarithmic function increases, the output increases at a slower rate (the slope is smaller; the graph becomes less steep) The relationships between f(x) = bx, g(x) = logb x and y = mx + b are seen in the table below FunctionDirectionGrowth RateAsymptoteDomainRangef(x) = bx b > 1 IncreasingFastestHorizontalAll realY > 0g(x) =logbx b > 1 IncreasingSlowestVerticalX > 0All realy = mx + b m > 0 increasingconstantnoneAll realAll real Homework: pg 649 654; problems 2, 6 Activity 5.13: The Elastic Ball SOLs: None Objectives: Students will be able to: Apply the log of a product property Apply the log of a quotient property Apply the log of a power property Discover change of base formula Vocabulary: None new Key Concept:  EMBED PowerPoint.Template.12  Activity: You are continuing your work on the development of the elastic ball. You are still investigating the question, If the ball is launched straight up, how far has it traveled vertically when it hits the ground for the 10th time? However, your supervisor tells you that you cannot count the initial launch distance. You must calculate only the rebound distance. Using some physical properties, timers and your calculator, you collect the following data.  Plot the data using STATPLOT What does the graph say about a model: linear, exponential or logarithmic? The data can be modeled (thru logarithmic regression) as T = 26.75 log N Plot both and see if the model is reasonable Product Examples: Log2 32 = Log (5st) = Log4 3 + log4 9 = Log2 7 + log2 11 = Quotient Examples: Log2 (2x/y) = Log (st) = Log4 3 - log4 9 = Log2 7 - log2 11 = Power Examples: Log2 32 = Ln (st)7 = Log4 "51 = 4Log2 3 = Power Examples: Log3 (5) = Log4 x = Log2 7x = Concept Summary: Properties of Logarithmic Functions: logb (A " B) = logb A + logb B logb (A / B) = logb A  logb B logb (A)p = p"logb A  Change bases: Remember: logb (x + y) `" logb x + logb y Homework: pg 662-665; problems 1, 2, 6-9 Activity 5.14: Prison Growth SOLs: None Objectives: Students will be able to: Solve exponential equations both graphically and algebraically Vocabulary: None new Key Concept: Graphical Approach Let Y1 = abx Let Y2 = c Find the intersection point of the two graphs Algebraic Approach Rewrite equation into form: bx = c/a (all positive) Take ln or log of both sides: ln bx = ln (c/a) Use properties of logs to simplify x ln b = ln c ln a Solve for the variable x = (ln c ln a) / ln b Activity: Your sister is a criminal justice major at WCC. The following statistics appeared in one of her required readings relating to the inmate population of US federal prisons (population in thousands). Year19751979198619901994199820002003Population20.121.531.847.876.295.5112.3158.0She asks you to help analyze the prison growth situation for a project in her criminology course. Enter the data and plot using STATPLOT. What form does the graph look like? Use an exponential regression model on your calculator to estimate the total inmate population. Y = abx Graph the STATPLOT and the Y1 = abx on the same graph Use the model to estimate the prison population in 2010 Solve 11.36(1.080)t = 180 algebraically and then graphically. 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If P0 represents the original amount of a radioactive substance, then the amount P present after a time t (usually measured in years) is modeled by P = P0ekt where k represents the rate of continuous decay Example 1: One type of uranium decays at a rate of 0.35% per day. If 40 pounds of this uranium was found today, how much will be left after 90 days? Example 2: Strontium-90 decays at a rate of 2.4% per year. If 10 grams of it were initially present, how much will be left after 20 years? Example 3: Strontium-90 decays at a rate of 2.4% per year. If 10 grams of it were initially present, how long until half of it is gone? (This is called half-life, a very important term with radioactive material) Concept Summary: Solving exponential problems in form abx = c Isolate exponential factor on one side of equation Take the log (or ln) of both sides of equation Apply log property: log bx = x log b to remove variable as an exponent Solve the resulting equation for the variable Homework: pg 669 72; problems 1, 4-10 Activity 5.15: Frequency and Pitch SOLs: None Objectives: Students will be able to: Solve logarithmic equations both graphically and algebraically Vocabulary: None new Key Concept: Graphical Approach Let Y1 = logb (f(x)) Let Y2 = c Find the intersection point of the two graphs Algebraic Approach Rewrite equation into form: logb (f(x)) = c (all positive) Rewrite step 1 in exponential form: f(x) = bc Solve the resulting equation from step 2 algebraically Check solution in the original equation Activity: Raising a musical note one octave has the effect of doubling the pitch, or frequency, of the sound. However, you do not perceive the note to sound twice as high, as you might predict. Perceived pitch is given by the function P(f) = 2410 log (0.0016f + 1) where P is the perceived pitch in mels (units of pitch) and f is the frequency in hertz.  Graph the function What is the perceived pitch, P, for an input of 10,000 hertz? Write an equation that can be used to determine what frequency, f, gives an output of 2000 mels. Solve it using the graphing approach Solve the equation, 2410 log (0.0016f + 1) = 2000 using an algebraic approach Use an algebraic approach to determine the frequency, f, that produces a perceived pitch of 3000 mels. 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FdXEMF+*@$??9DC*@$??9DC@ !!K@0 ee# ??9DC$$??9DC  Rp"Arial""AgG B`BBl""e"w !3l1T "́Nw !3l1hwݫw"YtgPCgPC`g&eNwĠe("aNw@"GDIC qm<"Ow-T X"UOwXT l4CPC&eNwĠdv%    TTa}1LPA#% ( " " FdXEMF+*@$??9DC*@$??9DC@ !!K@0 ee# ??9DC$$??9DC  Rp"Arial""AgG 8 CpBBl""e"w !3}1БwPw"́Nw !3}1hwݫw"YtgPCgPC`g&eNwĠe("aNw@"GDIC qm<"Ow,1T X"UOwXT l4CPC&eNwĠdv%    TT"LPp% ( " " FdXEMF+*@$??9DC*@$??9DC@ !!K@0 ee# ??9DC$$??9DC  Rp"Arial""AgG C`CBl""e"w !3"ww"́Nw !3"hwݫw"YtgPCgPC`g&eNwĠe("aNw@"GDIC qm<"Owp4T X"UOwXT l4CPC&eNwĠdv%    TZ1 L`) = p log% ( " " FdXEMF+*@$??9DC*@$??9DC@ !!K@0 ee# ??9DC$$??9DC  Rp"Arial""AgG UCApB$Bl""e"w !31@M (PL"́Nw !31hwݫw"YtgPCgPC`g&eNwĠe("aNw@"GDIC qm<"Ow7T X"UOwXT l4CPC&eNwĠdv%    TT[nw=LPb% ( " " FdXEMF+*@$??9DC*@$??9DC@ !!K@0 ee# ??9DC$$??9DC  Rp"Arial""AgG SC`BBl""e"w !3w=ww"́Nw !3w=hwݫw"YtgPCgPC`g&eNwĠe("aNw@"GDIC qm<"Ow(;T X"UOwXT l4CPC&eNwĠdv%    TXz1LPA !% ( " " 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FEMF+*@$??dDD*@$??dDD*@$??dDD*@$??dDD@ !!K@0 ee# ??dDD$$??dDD  Rp"Arial""AgG SCBl""e"w !3b"<"́Nw !3b"hwݫw"YtgPCgPC`g&eNwĠe("aNw@"GDIC qm<"OwPT "UOwT l4CPC&eNwĠdv%    T`g"LTlog% ( " " FdXEMF+*@$??dDD*@$??dDD@ !!K@0 ee# ??dDD$$??dDD  Rp"Arial""AgG 8`B@pB$Bl""e"w !3"DED"́Nw !3"hwݫw"YtgPCgPC`g&eNwĠe("aNw@"GDIC qm<"Ow TT X"UOwXT l4CPC&eNwĠdv%    TTL.LPb% ( " " FdXEMF+*@$??dDD*@$??dDD@ !!K@0 ee# ??dDD$$??dDD  Rp"Arial""AgG oB2CBl""e"w !3L."́Nw !3L.hwݫw"YtgPCgPC`g&eNwĠe("aNw@"GDIC qm<"OwdWT X"UOwXT l4CPC&eNwĠdv%    Tpgtl"LX(x) =  % ( " " FdXEMF+*@$??dDD*@$??dDD@ !!K@0 ee# ??dDD$$??dDD  Rp"Arial""AgG 8=[DCwCBl""e"w !3l"EDvi"́Nw !3l"hwݫw"YtgPCgPC`g&eNwĠe("aNw@"GDIC qm<"OwZT X"UOwXT l4CPC&eNwĠdv%    Tug/" Ld-----------% ( " " FEMF+*@$??dDD*@$??dDD*@$??dD (D*@$??dD (D@ !!K@0 ee# ??dD (D$$??dD (D  Rp"Arial""AgG ;[YCCBl""e"w !3"zi PM"́Nw !3"hwݫw"YtgPCgPC`g&eNwĠe("aNw@"GDIC qm<"OwLT "UOwT l4CPC&eNwĠdv%    T`"LTlog% ( " " FdXEMF+*@$??dD (D*@$??dD (D@ !!K@0 ee# ??dD (D$$??dD (D  Rp"Arial""AgG 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no x=!8'88=!:8"==8% ( " " FdXEMF+*@$??@DD*@$??@DD@ !!K@0   # ??@DD$$??@DD  Rp"ArialAócK\]DCC@Π v!^Vy(P \́w!^VyhwW8&XYtc ۲c `ҳc&ewawGDICXBw;OXUwXO  &ewdv%    TT<&" WMFC }KyLP-!% ( " " FdXEMF+*@$??@DD*@$??@DD@ !!K@0   # ??@DD$$??@DD  Rp"ArialAócK\eDDC@Π v!^y\́w!^yhwW8&XYtc ۲c `ҳc&ewawGDICXBw\OXUwXO  &ewdv%    T= Ky L`interceptdv=!8'88=!% ( " " FEMF+*@$??@DD*@$??@DC*@$??@DD*@$??@DC*@$??@DD*@$??@DD@ !!K@0   # ??@DD$$??@DD  Rp"ArialAócK8]pCC@Π v!^y^c (8\́w!^yhwW8&XYtc ۲c `ҳc&ewawGDICXBw!OUwO  &ewdv%    T|jkyL\y = 0, x5:88% ( " " FdXEMF+*@$??@DD*@$??@DD@ !!K@0   # ??@DD$$??@DD  Rp"ArialAócK]CCC@Π v!^y o+@\́w!^yhwW8&XYtc ۲c `ҳc&ewawGDICXBw%OXUwXO  &ewdv%    TTkOyLP-x!% ( " " FdXEMF+*@$??@DD*@$??@DD@ !!K@0   # ??@DD$$??@DD  Rp"ArialAócK^CDC@Π v!^Oy\́w!^OyhwW8&XYtc ۲c `ҳc&ewawGDICXBw)OXUwXO  &ewdv%    T kpyLaxis, is a horizontal asymptote88888=='2==!8885Y=!=!8% ( " " F EMF+*@$??@DD*@$??@DC*@$99*@$??@DC*@$??@DC*@$??@DC*@$??@DC*@$??@DC*@$??@DC*@$??@DC*@$??@DC*@$??@DC*@$??@D@D*@$??@D@D*@$??@D@D*@$??@DC*@$??@D@AD*@$??@D@AD*@$??@D@AD*@$??@DC*@$??@D@eD*@$??@DC*@$??@DD*@$??@DD*@$??@DD*@$??@DC*@$??@DD*@$??@DD*@$??@DD*@$??@DC*@$??@DD*@$??@DC*@$??@DD*@$??@DD*@$??@DD*@$??@DD*@$??@DD*@$??@DD*@$??@DD*@$??@DD*@$??@DD*@$??@DD*@$??@DD*@$??@DC*@$??@DD*@$??@DC*@$??@DD*@$??@DD*@$??@DD*@$??@DD*@$??@DD*@$??@DD*@$??@DD*@$??@DC*@$99*@$99+@ *@$99*@$99*@$99+@ *@$99*@$99*@$99+@ *@$99*@$99@@4F?@, H%J=J%J@( !b $$==_888% '%  V,3QWH#e?#e?UeHUe % % $$AA( F|pEMF+*@$99+@ *@$99*@$99*@$99+@ *@$99*@$99@@4F?@, H{J=J{J@$$==_888% %  V,3HX?X?YHY % % $$AA( F|pEMF+*@$99+@ *@$99*@$99*@$99+@ *@$99*@$99@@4F?@, H‚dJ=J‚dJ@$$==_888% %  V,3HL?L?LHL % % $$AA( F|pEMF+*@$99+@ *@$99*@$99*@$99+@ *@$99*@$99@&" WMFC ]@4F?@, H @J=J @J@$$==_888% %  V,3 He@?e@?@H@ % % $$AA( F|pEMF+*@$99+@ *@$99*@$99*@$99+@ *@$99*@$99@@4F?@ , H:zJ=J:zJ@ $$==_888% %  V,3@FH4?4?M4HM4 % % $$AA( F|pEMF+*@$99+@ *@$99*@$99*@$99+@ *@$99*@$99@ @4F?@ , H9J=J9J@  $$==_888% %  V,3H*?*?&+H&+ % % $$AA( F|pEMF+*@$99+@ *@$99*@$99*@$99+@ *@$99*@$99@ @4F?@ , HJ=JJ@  $$==_888% %  V,3H _? _?;_H;_ % % $$AA( F|pEMF+*@$99+@ *@$99*@$99*@$99+@ *@$99*@$99@@4F?@, HvJ=JvJ@$$==_888% %  V,3*1HR?R?RHR % % $$AA( F|pEMF+*@$99+@ *@$99*@$99*@$99+@ *@$99*@$99@@4F?@, H8RJ=J8RJ@$$==_888% %  V,3gmHF?F?FHF % % $$AA( F|pEMF+*@$99+@ *@$99*@$99*@$99+@ *@$99*@$99@@4F?@, H-J=J-J@$$==_888% %  V,3H@:?@:?r:Hr: % % $$AA( F|pEMF+*@$99+@ *@$99*@$99*@$99+@ *@$99*@$99@@4F?@, HVO J=JVO J@$$==_888% %  V,3H.?.?3.H3. % % $$AA( F|pEMF+*@$99+@ *@$99*@$99*@$99+@ *@$99*@$99@@4F?@, H{J=J{J@$$==_888% %  V,3H[?[?#\H#\ % % $$AA( F|pEMF+*@$99+@ *@$99*@$99*@$99+@ *@$99*@$99@@4F?@, H2mJ=J2mJ@$$==_888% %  V,3HO?O?OHO % % $$AA( F|pEMF+*@$99+@ *@$99*@$99*@$99+@ *@$99*@$99@@4F?@, Hn#IJ=Jn#IJ@$$==_888% %  V,36<HrC?rC?CHC % % $$AA( F|pEMF+*@$99+@ *@$99*@$99*@$99+@ *@$99*@$99@@4F?@, H$J=J$J@$$==_888% %  V,3rxH37?37?e7He7 % % $$AA( F|pEMF+*@$99+@ *@$99*@$99*@$99+@ *@$99*@$99@@4F?@, Hm+J=Jm+J@$$==_888% %  V,3 &Hb?b?HbHHb % % $$AA( F|pEMF+*@$99+@ *@$99*@$99*@$99+@ &" WMFC =*@$99*@$99@ @4F?@!, HJ=JJ@! $$==_888% %  V,3\bHU?U? VH V % % $$AA( F|pEMF+*@$99+@ *@$99*@$99*@$99+@ *@$99*@$99@"@4F?@#, HRm[J=JRm[J@#"$$==_888% %  V,3HI?I?IHI % % $$AA( F|pEMF+*@$99+@ *@$99*@$99*@$99+@ *@$99*@$99@$@4F?@%, H6J=J6J@%$$$==_888% %  V,3HN=?N=?=H= % % $$AA( F|pEMF+*@$99+@ *@$99*@$99*@$99+@ *@$99*@$99@&@4F?@', HdJ=JdJ@'&$$==_888% %  V,3H1?1?@1H@1 % % $$AA( F|pEMF+*@$99+@ *@$99*@$99*@$99+@ *@$99*@$99@(@4F?@), H]PJ=J]PJ@)($$==_888% %  V,3H;h?;h?mhHmh % % $$AA( FEMF+*@$99+@ *@$99*@$99*@$99+@ *@$99*@$99@*>F?d @ A?( ?@@@+, XIXJX II@+*$$==_888% %  [L ) =!YhG!@,!@,!Yh=!Yh ,!*s", , % % $$AA( " FEMF+*@$99+@ *@$99*@$99*@$99+@ *@$99*@$99*@$99+@ *@$99*@$99*@$?? CD*@$?? CD*@$?? CD*@$?? CD*@$?? CD*@$99*@$?? CD*@$?? CD*@$?? CD@ !!K@0   # ?? CD$$?? CD  Rp"ArialAócKM BB@??FAILPp!^hwW8&XYtc ۲c `ҳc&ewawGDICXBwlOUwO9wJw&ewdv%    TTQALPy"% ( " " FEMF+*@$?? CD*@$?? CD*@$99*@$?? CD*@$?? CD*@$?? CD*@$?? CD*@$?? CD*@$99*@$99+@ *@$99*@$99*@$99+@ *@$99*@$99*@$??`~DD*@$??`~DD*@$??`~DD*@$??`~DD*@$??`~DD*@$99*@$??`~DD*@$??`~DD*@$??`~DD@ !!K@0   # ??`~DD$$??`~DD  Rp"ArialAócK BB@Π v!^Ai\́w!^AhwW8&XYtc ۲c `ҳc&ewawGDICXBw\rOUwO  &ewdv%    TTHALPx%% ( " " FEMF+*@$??`~DD*@$??`~DD*@$99*@$??`~DD*@$??`~DD*@$??`~DD*@$??`~DD*@$??`~DD*@$99*@$99+@ *@$99*@$99@,>F?d @ A?( ?@@@-, H.RJ=J.RJ@-,!b ! $$==_888% %  [L04 a.>a.>bba=ha?Ib=*c=ha % % $$AA" ( FEMF+*@$99+@ *@$99*@$99*@$99+@ *@$99*@$99*@$&" WMFC 99+@ *@$99*@$99@.@4F?@/, >JI>JHJ@/.$$==_888% %  V,?*?Oh?Oh?* % % $$AA( F|pEMF+*@$99+@ *@$99*@$99*@$99+@ *@$99*@$99@0@4F?@1, HIHHJ@10$$==_888% %  V,ag^*^Oh,Oh,* % % $$AA( F|pEMF+*@$99+@ *@$99*@$99*@$99+@ *@$99*@$99@2@4F?@3, lJIlJHJ@32$$==_888% %  V,-*-Oh-Oh-* % % $$AA( F|pEMF+*@$99+@ *@$99*@$99*@$99+@ *@$99*@$99@4@4F?@5, 0+JI0+JHJ@54$$==_888% %  V,9*9Oh9Oh9* % % $$AA( F|pEMF+*@$99+@ *@$99*@$99*@$99+@ *@$99*@$99@6@4F?@7, IIIHJ@76$$==_888% %  V,*Oh_Oh_* % % $$AA( F|pEMF+*@$99+@ *@$99*@$99*@$99+@ *@$99*@$99@8@4F?@9, IIIHJ@98$$==_888% %  V,v|'*'Ohu'Ohu'* % % $$AA( F|pEMF+*@$99+@ *@$99*@$99*@$99+@ *@$99*@$99@:@4F?@;, LJILJHJ@;:$$==_888% %  V,8>3*3Oh3Oh3* % % $$AA( F|pEMF+*@$99+@ *@$99*@$99*@$99+@ *@$99*@$99@<@4F?@=, @HI@HHJ@=<$$==_888% %  V,a *a Oh/ Oh/ * % % $$AA( F|pEMF+*@$99+@ *@$99*@$99*@$99+@ *@$99*@$99@>@4F?@?,  7II 7IHJ@?>$$==_888% %  V,q*qOh?Oh?* % % $$AA( F|pEMF+*@$99+@ *@$99*@$99*@$99+@ *@$99*@$99@@4F?@, 7II7IHJ@$$==_888% %  V,SZ*OhOOhO* % % $$AA( F|pEMF+*@$99+@ *@$99*@$99*@$99+@ *@$99*@$99@@4F?@, XIIXIHJ@$$==_888% %  V,!*!Oho!Oho!* % % $$AA( F|pEMF+*@$99+@ *@$99*@$99*@$99+@ *@$99*@$99@@4F?@, JIJHJ@$$==_888% %  V,0*0Oh0Oh0* % % $$AA( F|pEMF+*@$99+@ *@$99*@$99*@$99+@ *@$99*@$99@@4F?@, 4JI4JHJ@$$==_888% %  V,<*|J`-IWJ5I1J-I<>J`bIJЖIJ0fIfJhIJiI\JILcJmIDJVI%X$Y$nZ|$ZK$w[$[#U\#\#]P#^]#]"]"p^%"^!W_g!_!(` z`\ ``.aZapaaDaaaaawa#aaZaaTaa^abb1b2b % % $$AA( " FH<EMF+*@$99+@ *@$99*@$99*@$99+@ +@ @ !@ @ 4@ @ *@$??!@ @ ! 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