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Exponential Functions Copyright Scott Storla 2014 Power functions vs. Exponential functions Power function Exponential function y x1 y 2x y x2 y 1 / 3 y x3 y 10 x y x1/ 2 y ex y x a where a is a real number. x y a x where a is a real number. (a 0, a 1) Copyright Scott Storla 2014 Definition – Exponential Function x An exponential function has the form f x Cb where b is a real number greater than 0 and not equal to 1 and C is a real number not equal to 0. Copyright Scott Storla 2014 The formula for continuous compounding, is a specific example of the more general exponential growth and decay formula. A Pe Future amount at time t. rt A t A0e Rate of growth or decay. kt Initial amount Copyright Scott Storla 2014 Vocabulary for exponential functions Copyright Scott Storla 2014 If as the value of x increases the value of y increases you have exponential growth. If as the value of x increases the value of y decreases you have exponential decay. Copyright Scott Storla 2014 Horizontal asymptote Let y denote a function. If as x or as x the values of y approach some fixed number L, then the line y L is a horizontal asymptote of the graph of y. The horizontal asymptote is y 0 Copyright Scott Storla 2014 Discuss whether the graph shows exponential growth or decay, the intercepts, the domain, the horizontal asymptote and the range. Copyright Scott Storla 2014 Discuss whether the graph shows exponential growth or decay, the intercepts, the domain, the horizontal asymptote and the range. Copyright Scott Storla 2014 Discuss whether the graph shows exponential growth or decay, the intercepts, the domain, the horizontal asymptote and the range. Copyright Scott Storla 2014 Discuss whether the graph shows exponential growth or decay, the intercepts, the domain, the horizontal asymptote and the range. Copyright Scott Storla 2014 A procedure to help graph exponential functions Procedure – Graphing an Exponential Function 1) The domain for the exponential function is all real numbers. 2) Find any x and y intercepts. 3) Find a few points on the graph. Make sure to include both positive and negative exponents. 4) Estimate the horizontal asymptote. 5) Draw the graph. Copyright Scott Storla 2014 Graph the exponential function. y 2x y e y e x x 1 y 2 x 3 Copyright Scott Storla 2014 The formula for continuous compounding, is a specific example of the more general exponential growth and decay formula. A Pe Future amount at time t. rt A t A0e Rate of growth or decay. kt Initial amount Copyright Scott Storla 2014 A t A0e kt A colony of bacteria is experiencing exponential growth which can be modeled by the function 𝑁 𝑡 = 50𝑒 0.04𝑡 where N is measured in grams and t is the number of hours. a) What is the general meaning of the y-intercept? b) What is the specific meaning of the y-intercept? c) What does the growth or decay constant imply about the future? d) What is the weight after 1 day? e) What is the doubling time for the bacteria? f) How long will it take for the bacteria to reach 1,000 grams? g) Compare the average rate of change for 10 hours to 11 hours with the average rate of change for 50 hours to 51 hours. Copyright Scott Storla 2014 A t A0e kt The healing time for a wound can be represented by the function 𝐴 𝑡 = 85𝑒 −0.38𝑡 where t is the number of days since healing began and A is the area of the wound in 𝑚𝑚2 . a) What is the general meaning of the y-intercept? b) What is the specific meaning of the y-intercept? c) What does the growth or decay constant imply about the future? d) When will the area of the wound be 20% of its initial area? e) What will the area of the wound be after 1 week? f) Find and discuss the meaning of the horizontal asymptote? g) Compare the average rate of change for 2 days to 3 days with the average rate of change for 7 days to 8 days. Copyright Scott Storla 2014 Assume the growth constant, k, for the population of the United States is 0.011. If the population was 281.4 million in 2000 estimate the population in 2020. A sample of the paint used in a cave painting in France is found to have lost 82% of its original carbon-14. Estimate the age of the painting. The value of k for Carbon-14 is – 0.000121. A cup of coffee contains approximately 96 mg of caffeine. When you drink the coffee, the caffeine is absorbed into the bloodstream and is eventually metabolized by the body. If the rate of decay is – 0.14, how many hours does it take for the amount of caffeine to be reduced to 12 mg? Copyright Scott Storla 2014 Carbon-14 is often used to date objects that were alive in the past. Between 1947 and 1956 the Dead Sea scrolls were discovered in 11 caves along the northwest shore of the Dead Sea. If the Dead Sea scrolls are authentic then they should date to around 2000 years old. If 78.5% of the original carbon-14 was left, was their age appropriate to being authentic? The value of k for Carbon-14 is -0.000121. Over the last 60 years a number of North American snail species have become extinct. If there are currently 650 species, and the extinction follows exponential decay, how many species were there originally? The value of k is -0.0014. Copyright Scott Storla 2014 Finding the rate of growth or decay Radioactive half-life is the time necessary for a radioactive substance to decay to one-half the original amount. Carbon-14 has a half life of 5715 years. Find the decay rate for cabon-14. From 1970 to 1980 the population of the United States in millions went from 203.3 to 226.5. Find the growth rate. $17,000 grows to $23,000 in 8.5 years. Find the growth rate. The half-life of silver-110 is 24.6 seconds. How long will it take for only 3% of the original sample of silver-110 to remain? From 1980 to 1990 the population of the United States in millions went from 226.5 to 248.7. Estimate the population in 2010. 200 bacteria grow to 600 bacteria in 2 hours. Find the number of bacteria in 2 days. Dinosaur bones were dated using potassium-40 which has a half-life of approximately 1.31 billion years. Analysis of certain rocks surrounding the bones found that 94% of the original potassium-40 was still present. Estimate the age of the bones. $350,000 is currently in an account which increased its value by 40% over the last 5 years. Estimate the number of years before there is $1,000,000 in the account. Copyright Scott Storla 2014 Compound Interest Copyright Scott Storla 2014 Compound Interest Compound interest occurs when interest is reinvested as principal and itself begins earning interest. The formula for yearly compounding is A P 1 r . t Find the amount at the end of five years if $8,000 is invested at 2.15%. Copyright Scott Storla 2014 Future Value The future value formula for compound interest is A P 1 r n nt The Vocabulary of Compounding Periods Word SemiYearly Quarterly Monthly annual Compounding periods per 1 2 4 12 year (n) Copyright Scott Storla 2014 Daily 360 or 365 A P 1 r n nt $30,000 is invested at 2% compounded quarterly. How much is the investment worth after 30 years? A 20 year old student inherits $10,000 and decides to invest the money for retirement at 5% annual interest compounded monthly. The student retires at 70. How much is the original $10,000 worth? An investment of $7,500 loses 4% per year for 3 years. If the investment was compounded semi-annually how much is the investment worth at the end of three years? Copyright Scott Storla 2014 Present Value A P 1 r A n P 1 r n nt nt A 1 r n nt Copyright Scott Storla 2014 P A 1 r n nt P In five years a couple wants $50,000 for a down payment. How much should they save today at 3% compounded monthly for that to happen? A great grandparent wanted to leave you a million dollars. How much would they need to have saved 150 years ago at 5% compounded quarterly for the account to hold a million dollars today? Copyright Scott Storla 2014 Property – The Power Property of Logarithms English: The logarithm of an argument written in exponential form can be written as two factors. One factor is the exponent of the argument. The other factor is the logarithm of the base of the argument. Example: ln2x x ln2 Algebra: logb xn n logb x x 0 and b 1 Copyright Scott Storla 2014 A P 1 r n nt How long will it take for $75,000 to become $92,000 at 4.3% compounded monthly? Find the time it takes at 1.7% compounded daily for $15,000 to become 18,000. Copyright Scott Storla 2014 Continuous Compounding A P 1 r n nt Let n 1 n n 360 1 1,000,000 360 1 1 1 1 1 1 n 1 360 Answer 2 2.714516 A Pe rt Copyright Scott Storla 2014 1000000 1 1 1000000 2.7182804 A Pe rt In 1800 Great Great Grandma put $50 in an account paying 5.5% interest compounded continuously? How much was in the account in 1900? How much was in the account in 2000? How much will be in the account this year? I would like to have $25,000 in an account in 17 years. What rate would I need to make this happen if I have $12,800 to invest? Assume continuous compounding. Copyright Scott Storla 2014 To retire in 14 years I want to have $1,500,000 in savings. If I’m able to get 4.2% compounded continuously how much do I need to have saved by today to make this happen? Upon retiring I don’t want to use any of my principal. How much in interest will I be able to live on each subsequent year? Assume 4.2% compounded continuously. Unfortunately . Copyright Scott Storla 2014