1 / 17

EXPONENTIAL FUNCTIONS

EXPONENTIAL FUNCTIONS. 1 + r > 1. A quantity is growing exponentially if it increases by the same percent in each unit of time. This is called exponential growth. C is the initial amount. (1 + r ) is the growth factor. y = C (1 + r ) t. r is the growth rate.

madge
Télécharger la présentation

EXPONENTIAL FUNCTIONS

An Image/Link below is provided (as is) to download presentation Download Policy: Content on the Website is provided to you AS IS for your information and personal use and may not be sold / licensed / shared on other websites without getting consent from its author. Content is provided to you AS IS for your information and personal use only. Download presentation by click this link. While downloading, if for some reason you are not able to download a presentation, the publisher may have deleted the file from their server. During download, if you can't get a presentation, the file might be deleted by the publisher.

E N D

Presentation Transcript


  1. EXPONENTIAL FUNCTIONS

  2. 1 +r > 1 A quantity is growing exponentially if it increases by the same percent in each unit of time. This is called exponential growth. C is the initial amount (1 + r) is the growth factor y = C(1 + r)t r is the growth rate t is the time period

  3. You deposit $500 in an account that pays 8% annual interest compounded yearly. What is the account balance after 6 years? y = C(1 + r)t y = 500(1 + 0.08)6 Note: always convert the percentage into a decimal. = 500(1.08)6  793.437 The balance after 6 years will be about $793.44.

  4. Graphically 793.437 t The x-axis usually represents time

  5. A savings account certificate of $1000 pays 6.5% annual interest compounded yearly. What is the balance when the certificate matures after 5 years? y = C(1 + r)t y = 1000(1 + 0.065)5 = 1000(1.065)5  1370.09 The balance of the certificate after 5 years is about $1370.09.

  6. A newly hatched channel catfish typically weighs about 0.3 grams. During the first six weeks of life, its growth is approximately exponential, increasing by about 10% each day. Find the weight at the end of six weeks. y = C(1 + r)t y = 0.3(1 + .10)42 Note: since the problem stated that the rate of increase was daily, we must let t be represented in days. y = 0.3(1.1)42 y  16.42910977 The weight is about 16.4 grams.

  7. An experiment started with 100 bacteria. They double in number every hour. Find the number of bacteria after 8 hours. y = C(1 + r)t y = 100(1 + 1)8 Note: since the bacteria doubled every hour, they rate of increase was 100% or 1. y = 100(2)8 y = 25,600 The number of bacteria after 8 hours is 25,600.

  8. A population of 20 rabbits is released into a wild-life region. The population triples each year for 5 years. • What is the percent increase each year? • What is the population after 5 years? Solution a. The population triples each year, so the growth factor is 3. 1 + r = 3 • So, the growth factor rate, r, is 2 and the percent increase each year is 200%.

  9. A population of 20 rabbits is released into a wild-life region. The population triples each year for 5 years. • What is the percent increase each year? • What is the population after 5 years? Solution There will be about 4860 rabbits after 5 years. • P = C(1 + r)t • = 20(1 + 2)5 • = 20(3)5 • = 4860

  10. Graphically 4860 t

  11. A company starts with 50 employees and after one year has 75. It increases the employees at the same rate every year for 4 years. • What is the percent of increase each year? • Find the number of employees after 4 years? solution • After one year you have 75 employees which means there • are 25 more employees than the company started with. • Since there were 50 employees and 25 were added within • one year, the percent increase must be 50%. Try this example on a sheet of paper before clicking.

  12. A company starts with 50 employees and after one year has 75. It increases the employees at the same rate every year for 4 years. • What is the percent of increase each year? • Find the number of employees after 4 years? solution There will be 253 employees at the end of 4 years. • P = C(1 + r)t • = 50(1 + 0.5)4 • = 50(1.5)4 •  253

  13. 0 < 1 – r < 1 A quantity is decaying exponentially if it decreases by the same percent in each unit of time. This is called exponential decay. C is the initial amount (1 – r) is the decay factor y = C(1 r)t r is the decay rate t is the time period

  14. From 1982 through 1997, the purchasing power of a dollar decreased by about 3.5% per year. Using 1982 as the base for comparison, what was the purchasing power of a dollar in 1997? solution Let y represent the purchasing power and let t = 0 represent the year 1982. The initial amount is $1. Use an exponential decay model (Notice the dollar decreased, therefore, we use a decay model). y = C(1 –r)t Because 1997 is 15 years after 1982, substitute 15 for t. y = 1(1 – 0.035)15  0.59

  15. You bought a used car for $18,000. The value of the car will be less each year because of depreciation. The car depreciates (loses value) at the rate of 12% per year. • Write an exponential decay model to represent this situation. • Estimate the value of your car in 8 years. solution • The initial value C is $18,000. The decay rate r is 0.12. Let • tbe the age of the car in years. y = 18,000(1 – 0.12)t y = 18,000(0.88)t

  16. You bought a used car for $18,000. The value of the car will be less each year because of depreciation. The car depreciates (loses value) at the rate of 12% per year. • Write an exponential decay model to represent this situation. • Estimate the value of your car in 8 years. solution b. To find the value in 8 years, substitute 8 for t. y = 18,000(0.88)8 6473 The value of your car in 8 years will be about $6473.

  17. Checkpoint A house costs $140,000 in the year 2002. It appreciates 4.5% annually. • Write a model to represent the value of the house • after t years. • b. Find the value of the house after 20 years. The amount of a particular medicine in the body decreases at a rate of 40% an hour. You are given 250mg. • Write a model to represent the amount of medicine • in the body after t hours. • b. Find the amount of medicine after 5 hours.

More Related