Exponential Growth and Decay 30 April 2012 Exponential Growth and Decay 30 April 2012 1/24
This week we’ll talk about a few situations which behave mathematically like compound interest. They include population growth, radioactive decay, and google map zooming. Exponential Growth and Decay 30 April 2012 2/24
This week we’ll talk about a few situations which behave mathematically like compound interest. They include population growth, radioactive decay, and google map zooming. To help make the connection with interest rates, we discuss one more idea of compound interest. Exponential Growth and Decay 30 April 2012 2/24
If you have money in a savings account, then the amount of interest you make per period is proportional to the amount of money in the account. The constant ratio of interest to principal is the interest rate paid to you. Exponential Growth and Decay 30 April 2012 3/24
If you have money in a savings account, then the amount of interest you make per period is proportional to the amount of money in the account. The constant ratio of interest to principal is the interest rate paid to you. A consequence of this proportionality is that if you double how much money you have in the account, you’ll double the interest you get. Exponential Growth and Decay 30 April 2012 3/24
Continuous Compounding Let’s suppose we invest $100 in a savings account paying 5% per year. If the interest is compounded yearly, after one year we’d have $100 · (1 + . 05) = $105 . Exponential Growth and Decay 30 April 2012 4/24
Continuous Compounding Let’s suppose we invest $100 in a savings account paying 5% per year. If the interest is compounded yearly, after one year we’d have $100 · (1 + . 05) = $105 . If interest is compounded quarterly, then we get 5% / 4 = 1 . 25% per quarter. There are 4 quarters in a year, so after one year we’d have $100 · (1 + . 05 / 4) 4 = $105 . 09 Exponential Growth and Decay 30 April 2012 4/24
Continuous Compounding Let’s suppose we invest $100 in a savings account paying 5% per year. If the interest is compounded yearly, after one year we’d have $100 · (1 + . 05) = $105 . If interest is compounded quarterly, then we get 5% / 4 = 1 . 25% per quarter. There are 4 quarters in a year, so after one year we’d have $100 · (1 + . 05 / 4) 4 = $105 . 09 If interest is compounded monthly, after one year we’d have $100 · (1 + . 05 / 12) 12 = $105 . 12 Exponential Growth and Decay 30 April 2012 4/24
If we compound daily, after one year we’d have $100 · (1 + . 05 / 365) 365 = $105 . 13 Exponential Growth and Decay 30 April 2012 5/24
If we compound daily, after one year we’d have $100 · (1 + . 05 / 365) 365 = $105 . 13 If we compound hourly, after one year we’d have $100 · (1 + . 05 / 8760) 8760 = $105 . 13 Exponential Growth and Decay 30 April 2012 5/24
If we compound daily, after one year we’d have $100 · (1 + . 05 / 365) 365 = $105 . 13 If we compound hourly, after one year we’d have $100 · (1 + . 05 / 8760) 8760 = $105 . 13 By compounding more and more frequently, we are increasing how much we get, but only by a little. Exponential Growth and Decay 30 April 2012 5/24
If we compound daily, after one year we’d have $100 · (1 + . 05 / 365) 365 = $105 . 13 If we compound hourly, after one year we’d have $100 · (1 + . 05 / 8760) 8760 = $105 . 13 By compounding more and more frequently, we are increasing how much we get, but only by a little. The most extreme notion is called compounding continuously. The name gives a rough idea of what this means. Exponential Growth and Decay 30 April 2012 5/24
There is a number, usually denoted by e , whose value is approximately 2 . 7, which helps to calculate continuous compounding. If you invest P 0 in an account paying r % per year, compounded continuously.after t years, the amount of money you’ll have is P 0 · e rt Exponential Growth and Decay 30 April 2012 6/24
There is a number, usually denoted by e , whose value is approximately 2 . 7, which helps to calculate continuous compounding. If you invest P 0 in an account paying r % per year, compounded continuously.after t years, the amount of money you’ll have is P 0 · e rt On a calculator, or a spreadsheet, the function e x is often written exp( x ). Some calculators will have an exp button, and some will have a button for the number e . Exponential Growth and Decay 30 April 2012 6/24
When a quantity grows at a rate proportional to its size, then the equation P = P 0 · e rt governs the size, where P 0 represents the size at some initial time, r is the growth rate, and t is the amount of time past the initial time. Exponential Growth and Decay 30 April 2012 7/24
When a quantity grows at a rate proportional to its size, then the equation P = P 0 · e rt governs the size, where P 0 represents the size at some initial time, r is the growth rate, and t is the amount of time past the initial time. The situation of something growing proportionally to its size occurs in a number of situations. When this happens the quantity is said to grow exponentially. Exponential Growth and Decay 30 April 2012 7/24
For example, we can apply this idea to compound interest, where P 0 is our initial deposit, r is our annual interest rate (compounding continuously) and t is the number of years we’ve left the money in the bank. Exponential Growth and Decay 30 April 2012 8/24
For example, we can apply this idea to compound interest, where P 0 is our initial deposit, r is our annual interest rate (compounding continuously) and t is the number of years we’ve left the money in the bank. For example, if we deposit $100 in the bank paying 5% compounded continuously, after 1 year we’ll have $100 · e 0 . 05 · 1 = $100 · e 0 . 05 = $105 . 13 Exponential Growth and Decay 30 April 2012 8/24
For example, we can apply this idea to compound interest, where P 0 is our initial deposit, r is our annual interest rate (compounding continuously) and t is the number of years we’ve left the money in the bank. For example, if we deposit $100 in the bank paying 5% compounded continuously, after 1 year we’ll have $100 · e 0 . 05 · 1 = $100 · e 0 . 05 = $105 . 13 Compounding 5% continuously is equivalent to 5.13% compounded yearly. Exponential Growth and Decay 30 April 2012 8/24
Clicker Question If you deposit $100 in a savings account which pays an annual rate of 6% interest, compounded continuously, how much money will you have in 3 years? To enter the expression P 0 · e rt on a calculator, the following steps most likely will work ( ) = P 0 e r t × ∧ × or x y ( ) = P 0 e r t × × Exponential Growth and Decay 30 April 2012 9/24
Answer The amount you’ll have in 3 years is $100 · e 0 . 06 · 3 = $100 · e 0 . 18 = $119 . 72 Exponential Growth and Decay 30 April 2012 10/24
Answer The amount you’ll have in 3 years is $100 · e 0 . 06 · 3 = $100 · e 0 . 18 = $119 . 72 The webpage http://ultimatecalculators.com does both standard compound interest and continuous compounding. Exponential Growth and Decay 30 April 2012 10/24
Population Growth A first assumption about population growth is often that the rate of births and deaths is proportional to the size of the population. That is, if the population doubles, then the number of births and deaths each double. The growth rate is then proportional to the size of the population. Exponential Growth and Decay 30 April 2012 11/24
Population Growth A first assumption about population growth is often that the rate of births and deaths is proportional to the size of the population. That is, if the population doubles, then the number of births and deaths each double. The growth rate is then proportional to the size of the population. This is a reasonable assumption for many populations, including human populations, although it can be simplistic, especially when resources are scarce. Exponential Growth and Decay 30 April 2012 11/24
Population Growth A first assumption about population growth is often that the rate of births and deaths is proportional to the size of the population. That is, if the population doubles, then the number of births and deaths each double. The growth rate is then proportional to the size of the population. This is a reasonable assumption for many populations, including human populations, although it can be simplistic, especially when resources are scarce. Assuming this model, population would be governed by the equation P = P 0 · e rt where P 0 is the population at some given time, r is the annual population growth rate, and t is the number of years past the given time. Exponential Growth and Decay 30 April 2012 11/24
An Example In 1990 the world population was estimated to be 5.3 billion and increasing at the rate of 1.7% per year. What would the world population be in 2000? Exponential Growth and Decay 30 April 2012 12/24
An Example In 1990 the world population was estimated to be 5.3 billion and increasing at the rate of 1.7% per year. What would the world population be in 2000? We can view the initial population (in 1990) as 5.3 (measured in billions). If t is the number of years past 1990, then the world population, if it grows at 1 . 7% per year, would be given by the equation P = 5 . 3 · e . 017 t Exponential Growth and Decay 30 April 2012 12/24
Recommend
More recommend
