Example 1

India's population was 1.14 billion in the year 2008 and is growing by about 1.34% each year. Write an exponential function for India's population, and use it to predict the population in 2020. Using 2008 as our starting time ( ), our initial population will be 1.14 billion. Since the percent growth rate was 1.34%, our value for is 0.0134. Using the basic formula for exponential growth we can write the formula, f(t) = 1.14(1 + 0.0134)^t To estimate the population in 2020, we evaluate the function at , since 2020 is 12 years after 2008:

Example 2

A certificate of deposit (CD) is a type of savings account offered by banks, typically offering a higher interest rate in return for a fixed length of time you will leave your money invested. If a bank offers a 24 month CD with an annual interest rate of 1.2% compounded monthly, how much will a $1000 investment grow to over those 24 months? First, we must notice that the interest rate is an annual rate, but is compounded monthly, meaning interest is calculated and added to the account monthly. To find the monthly interest rate, we divide the annual rate of 1.2% by 12 since there are 12 months in a year: 1.2%/12 = 0.1%. Each month we will earn 0.1% interest. From this, we can set up an exponential function, with our initial amount of $1000 and a growth rate of , and our input measured in months: [latex-display]\displaystyle{f{{({m})}}}={1000}{({1}+\frac{{0.012}}{{12}})}^{{m}}={1000}{({1.001})}^{{m}}[/latex-display] After 24 months, the account will have grown to .

Example 3

Bismuth-210 is an isotope that radioactively decays by about 13% each day, meaning 13% of the remaining Bismuth-210 transforms into another atom (polonium-210 in this case) each day. If you begin with 100 mg of Bismuth-210, how much remains after one week? With radioactive decay, instead of the quantity increasing at a percent rate, the quantity is decreasing at a percent rate. Our initial quantity is mg, and our growth rate will be negative 13%, since we are decreasing: . This gives the equation This can also be explained by recognizing that if 13% decays, then 87% remains. After one week, 7 days, the quantity remaining would be mg of Bismuth-210 remains.

Euler's Number: e

Because e is often used as the base of an exponential, most scientific and graphing calculators have a button that can calculate powers of e, usually labeled exp(x). Some computer software instead defines a function exp(x), where exp(x) = . Since calculus studies continuous change, we will almost always use the -based form of exponential equations in this course.

Continuous Growth Formula

Continuous growth can be calculated using the formula where

• is the starting amount,
• is the continuous growth rate.

Example 4

Radon-222 decays at a continuous rate of 17.3% per day. How much will 100mg of Radon-222 decay to in 3 days? Since we are given a continuous decay rate, we use the continuous growth formula. Since the substance is decaying, we know the growth rate will be negative: , mg of Radon-222 will remain.

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Shana Calaway, Dale Hoffman, and David Lippman, Business Calculus, " 1.7: Exponential Functions," licensed under a CC-BY license.