4 1 Exponential Functions; Compound Interest

This example demonstrates how the formula for compound interest can be used to derive the power series definition of the exponential function. The power series of the exponential function is shown below. Suppose that you can invest money at 9% interest compounded daily. How much should you invest in order to have $5,000 in 7 years?

Given that carbon-14 has a half-life of 5,730 years, estimate the age of the papyrus. A new MP3 player was purchased for $320 and in 1 year it was selling used online for $210. If the value continues to decrease exponentially at this rate, determine the value of the MP3 player 3 years after it was purchased. In 2000, the population of the United States was estimated to be 282 million people and in 2010 the estimate was 309 million people. If the population of the United States grows exponentially, estimate the population in 2020. Estimate the time it will take for the population to reach 25,000 cells.

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Start with the compound interest formula shown below. In the discussion that follows, we will compute the account balance at the end of each month. Since one month is \(\frac\) of a year, \(P(\frac)\) represents the balance at the end of the first month, \(P(\frac)\) represents the balance at the end of the second month, etc. If you already have a bank account or if you plan to have one in the future, then this tutorial is a must see!

How much should you invest in order to have $17,000 in 13 years? Suppose that you can invest money at 8% interest compounded continuously. How much should you invest in order to have $10,000 in 6 years?

Break Time!

The Rhind Mathematical Papyrus is considered to be the best example of Egyptian mathematics found to date. This ancient papyrus was found to contain 64% of the carbon-14 normally found in papyrus.

The frequency could be yearly, half-yearly, quarterly, monthly, weekly, daily, or continuously . As you review both formulas shown below, notice how each of these formulas represent exponential models. Compound interest can significantly boost investment returns over the long term.

Annual equivalent rate

Write each answer as a dollar amount rounded to the nearest cent. It’s quite interesting to note the effect of different interest rates on the amount earned. Richard Witt’s book Arithmeticall Questions, published in 1613, was a landmark in the history of compound interest. It was wholly devoted to the subject , whereas previous writers had usually treated compound interest briefly in just one chapter in a mathematical textbook.

As the frequency of compounding interest increases, so does the accumulated balance. Let’s take that Exponential Functions: Compound Interest $2500 and figure the amount you would have after 40 years at each of the following interest rates.

continuously Compounded Interest

More frequent compounding of interest is beneficial to the investor or creditor. The younger you are, the more that compound interest will earn you across your life span. Anderson is CPA, doctor of accounting, and an accounting https://simple-accounting.org/ and finance professor who has been working in the accounting and finance industries for more than 20 years. Her expertise covers a wide range of accounting, corporate finance, taxes, lending, and personal finance areas.

It is fairly simple and also allows inputs of monthly additional deposits to the principal, which is helpful for calculating earnings when additional monthly savings are being deposited. Consider a mutual fund investment opened with an initial $5,000 and an annual addition of $2,400. With an average annual return of 12% over 30 years, the future value of the fund is $798,500.

Exponential Growth, Decay, and Compound Interest Notes & Practice

Make a bar graph in excel for investing $10,000 once a year for 30 years. If you do not have excel, you can download either Open Office or Gnumeric, both are free spreadsheet softwares. Assume the interest is compounded continuously at a rate of 6%. This concludes our study of the exponential and logarithmic function.

Consider the element Uranium that undergoes radioactive decay at the rate of 12% of the material per year. This decay rate, 12%, is independent of the amount of material present. Find the time taken for half of Uranium to disappear, or the half life. While the time required for one trait to become prevalant in a population is greater than one lifetime, it is often no more than a few hundred generations. This tells us that small differences among individuals when compounded by natural selection over time really do add up to the large differences we see among species.