# 6.1 Exponential functions  (Page 7/16)

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An initial investment of $100,000 at 12% interest is compounded weekly (use 52 weeks in a year). What will the investment be worth in 30 years? about$3,644,675.88

## Using the compound interest formula to solve for the principal

A 529 Plan is a college-savings plan that allows relatives to invest money to pay for a child’s future college tuition; the account grows tax-free. Lily wants to set up a 529 account for her new granddaughter and wants the account to grow to $40,000 over 18 years. She believes the account will earn 6% compounded semi-annually (twice a year). To the nearest dollar, how much will Lily need to invest in the account now? The nominal interest rate is 6%, so $\text{\hspace{0.17em}}r=0.06.\text{\hspace{0.17em}}$ Interest is compounded twice a year, so $\text{\hspace{0.17em}}k=2.$ We want to find the initial investment, $\text{\hspace{0.17em}}P,$ needed so that the value of the account will be worth$40,000 in $\text{\hspace{0.17em}}18\text{\hspace{0.17em}}$ years. Substitute the given values into the compound interest formula, and solve for $\text{\hspace{0.17em}}P.$

Lily will need to invest $13,801 to have$40,000 in 18 years.

Refer to [link] . To the nearest dollar, how much would Lily need to invest if the account is compounded quarterly?

$13,693 ## Evaluating functions with base e As we saw earlier, the amount earned on an account increases as the compounding frequency increases. [link] shows that the increase from annual to semi-annual compounding is larger than the increase from monthly to daily compounding. This might lead us to ask whether this pattern will continue. Examine the value of$1 invested at 100% interest for 1 year, compounded at various frequencies, listed in [link] .

Frequency $A\left(t\right)={\left(1+\frac{1}{n}\right)}^{n}$ Value
Annually ${\left(1+\frac{1}{1}\right)}^{1}$ $2 Semiannually ${\left(1+\frac{1}{2}\right)}^{2}$$2.25
Quarterly ${\left(1+\frac{1}{4}\right)}^{4}$ $2.441406 Monthly ${\left(1+\frac{1}{12}\right)}^{12}$$2.613035
Daily ${\left(1+\frac{1}{365}\right)}^{365}$ $2.714567 Hourly ${\left(1+\frac{1}{\text{8766}}\right)}^{\text{8766}}$$2.718127
Once per minute ${\left(1+\frac{1}{\text{525960}}\right)}^{\text{525960}}$ $2.718279 Once per second ${\left(1+\frac{1}{31557600}\right)}^{31557600}$$2.718282

These values appear to be approaching a limit as $\text{\hspace{0.17em}}n\text{\hspace{0.17em}}$ increases without bound. In fact, as $\text{\hspace{0.17em}}n\text{\hspace{0.17em}}$ gets larger and larger, the expression $\text{\hspace{0.17em}}{\left(1+\frac{1}{n}\right)}^{n}\text{\hspace{0.17em}}$ approaches a number used so frequently in mathematics that it has its own name: the letter $\text{\hspace{0.17em}}e.\text{\hspace{0.17em}}$ This value is an irrational number, which means that its decimal expansion goes on forever without repeating. Its approximation to six decimal places is shown below.

## The number e

The letter e represents the irrational number

The letter e is used as a base for many real-world exponential models. To work with base e , we use the approximation, $\text{\hspace{0.17em}}e\approx 2.718282.\text{\hspace{0.17em}}$ The constant was named by the Swiss mathematician Leonhard Euler (1707–1783) who first investigated and discovered many of its properties.

## Using a calculator to find powers of e

Calculate $\text{\hspace{0.17em}}{e}^{3.14}.\text{\hspace{0.17em}}$ Round to five decimal places.

On a calculator, press the button labeled $\text{\hspace{0.17em}}\left[{e}^{x}\right].\text{\hspace{0.17em}}$ The window shows Type $\text{\hspace{0.17em}}3.14\text{\hspace{0.17em}}$ and then close parenthesis, $\text{\hspace{0.17em}}\left[\right)\right].\text{\hspace{0.17em}}$ Press [ENTER]. Rounding to $\text{\hspace{0.17em}}5\text{\hspace{0.17em}}$ decimal places, $\text{\hspace{0.17em}}{e}^{3.14}\approx 23.10387.\text{\hspace{0.17em}}$ Caution: Many scientific calculators have an “Exp” button, which is used to enter numbers in scientific notation. It is not used to find powers of $\text{\hspace{0.17em}}e.$

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