Problem 70 - College Algebra and Calculus: An Applied Approach

Question

You deposit in an account that is compounded quarterly at an annual rate of \(r\) (in decimal form). The balance \(A\) after 10 years is \(A=2000\left(1+\frac{r}{4}\right)^{40}\) (a) Complete the table. $$ \begin{array}{|l|l|l|l|l|l|} \hline r & 0.059 & 0.0599 & 0.06 & 0.0601 & 0.061 \\ \hline A & & & & & \\ \hline \end{array} $$ (b) Does the limit of \(A\) exist as the interest rate approaches \(6 \%\) ? If so, what is the limit?

Step-by-Step Solution

Answer

The table entries for \(A\) can be computed as: for \(r=0.059\), \(A=\$3309.21\); for \(r=0.0599\), \(A=\$3315.91\); for \(r=0.06\), \(A=\$3317.20\); for \(r=0.0601\), \(A=\$3318.49\); and for \(r=0.061\), \(A=\$3323.60\). As the interest rate \(r\) approaches \(0.06\) or \(6\% \), the account balance \(A\) appears to conform to a certain value, which in this case is approximately \$3317.20 suggesting that a limit at \(6\%\) does indeed exist.

Step 1: Calculate Account Balance for Given Interest Rates

Substitute the values of \(r\) into the given equation \(A=2000\left(1+\frac{r}{4}\right)^{40}\) for each given value of \(r = 0.059, 0.0599, 0.06, 0.0601, 0.061\). Get the account balance \(A\) for each.

Step 2: List the Calculated Account Balances

After calculating, put down the corresponding values of \(A\) in the table. It helps to observe how the account balance \(A\) changes with the change in interest rate \(r\).

Step 3: Find the Limit of Account Balance as r Approaches 0.06

Analyze the amounts calculated for \(A\) as \(r\) gets closer to 0.06. If the values of \(A\) approach a certain value or become stable, we might conclude that the limit exists. Use the concept of limit to calculate if necessary.

Key Concepts

Quarterly Compounding

Quarterly compounding refers to the frequency with which interest is calculated and added to your account balance. When compounding occurs quarterly, the bank calculates interest four times a year.

To better understand this, imagine you deposit money into a savings account. If the interest is compounded quarterly, each three-month period your account earns interest. This interest is added to your principal, becoming part of the new balance that earns interest in the next period.

• This increases the total amount you earn because interest is calculated on a larger balance as the year progresses.
• The formula for calculating the balance with quarterly compounding is: \( A = P \left(1 + \frac{r}{4}\right)^{4t} \) where \(P\) is the principal amount, \(r\) is the annual interest rate, and \(t\) is the time in years.

As demonstrated in our exercise, you substitute the various interest rates into similar formulas to find the balance at the end of the given period.

Annual Interest Rate

The annual interest rate is the percentage increase on your deposited or borrowed amount for a whole year. It is often expressed as a percentage.

The exercises use this concept in decimal format to simplify calculations. Converting from percentage to decimal is done by dividing the percentage by 100.

• For example, a 6% interest rate becomes \( r = 0.06 \).
• The rate given indicates how much the initial amount will grow over a period of one year if no withdrawals or additional deposits are made.

When examining different annual interest rates in our exercise, it is important to see how changes, even as small as 0.0001, can affect the balance over several years when compounded quarterly. This demonstrates the sensitivity of compound interest calculations to changes in the interest rate.

Limit of a Function

The concept of the limit of a function is critical when analyzing how a function behaves as its input approaches a certain value. In calculus, the limit helps us understand what value, if any, a function is approaching.

In our problem, we looked at the limit of the balance \( A \) as the interest rate \( r \) approaches 6% (or 0.06 as a decimal). This involves checking if the calculated balance \( A \) stabilizes around a particular value when \( r \) gets closer to 0.06.

• Using the limit involves substituting values closer and closer to 0.06 into the equation for \( A \).
• If the resulting values of \( A \) converge to a single number, the limit of \( A \) exists at that rate.

By using this property, one can observe that minor variations in the interest rate, especially when compounded over time, can still lead to significant changes in the balance. This highlights the usefulness of understanding limits to predict financial outcomes.