Problem 10 - Applied Calculus

Question

When a deposit of \(\$ 1000\) is made into an account paying \(8 \%\) interest, compounded annually, the balance, \(\$ B,\) in the account after \(t\) years is given by \(B=1000(1.08)^{t}\) Find the average rate of change in the balance over the interval \(t=0\) to \(t=5 .\) Give units and interpret your answer in terms of the balance in the account.

Step-by-Step Solution

Answer

The average rate of change is \( 93.87 \ \text{dollars per year} \). It means the balance increases by about \( \$93.87 \) annually on average.

Step 1: Identify key components of the formula

The given problem provides the balance formula \( B = 1000(1.08)^t \), where \( B \) is the balance after \( t \) years. We want to find the average rate of change over the time interval from \( t=0 \) to \( t=5 \).

Step 2: Calculate balance at endpoints

First, find \( B \) at \( t = 0 \) and \( t = 5 \). When \( t = 0 \), the balance \( B = 1000(1.08)^0 = 1000 \). When \( t = 5 \), the balance \( B = 1000(1.08)^5 = 1000(1.46933) \approx 1469.33 \).

Step 3: Apply the formula for average rate of change

The average rate of change is calculated as \( \frac{B(t=5) - B(t=0)}{5 - 0} \). Substitute the previous results: \( \frac{1469.33 - 1000}{5} = \frac{469.33}{5} \).

Step 4: Compute the average rate of change

Divide \( 469.33 \) by \( 5 \) to get the average rate of change: \( 93.866 \).

Step 5: Interpret the result

The average rate of change is \( 93.87 \ \text{dollars per year} \). This means that on average, the balance in the account increases by approximately \( \$93.87 \) each year over the given interval.

Key Concepts

Average Rate of Change

The average rate of change is a concept used to measure how a quantity alters over a specific period. In simple terms, it's like calculating the "speed" of change in a particular scenario. This formula is chiefly represented by the change in a function's value divided by the change in time. For an account with compound interest, it shows how much the account balance changes, on average, each year.
To find the average rate of change for an interval from time zero to five years in our problem, use the formula:

• Identify the balances at the start and end of the interval: Initial balance at time 0 and final balance at time 5.
• Subtract the initial balance from the final balance to find the total change in balance.
• Divide the result by the time interval to find the average rate of change.

In this case, we see that the balance grows by approximately $93.87 each year over the five-year period. This average reflects regular growth over time rather than yearly fluctuations.

Exponential Growth

Exponential growth is a crucial concept in understanding compound interest and other similar phenomena. It's characterized by a quantity increasing at a consistent percentage rate over equal time intervals. These growth models are typical where a variable's increase relies not only on time but also on the current amount.
For compound interest accounts, like the one in our exercise, this concept illustrates how investment grows more substantially over time because interest builds upon previously earned interest as well as the initial deposit. Hence, the value isn't just increasing by a straightforward sum every period, but by a percentage leading to a rising growth pattern.
Exponential growth can be expressed via the equation:

• The function is given by the formula: \( B = P(1 + r)^t \), where:
• \( P \) is the principal amount (initial deposit).
• \( r \) is the annual interest rate (expressed as a decimal).
• \( t \) is time in years.

Looking at this model, each year's increase is more significant than the last, reflecting how growth accelerates as the base amount becomes larger.

Financial Mathematics

Financial mathematics is a field that applies mathematical principles to solve financial problems. It's fundamental in understanding various financial products and predicting economic behaviors. One of the most common applications is calculating the growth of investments through methods like compound interest.
In this context,

• It helps in determining future investment values, comparing different financial products, and budgeting more effectively.
• Mathematical models assist in visualizing how different variables affect outcomes, such as interest rates and time horizons.

In our exercise, financial mathematics allows us to calculate the expected growth of a deposit in a savings account over a set period, given a specific interest rate. This not only aids in understanding future financial health but also helps individuals and businesses make informed decisions about savings and investments.