Question

A deposit of \(\$ 2250\) is made in a savings account at an annual interest rate of \(6 \%\), compounded continuously. Find the average balance in the account during the first 5 years.

Short answer

To obtain the average balance subtract the second term from the first term and simplify for B. B is the average balance in the account during the first 5 years.

Step-by-step solution

Plug in the given values into the compound interest formula

Using the given values, \(A = \$2250 \times e^{0.06t}\) where t is the time in years.

Calculate the average balance using integral

The average balance B during the first 5 years is determined by the equation \(B=(1/5)\int_0^5 A(t) dt\), substituting for \(A(t)\) from step 1 into equation would give \(B=(1/5)\int_0^5 \$2250e^{0.06t} dt\)

Solve the integral

Solving this equation involves calculating the integral of \( \$2250e^{0.06t} \) which is \(\$2250* e^{0.06t}/0.06\) evaluated from 0 to 5.

Calculate the average balance

Substitute the limits into the evaluated integral to find the average balance. \(B = \frac{1}{5} ([\$2250* e^{0.06*5}/0.06] - [\$2250* e^{0.06*0}/0.06])\), simplifying this gives the average balance.

Key concepts

Continuous Compound Interest

When a deposit is made into a savings account with an interest rate that is compounded continuously, the balance in the account grows at an exponential rate. This concept is a cornerstone of financial mathematics and plays a critical role in calculating the future value of investments. Unlike simple interest, wherein the interest is calculated periodically, continuous compounding assumes that the interest is being calculated and added to the account balance at every possible moment in time.

The formula for continuous compound interest is given by the equation
\[A = Pe^{rt}\]
where

• \(A\) is the amount of money accumulated after time \(t\), including interest,
• \(P\) is the principal amount (the initial deposit),
• \(r\) is the annual interest rate (expressed as a decimal), and
• \(e\) is the base of the natural logarithm, approximately 2.71828.

Continuous compounding can greatly affect the growth of the investment because as the frequency of compounding interest increases, so does the final balance of the account. Therefore, it represents the upper limit of compounding frequency and the most advantageous scenario for an investor seeking to maximize their returns over time.

Exponential Functions

Exponential functions are mathematical expressions that describe situations where a quantity grows at a rate proportional to its current value, such as in continuously compounded interest. These functions take the form \[ f(x) = ab^{x} \]where

• \(a\) is a constant that represents the initial value,
• \(b\) is the base of the exponential function, determining the growth rate,
• and \(x\) is the exponent, usually corresponding to time.

In the context of continuous compounding, the exponential function is particularized with the base \(e\), which is Euler's number, and the exponent includes the product of the interest rate and time, making the function
\[A(t) = Pe^{rt}\]This mathematical model is widely used not only in finance but also in sciences like biology, for modeling population growth, or physics, for decay processes. Exponential growth is characterized by its accelerating growth pattern, meaning the larger the value gets, the faster it grows, which highlights its stark difference from linear growth.

Integral Calculus

Integral calculus is a branch of mathematics that is concerned with the accumulation of quantities and the areas under and between curves. When we want to determine the average value of a continuous function over a particular interval, integral calculus provides us with the tools to do just that. The average balance of a continuously compounded account over a certain time period can be obtained by finding the integral of the balance function over that period and then dividing by the interval's length.

The process can be described in the following steps: First, we identify the function to integrate, which in the case of our compound interest problem, is the exponential function \(A(t) = Pe^{rt}\). Next, we compute the definite integral over the desired time interval. The definite integral of an exponential function like \(A(t)\) is: \[ \int Pe^{rt} dt = \frac{P}{r}e^{rt} + C \]where \(C\) is the constant of integration. However, when evaluating a definite integral from \(a\) to \(b\), the constant cancels out, and we're left with: \[ \int_a^b Pe^{rt} dt = \left[\frac{P}{r}e^{rt}\right]_a^b = \frac{P}{r}(e^{rb} - e^{ra}) \]Finally, to find the average value, we divide the result by the interval length. In this way, integral calculus not only gives us the total accumulated value but also allows us to gain deeper insights into the behavior of functions over intervals, which is essential in various fields, including economics, physics, engineering, and beyond.