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Differential equations describe how a quantity changes over time. In our exercise, the differential equation is: \(A'(t) = 0.045A(t)\). This describes how the balance \(A(t)\) in a savings account changes with time. The prime symbol \('A'(t)\) signifies a derivative, meaning the rate of change of \(A(t)\). Essentially, this equation states that the rate of growth of the balance is proportional to the current balance.
Importance: Differential equations like this are essential in modeling situations where growth is exponential, such as population growth, radioactive decay, and financial investments. Understanding how to set up and solve these equations helps in predicting future values of a changing quantity.

The initial deposit refers to the amount of money first deposited into the savings account. In our problem, this is denoted by \(A(0)\). According to the initial condition given, \(A(0) = 3000\), the initial deposit is 3000 dollars.
Why it matters: Knowing the initial deposit is crucial for predicting future values in a savings account. It's the foundation on which the interest is calculated and compounded. Without this piece of information, it would be impossible to accurately determine the growth of the balance over time.

The interest rate determines how quickly the balance in the account grows. In our differential equation, \(A'(t) = 0.045A(t)\), the interest rate is represented by the constant 0.045, or 4.5% per year. This rate shows that for every dollar in the account, 0.045 dollars are added each year.
Key points:

• An interest rate can vary or be fixed over different periods.
• The higher the interest rate, the faster the account balance grows.
• Financial institutions use interest rates to incentivize savings and offer returns on deposits.

Exponential growth occurs when the growth rate of a quantity is proportional to its current value. In the context of our exercise, the formula \(A(t) = A(0)e^{0.045t}\) illustrates exponential growth. Here, \(A(0)\) is our initial deposit of 3000 dollars, and \(e^{0.045t}\) shows how the balance grows over time.
Explanation:

• Exponential functions grow faster than linear functions.
• Exponential growth is common in many natural and financial systems.
• It's crucial for understanding long-term behaviors of savings and investments.

Example: After 5 years, the balance becomes approximately 3756.97 dollars, showing significant growth from the initial 3000 dollars due to the compounding effect of exponential growth.