Let A(t)=5000 e^.04 t be the balance in a savings account...

Let $A(t)=5000 e^{.04 t}$ be the balance in a savings account after $t$ years. (a) How much money was originally deposited? (b) What is the interest rate? (c) How much money will be in the account after 10 years? (d) What differential equation is satisfied by $y=A(t)$ ? (e) Use the results of parts (c) and (d) to determine how fast the balance is growing after 10 years. (f) How large will the balance be when it is growing at the rate of $\$ 280$ per year?

Let $A(t)=5000 e^{.04 t}$ be the balance in a savings account after $t$ years.
(a) How much money was originally deposited?
(b) What is the interest rate?
(c) How much money will be in the account after 10 years?
(d) What differential equation is satisfied by $y=A(t)$ ?
(e) Use the results of parts (c) and (d) to determine how fast the balance is growing after 10 years.
(f) How large will the balance be when it is growing at the rate of $\$ 280$ per year?

Key Concepts

Exponential Growth Model

Exponential growth models describe processes that increase at a rate proportional to their current value. These models are commonly used in finance, biology, and physics to capture scenarios where a quantity grows continuously over time based on its present size. The standard form of an exponential growth function is A(t) = A(0) e^(rt), where A(0) is the initial amount and r is the constant growth rate.

Initial Value and Continuous Compounding

In the context of exponential functions, the initial value (or initial condition) represents the quantity present at time zero. In continuous compounding, this initial amount is compounded at every moment, leading to the formula A(t) = A(0) e^(rt). Understanding the initial value is crucial since it serves as the base from which all subsequent growth is measured.

Interest Rate in Exponential Models

The rate in an exponential growth function, often represented as r in the function A(t) = A(0) e^(rt), denotes the continuous growth rate. In financial situations, this rate corresponds to the continuously compounded interest rate, which determines how quickly an investment grows over time.

Differential Equations in Continuous Growth

The differential equation associated with an exponential growth model is dy/dt = ry. This equation states that the instantaneous rate of change of the quantity is directly proportional to the current amount. Solving this differential equation yields the exponential growth function, thereby linking differential calculus with real-world growth phenomena.

Rate of Change and Derivatives

The derivative of an exponential function, such as dA/dt for A(t) = A(0) e^(rt), provides the instantaneous rate at which the variable grows at any given moment. Evaluating the derivative at specific time points allows one to determine how fast the quantity is increasing, a key concept when analyzing the dynamics of systems modeled by exponential functions.

Solving Exponential Equations for Specific Conditions

Exponential equations can be manipulated to solve for different variables under given conditions. For instance, one might solve for time, initial amount, or the corresponding quantity when the growth rate (or derivative) reaches a particular value. This involves using logarithms or algebraic rearrangements to isolate the variable of interest, demonstrating the flexibility of exponential models in practical problem-solving.

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