The projected rate of increase in enrollment at a new...

The projected rate of increase in enrollment at a new college is estimated by
$$
\frac{d E}{d t}=5,000(t+1)^{-3 / 2} \quad t \geq 0
$$
where $E(t)$ is the projected enrollment in $t$ years. If enrollment is 2,000 now $(t=0),$ find the projected enrollment 15 years from now.

Key Concepts

Differential Equations

Differential equations are mathematical expressions that establish a relationship between a function and its derivatives. They are used to model and solve problems where a rate of change is known or must be determined, allowing one to reconstruct the original function describing a system or process from its instantaneous rate of change.

Initial Value Problems

An initial value problem is a type of differential equation that includes an initial condition, which specifies the value of the function at a particular point. This additional information is essential for determining a unique solution to the differential equation that accurately models the behavior of the system over time.

Integration Techniques

Integration is the process of finding the antiderivative of a function, essentially 'undoing' differentiation. In contexts where the rate of change is given, integration is used to determine the aggregate quantity or the original function, reflecting how a system accumulates or evolves over time.

Power Rule for Integration

The power rule for integration is a fundamental technique for integrating functions of the form (t + constant)^n. It states that the integral is given by (t + constant)^(n+1) divided by (n+1), provided n is not equal to -1. This rule simplifies the integration of power functions and is widely used in solving differential equations with power law derivatives.

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