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Understanding the derivative of a function is crucial in calculus, as it represents the rate at which a function is changing at any given point. For instance, in the context of our exercise
, the function describing the percentage of foreign-born medical residents over time can be analyzed using its derivative.

The process of finding the derivative involves applying rules of differentiation. The power rule, one of the most fundamental rules, states that for a function of the form \( c \times t^n \), where \( c \) is a constant and \( n \) is a positive integer, the derivative is \( c \times n \times t^{(n-1)} \). In our example, by applying the power rule to each term in the given function, the derivatives of \( t^3 \), \( t^2 \), and \( t \) were calculated, producing a new function that describes how the percentage rate of foreign-born medical residents changes over time.

This step is important because it sets the stage for finding the critical points, which are essential for determining where the function attains its maximum or minimum values, or where its slope is zero.

The quadratic formula is a powerful tool for solving equations of the second degree, which are in the form \( at^2 + bt + c = 0 \). The formula gives the solutions for \( t \) as \[ t = \frac{-b \begin{document} \begin{document} \begin{document} \begin{document} \begin{document} \begin{document} \begin{document} \begin{document} \begin{document} \begin{document} \begin{document} \begin{document} \begin{document} \begin{document} \begin{document} \begin{document} \begin{document} \begin{document} \begin{document} \begin{document} \begin{document} \begin{document} \begin{document} \begin{document} \begin{document} \begin{document} \begin{document} \begin{document} \pm \begin{document} \sqrt{b^{2} - 4ac}}{2a} \], where \( a \begin{document} \begin{document} \begin{document} \begin{document} \begin{document} \begin{document} \begin{document} \begin{document} \begin{document} \begin{document} \begin{document} \begin{document} \begin{document} \begin{document} \begin{document} \begin{document} \begin{document} \begin{document} \begin{document} \begin{document} \begin{document} \begin{document} \begin{document} \begin{document} \begin{document} \begin{document} \begin{document} \), \( b \), and \( c \) are constants from the quadratic equation. In our exercise, after setting the derivative of the function equal to zero, we obtained a quadratic equation. Using the coefficients from this equation, the quadratic formula was used to find the values of \( t \) corresponding to the critical points. These critical points indicate where the function might have a highest or lowest value, which is pivotal in determining when the percentage of foreign-born medical residents was the lowest.

Minimization in calculus involves finding the point at which a function takes on its minimum value. This concept is particularly valuable in various applications such as economics, physics, and, as in our exercise, analyzing trends over time.

Once the critical points of a function are identified, we need to determine whether these points correspond to a minimum, a maximum, or neither. For our problem, we found two critical points by setting the derivative of the function equal to zero. However, identifying these points is only part of the process. We must assess the intervals around these points to confirm if they indeed correspond to the lowest value of the function within a specific range, in this case, the early 1970s.

To find a minimum, we examine the sign of the derivative on both sides of the critical point or use the second derivative test. If the first derivative changes from negative to positive at the critical point, or if the second derivative is positive, we typically have a minimum. In the exercise, after calculating the critical points, we confirm that the point t = 5.906, which corresponds to the early 1970s, represents when the function P(t) has its minimum value, indicating the lowest percentage of foreign-born medical residents.