Problem 103 - Applied Mathematics: For the Managerial, Life, and Social Sciences

Question

Medicaid spending on drugs in Massachusetts started slowing down in part after the state demanded that patients use more generic drugs and limited the range of drugs available to the program. The annual pharmacy spending (in millions of dollars) from 1999 through 2004 is given by \(S(t)=-1.806 t^{3}+10.238 t^{2}+93.35 t+583 \quad(0 \leq t \leq 5)\) where \(t\) is measured in years with \(t=0\) corresponding to the beginning of \(1999 .\) Find the inflection point of \(S\) and interpret your result.

Step-by-Step Solution

Answer

The inflection point of the function \(S(t)\) occurs approximately at \((1.889, 665.166)\), which represents the end of 2000 and an annual pharmacy spending of approximately 665.166 million dollars. This means that the rate at which Medicaid spending changed initially was increasing, but started to slow down around the end of 2000. This implies that the measures taken by the state to demand more generic drugs and limit the range of drugs available to the program began to have an effect on slowing down the spending growth around the end of 2000.

Step 1: Find the first derivative of S(t)

In order to find the inflection point, we first have to find the first derivative of \(S(t)\) with respect to \(t\). The first derivative will tell us the rate of change in spending. Using the power rule for differentiation, we obtain: \(S'(t) = \frac{d}{dt} (-1.806 t^{3}+10.238 t^{2}+93.35 t+583) = -3(1.806) t^{2}+2(10.238) t+93.35\) which simplifies to: \(S'(t) = -5.418 t^{2}+20.476 t+93.35\)

Step 2: Find the second derivative of S(t)

To determine where the inflection point is, we need to find the second derivative of \(S(t)\), which is the derivative of the first derivative \(S'(t)\). The second derivative will tell us the rate of change in the rate of change in spending (i.e., the concavity of the function). Taking the derivative of \(S'(t)\), we get: \(S''(t) = \frac{d}{dt} (-5.418 t^{2}+20.476 t+93.35) = -2(5.418) t+20.476\) which simplifies to: \(S''(t) = -10.836 t+20.476\)

Step 3: Solve for t when the second derivative is equal to 0

At the inflection point, the second derivative is equal to 0. Thus, we need to solve for \(t\) in the equation \(S''(t) = 0\): \(-10.836 t+20.476=0\) Now, isolate \(t\): \(t = \frac{20.476}{10.836}\) \(t \approx 1.889\)

Step 4: Find the inflection point

To find the inflection point, substitute the value of \(t\) we found in the previous step into the original function \(S(t)\): \(S(1.889) \approx -1.806 (1.889)^{3}+10.238 (1.889)^{2}+93.35 (1.889)+583\) \(S(1.889) \approx 665.166\) Thus, the inflection point is approximately at \((1.889, 665.166)\).

Step 5: Interpret the result

The inflection point occurs around 1.889 years (which represents the end of 2000) since the beginning of 1999, and at that time, the annual pharmacy spending was approximately 665.166 million dollars. This means that the rate at which Medicaid spending changed initially was increasing, but started to slow down around the end of 2000. This implies that the measures taken by the state to demand more generic drugs and limit the range of drugs available to the program began to have an effect on slowing down the spending growth around the end of 2000.

Key Concepts

Derivative Calculations

When solving any real-world problem involving change, like pharmacy spending over time, derivative calculations are indispensable. Derivatives help us understand how a function's output changes concerning its input. In simpler terms, it describes how fast or slow something is changing.

To discover the inflection point of a function, we first need the first derivative. The first derivative of a function, here represented as \(S(t)\), provides us the slope of the tangent line at any given point. It tells us the rate of change of the original function, which for this problem indicates how quickly spending is increasing or decreasing.

Using the power rule for differentiation, we calculate the first derivative:

• Start with the function: \(S(t)=-1.806 t^{3}+10.238 t^{2}+93.35 t+583\).
• Apply the power rule to each term: bring down the exponent as a coefficient and subtract one from the exponent.
• The result is: \(S'(t)=-5.418 t^{2}+20.476 t+93.35\).

The derivative tells us how spending changes annually. To find where this change in spending behavior shifts, we move towards the second derivative.

Polynomial Function Analysis

Polynomials are mathematical expressions involving variables and constants, combined using addition, subtraction, multiplication, and positive integer exponents. Polynomial functions like \(S(t)\) can take many shapes, characterized by the degree, sign, and magnitude of their coefficients.

In our example, \(S(t)\) is a cubic polynomial, identifiable by the highest power of the variable \(t\) being three. Cubic functions are known for their characteristic curve that can change direction up to two times, leading to potential local maxima, minima, and points of inflection.

Analyzing such functions involves understanding these key aspects:

• The degree of the polynomial tells us the number of turning points possible. Here, a maximum of two.
• The coefficients affect the "width" and "orientation" of the graph.
• The leading term defines the end behavior, meaning how the function behaves as \(t\) tends towards infinity.

To deeply understand this function, we compute its derivatives to pin down where these directional changes occur.

Concavity and Convexity in Graphs

Concavity and convexity describe how a function curves. To determine these characteristics in a function, we look at the second derivative. When we discuss concavity:

• A function is concave up (like a cup) when its second derivative is positive, indicating that the tangent line lies below the curve.
• When the second derivative is negative, the function is concave down (like an arch), meaning tangents are above the curve.

The inflection point is the critical point where the curve changes concavity from up to down or down to up.

For the function \(S(t)\), we calculate the second derivative as \(S''(t) = -10.836 t + 20.476\). Setting it to zero helps find the potential inflection points:

• Solving \(-10.836 t + 20.476 = 0\) yields \(t \approx 1.889\).

Thus, the inflection point occurs at approximately \(t = 1.889\), shifting the spending curve’s concavity. At this point, the initial rise slows as spending management measures take effect.