Problem 48 - Calculus and its applications

Question

Pharmaceutical firms invest significantly in testing new medications. After a drug is approved by the Federal Drug Administration, it still takes time for physicians to fully accept and start prescribing it. The acceptance by physicians approaches a limiting value of \(100 \%,\) or \(1,\) after \(t\) months. Suppose that the percentage \(P\) of physicians prescribing a new cancer medication after \(t\) months is approximated by $$ P(t)=100\left(1-e^{-0.4 t}\right) $$ a) What percentage of doctors are prescribing the medication after 0 months? 1 month? 2 months? 3 months? 5 months? 12 months? 16 months? b) Find \(P^{\prime}(7)\), and interpret its meaning. c) Sketch a graph of the function.

Step-by-Step Solution

Answer

After 0 months: 0%, 1 month: 32.97%, 2 months: 55.07%, 3 months: 69.88%, 5 months: 86.47%, 12 months: 99.18%, 16 months: 99.83%. The rate at 7 months is 2.432%/month. Graph is exponential, leveling at 100%.

Step 1: Determine P(t) at various time points

To calculate the percentage of physicians prescribing the medication after different months, substitute the given values of months (0, 1, 2, 3, 5, 12, 16) into the function \( P(t) = 100(1-e^{-0.4t}) \) and evaluate.1. For \( t = 0 \): \[ P(0) = 100(1-e^{0}) = 100(1-1) = 0 \]% 2. For \( t = 1 \): \[ P(1) = 100(1-e^{-0.4 \times 1}) = 100(1-e^{-0.4}) \approx 100(1-0.6703) = 32.97 \]%3. For \( t = 2 \): \[ P(2) = 100(1-e^{-0.8}) \approx 100(1-0.4493) = 55.07 \]%4. For \( t = 3 \): \[ P(3) = 100(1-e^{-1.2}) \approx 100(1-0.3012) = 69.88 \]%5. For \( t = 5 \): \[ P(5) = 100(1-e^{-2}) \approx 100(1-0.1353) = 86.47 \]%6. For \( t = 12 \): \[ P(12) = 100(1-e^{-4.8}) \approx 100(1-0.0082) = 99.18 \]%7. For \( t = 16 \): \[ P(16) = 100(1-e^{-6.4}) \approx 100(1-0.0017) = 99.83 \]%This step calculates the specific percentages of doctors prescribing the medication at each specified month.

Step 2: Find the derivative P'(t)

We need to find the derivative of the function \( P(t) = 100(1-e^{-0.4t}) \) to determine \( P'(7) \).First, differentiate \( P(t) \):\[ P'(t) = \frac{d}{dt}[100(1-e^{-0.4t})] = 100 \cdot (-(-0.4)e^{-0.4t}) = 40e^{-0.4t} \]This is the derivative function, which represents the rate of change of the percentage of physicians prescribing the medication over time.

Step 3: Evaluate the derivative at t = 7

Substitute \( t = 7 \) into the derivative function \( P'(t) = 40e^{-0.4t} \) and solve for \( P'(7) \).\[ P'(7) = 40e^{-0.4 \times 7} = 40e^{-2.8} \approx 40 \times 0.0608 = 2.432 \]Interpretation: At \( t = 7 \) months, the rate at which physicians increase their prescription of the medication is approximately 2.432% per month.

Step 4: Sketch the graph of P(t)

The function \( P(t) = 100(1-e^{-0.4t}) \) represents an exponential growth model bounded by 100%. To sketch the graph:- The y-axis represents \( P(t) \), the percentage of prescribing doctors, and the x-axis represents time \( t \) in months.- The curve starts at \( P(0) = 0 \)%.- As \( t \) increases, \( P(t) \) approaches 100% (horizontal asymptote at 100%).- The curve is steep at first, gradually leveling off as \( t \) increases.Key points from Step 1 (0, 1, 2, 3, 5, 12, 16 months) can be plotted to assist the sketch with approximate values from those calculations.

Key Concepts

Pharmaceutical Rates of Change

In pharmaceutical research, understanding how quickly new medications are accepted by healthcare professionals is crucial. This can be observed through the rate of change in the percentage of physicians prescribing a medication over time. For a new cancer medication, we define this rate of change using a mathematical function. The function provided is an example of exponential growth: \( P(t) = 100(1-e^{-0.4t}) \). This reflects how physicians gradually start prescribing the medication as more information becomes available and confidence in its efficacy grows.

Calculating the rate of change involves finding the derivative of this function, \( P'(t) \), which tells us how fast the prescribing rate is changing at any given moment. With the derivative \( P'(t) = 40e^{-0.4t} \), you can evaluate specific points in time, like month 7, to understand the dynamics of adoption:

• At month 7, \( P'(7) \approx 2.432\%\), indicating the percentage of physicians adopting the medication is increasing by approximately 2.432% per month.

Such insights are vital as they help pharmaceutical companies and healthcare policymakers assess how quickly a medication penetrates the market.

Derivative Interpretation

Derivatives offer insight beyond mere changes in values; they capture the dynamics behind these changes. Here, the derivative \( P'(t) = 40e^{-0.4t} \) represents the rate at which the prescribing percentage is growing. The exponential decay component \( e^{-0.4t} \) signifies that over time, the initial rapid growth in the prescription rate decreases as more physicians begin prescribing.

Interpreting derivatives in this context highlights:

• Initial rapid adoption slowed by saturation as more physicians prescribe the drug.
• The derivative value decreases over time, indicating a slowing in the rate of change.

At the start (e.g., month 0), the rate of adoption is the fastest, which matches real-world scenarios where early adopters lead the charge. As this rate decreases, it's a sign that the drug is nearing its maximum market penetration, highlighting effective adoption.

Function Graph Sketching

Graphing the function \( P(t) = 100(1-e^{-0.4t}) \) visually conveys how the percentage of prescribing doctors changes over time. This exponential growth curve rises steeply initially, indicating fast early adoption. However, as \( t \) (time in months) continues, the curve begins to flatten, approaching a horizontal asymptote at 100%. This asymptote signifies the maximum possible percentage of adoption, as 100% is when all targeted physicians are prescribing the drug.

• Key points like those calculated at months 0, 1, 2, 3, 5, 12, and 16 serve as markers for sketching this curve, showing steady growth toward saturation.
• The graph effectively communicates that while growth is rapid initially, the pace slows as \( t \) increases.
• Insights from such a visual representation help understand how the prescribing behavior transitions over time from rapid acceptance to a plateau.

Utilizing graph sketching complements numerical analysis by providing a more intuitive grasp of how physicians' acceptance evolves.