Problem 4 - Calculus for the Life Sciences: A Modeling Approach Volume I

Question

Suppose penicillin concentration is given by \(C(t)=0.4 t e^{-0.5 t} \mu \mathrm{gm} / \mathrm{ml} t\) hours after ingestion of a penicillin pill. For what time period is the concentration decreasing? What is the maximum penicillin concentration?

Step-by-Step Solution

Answer

The concentration decreases for \( t > 2 \) and peaks at \( t = 2 \) with a maximum of about \( 0.2943 \mu \text{gm/ml} \).

Step 1: Understanding the Function

The function provided is \( C(t) = 0.4 t e^{-0.5 t} \), which describes the concentration of penicillin in the bloodstream over time. The task is to determine when this concentration is decreasing, and find the time at which it is maximized.

Step 2: Find the Derivative

To analyze whether the concentration is increasing or decreasing, we need to find the first derivative of \( C(t) \). Use the product rule for differentiation: \( u = 0.4t \) and \( v = e^{-0.5t} \). The derivative of \( u \) is \( 0.4 \) and the derivative of \( v \) is \( -0.5e^{-0.5t} \). Applying the product rule, \( C'(t) = 0.4e^{-0.5t} + 0.4t(-0.5)e^{-0.5t} \). This simplifies to \( C'(t) = e^{-0.5t}(0.4 - 0.2t) \).

Step 3: Determine Decreasing Intervals

A function is decreasing when its derivative is less than zero. So, set \( C'(t) < 0 \): \( e^{-0.5t}(0.4 - 0.2t) < 0 \). Since \( e^{-0.5t} > 0 \) for all \( t \geq 0 \), solve \( 0.4 - 0.2t < 0 \). This gives \( 0.2t > 0.4 \) or \( t > 2 \). Therefore, the concentration is decreasing when \( t > 2 \).

Step 4: Find Critical Points

To find when the concentration is at its maximum, set the derivative \( C'(t) = 0 \): \( e^{-0.5t}(0.4 - 0.2t) = 0 \). Solving \( 0.4 - 0.2t = 0 \) gives \( t = 2 \). This is the critical point where a change in increasing/decreasing behavior might occur.

Step 5: Evaluate Maximum Concentration

To ensure this critical point is a maximum, we can test the intervals around \( t = 2 \) or use the second derivative test. With a single critical value, the change from increasing to decreasing confirms a maximum at \( t = 2 \). Substituting \( t = 2 \) back into the original function, \( C(2) = 0.4 \times 2 \times e^{-0.5\times2} = 0.8 \times e^{-1} \approx 0.8 \times 0.3679 \approx 0.2943 \). Thus, the maximum concentration is approximately \( 0.2943 \mu \text{gm/ml} \) at \( t = 2 \) hours.

Key Concepts

Derivative

Derivatives are a fundamental concept in calculus. They measure how a function changes as its input changes. A derivative represents the slope of the tangent line to the curve at a given point. This concept helps us understand behaviors like increasing or decreasing trends of functions, such as the concentration of penicillin over time.

• The notation used for the derivative of a function \( C(t) \) is \( C'(t) \).
• It can be found using different rules, such as the product rule, which applies when multiplying two functions together.

In our problem, the function representing penicillin concentration is \( C(t) = 0.4t e^{-0.5t} \). By applying the product rule, which involves taking the derivative of the product of two functions, we find the derivative \( C'(t) = e^{-0.5t}(0.4 - 0.2t) \).
This expression helps us determine where the function either increases or decreases over time. Knowing how to compute derivatives is essential for finding such intervals.

Maxima and Minima

Maxima and minima refer to the highest and lowest points on a graph of a function. A maximum point indicates where the function reaches its highest value, while a minimum point shows its lowest value in a certain interval.

• Critical points are found by setting the derivative \( C'(t) \) equal to zero.
• These points are where the function changes from increasing to decreasing or vice versa.

In the example provided, the derivative \( C'(t) \) becomes zero when \( t = 2 \), making it a critical point. We verify whether this point is a maximum by examining the behavior of the original function around the critical point. If the function changes from increasing to decreasing at \( t = 2 \), this confirms a maximum.
By substituting \( t = 2 \) back into the original function \( C(t) \), we find that the maximum penicillin concentration is about 0.2943 \( \mu \text{gm/ml} \). This highlights how the concept of maxima allows us to find where certain conditions reach their peak.

Exponential Functions

Exponential functions are a type of mathematical function that has the form \( f(t) = ae^{bt} \), where \( e \) is Euler's number (approximately 2.71828), and \( a \) and \( b \) are constants. These functions are significant in many real-world applications such as population growth, radioactive decay, and the concentration of drugs in the bloodstream.

• The base \( e \) makes exponential functions grow or decay rapidly depending on the sign of \( b \).
• A negative exponent, as in \( e^{-0.5t} \), signifies a rapid decrease or decay over time.

In our problem, the penicillin concentration function \( C(t) = 0.4t e^{-0.5t} \) includes an exponential component that decreases over time due to the negative exponent \(-0.5 \). This results in the concentration decreasing after it reaches its peak at \( t = 2 \) hours.
Understanding exponential functions is crucial to predicting how quantities change over time, particularly in fields like pharmacology and biology.