Problem 21 - Applied Calculus

Question

After a foreign substance is introduced into the blood, the rate at which antibodies are made is given by \(r(t)=\frac{t}{t^{2}+1}\) thousands of antibodies per minute, where time, \(t\), is in minutes. Assuming there are no antibodies present at time \(t=0,\) find the total quantity of antibodies in the blood at the end of 4 minutes.

Step-by-Step Solution

Answer

Approximately 1240.5 thousand antibodies are produced.

Step 1: Understand the Problem

We are provided with the rate function \( r(t) = \frac{t}{t^2 + 1} \) which gives the rate at which antibodies are produced in thousands per minute. We need to find the total quantity of antibodies produced in the first 4 minutes.

Step 2: Set Up the Integral

Since the rate function \( r(t) \) represents the derivative of the antibody count, the total quantity of antibodies over a time interval can be found by integrating this function with respect to time. We need to find \( \int_{0}^{4} r(t) \, dt = \int_{0}^{4} \frac{t}{t^2 + 1} \, dt \).

Step 3: Solve the Integral

The integral \( \int \frac{t}{t^2 + 1} \, dt \) can be solved using substitution. Let \( u = t^2 + 1 \). Then \( du = 2t \, dt \) or \( t \, dt = \frac{1}{2} du \). The limits change to 1 when \( t = 0 \) and 17 when \( t = 4 \). Thus, the integral becomes: \[ \int \frac{1}{2u} \, du = \frac{1}{2} \ln|u| + C. \] Evaluating from 1 to 17 gives \[ \frac{1}{2} \ln|17| - \frac{1}{2} \ln|1|. \]

Step 4: Compute the Value

Simplifying \( \frac{1}{2} (\ln 17 - \ln 1) \) gives \( \frac{1}{2} \ln 17 \). Since \( \ln 1 = 0 \), we find \[ \frac{1}{2} \ln 17 \approx 1.2405. \] Thus, approximately 1240.5 antibodies are made.

Step 5: Interpret the Result in Context

The integral result, \(1.2405\), represents thousands of antibodies. Therefore, the total number of antibodies produced in 4 minutes is approximately 1240.5.

Key Concepts

Antibody Production

Antibodies are a crucial part of our immune system. When a foreign substance, like a virus or bacteria, enters the body, the immune system immediately responds by producing antibodies. Antibodies are proteins that detect and neutralize foreign objects.
These proteins attach themselves to unwanted invaders, helping the body recognize and combat these threats.

• The production of antibodies begins soon after exposure to an antigen.
• The number of antibodies produced can vary depending on several factors, including the type of invader and the body's immune response.

To quantify the production of antibodies, we can model this process using mathematical functions. These functions describe how the production rate changes over time, allowing us to calculate the total number of antibodies generated after a certain period.

Rate Function

In mathematics, a rate function helps us understand how a quantity changes over time. For antibody production, the rate function mathematically represents how fast antibodies are being produced in the body.
The rate function given as \( r(t) = \frac{t}{t^2 + 1} \) tells us the antibodies production rate per minute.

• Here, 't' represents time in minutes.
• The function is derived based on biological understanding of the immune response.

By observing this rate function, you can see that the initial rate starts at zero and gradually increases before stabilizing as time progresses.
This rate function is crucial in determining the cumulative amount of antibodies, which involves integrating it over a given time interval.

Substitution Method

The substitution method is a powerful technique in calculus used to simplify the process of integration, making it easier to handle complex functions.
When dealing with the rate function \( r(t) = \frac{t}{t^2 + 1} \), substitution simplifies the integration process by changing variables.

• We set \( u = t^2 + 1 \), where \( du = 2t \, dt \).
• This substitution leads to a new integrand that's more straightforward: \( \int \frac{1}{2u} \, du \).

The limits of integration transform accordingly. For instance, when \( t = 0 \), \( u = 1 \), and for \( t = 4 \), \( u = 17 \).
Solving the integral using this method leads to a logarithmic function, which ultimately provides us with the total number of antibodies produced over the time interval.