Problem 50 - Thomas Calculus

Question

The reaction of the body to a dose of medicine can sometimes be represented by an equation of the form $$ R=M^{2}\left(\frac{C}{2}-\frac{M}{3}\right) $$ where \(C\) is a positive constant and \(M\) is the amount of medicine absorbed in the blood. If the reaction is a change in blood pressure, \(R\) is measured in millimeters of mercury. If the reaction is a change in temperature, \(R\) is measured in degrees, and so on. Find \(d R / d M\). This derivative, as a function of \(M\), is called the sensitivity of the body to the medicine. In Section \(4.5,\) we will see how to find the amount of medicine to which the body is most sensitive.

Step-by-Step Solution

Answer

\( \frac{dR}{dM} = MC - M^2 \)

Step 1: Identify the Function to Differentiate

The given function represents the reaction of the body: \( R = M^2 \left( \frac{C}{2} - \frac{M}{3} \right) \). We need to find the derivative \( \frac{dR}{dM} \).

Step 2: Apply the Product Rule

Since \( R \) is a product of two functions, \( M^2 \) and \( \left( \frac{C}{2} - \frac{M}{3} \right) \), we use the product rule: If \( u = M^2 \) and \( v = \left( \frac{C}{2} - \frac{M}{3} \right) \), then \( \frac{dR}{dM} = u'v + uv' \).

Step 3: Differentiate Each Component

Now, calculate the derivatives: - Derivative of \( u = M^2 \) is \( u' = 2M \).- Derivative of \( v = \frac{C}{2} - \frac{M}{3} \) is \( v' = -\frac{1}{3} \), because \( \frac{C}{2} \) is a constant.

Step 4: Substitute into Product Rule Formula

Substitute the derivatives into the product rule formula: \( \frac{dR}{dM} = (2M) \left( \frac{C}{2} - \frac{M}{3} \right) + M^2 \left( -\frac{1}{3} \right) \).

Step 5: Simplify the Expression

Simplify the expression using distribution and combining like terms:\( \frac{dR}{dM} = 2M \left( \frac{C}{2} - \frac{M}{3} \right) - \frac{M^2}{3} \).Performing the multiplication, we have:\( \frac{dR}{dM} = MC - \frac{2M^2}{3} - \frac{M^2}{3} \).Combine the \( M^2 \) terms:\( \frac{dR}{dM} = MC - \frac{3M^2}{3} = MC - M^2 \).

Key Concepts

Product Rule in Differentiation

The product rule is an essential tool in calculus for finding the derivative of a product of two functions. When you multiply two functions together and want to find out how the product changes, the product rule is your friend. Imagine you have two functions, \( u \) and \( v \). To find the derivative of their product \( uv \), you would use the formula:

• \( (uv)' = u'v + uv' \).

This means you take the derivative of \( u \) (written as \( u' \)) and multiply it by \( v \), then take the derivative of \( v \) and multiply it by \( u \), then add these two results together.
In the context of our exercise, we used this rule because the reaction equation \( R = M^2 \left( \frac{C}{2} - \frac{M}{3} \right) \) is the product of two functions, \( M^2 \) and \( \frac{C}{2} - \frac{M}{3} \). Differentiating each part separately and combining them is how we found the derivative of the whole product.

Understanding Derivatives

A derivative is a concept that measures how a function changes as its input changes. It is the mathematical way to represent the rate of change. If you think about driving a car, the derivative of your position over time is your speed.
For any function \( f(x) \), the derivative \( f'(x) \) tells us how sensitive \( f(x) \) is to changes in \( x \). In our exercise, we are interested in \( \frac{dR}{dM} \), which shows how the body's reaction \( R \) changes as the amount of medicine \( M \) changes.
Calculating the derivative involves applying rules like the power rule or product rule to the function. In our solution, we found how the body's reaction changes concerning two different parts of the function: \( M^2 \) and \( \frac{C}{2} - \frac{M}{3} \). By applying these rules, we can accurately find how small adjustments in \( M \) affect \( R \).

Function Sensitivity

The concept of function sensitivity refers to how responsive a function is to changes in its input. In medical terms, it tells us how a change in the dose of medicine impacts the body's reaction, such as blood pressure or temperature.
For the function given by \( R = M^2 \left( \frac{C}{2} - \frac{M}{3} \right) \), the sensitivity is represented by the derivative \( \frac{dR}{dM} \). This derivative is a measure of how much the reaction, \( R \), changes when there is a slight change in the amount of medicine, \( M \).
Understanding sensitivity is crucial because it helps determine the optimal amount of medicine needed for the best therapeutic effect. In calculus, finding the derivative gives a quantitative measure of this sensitivity.