Problem 67 - Calculus for Biology and Medicine

Question

Assume that the radius \(r\) and the surface area \(S=4 \pi r^{2}\) of a sphere are differentiable functions of \(t .\) Express \(d S / d t\) in terms of \(d r / d t\)

Step-by-Step Solution

Answer

\( \frac{dS}{dt} = 8 \pi r \frac{dr}{dt} \)

Step 1: Identify Known Functions

We are given that the surface area of a sphere is a function of its radius, given by the formula \( S = 4 \pi r^2 \). Also, both the radius \( r \) and the surface area \( S \) are functions of time \( t \).

Step 2: Differentiate with Respect to Time

We are required to find \( \frac{dS}{dt} \). To do this, we need to differentiate the surface area formula \( S = 4 \pi r^2 \) with respect to time \( t \).

Step 3: Apply the Chain Rule

To find the derivative \( \frac{dS}{dt} \), apply the chain rule. Derivative of \( r^2 \) with respect to \( t \) is \( 2r \frac{dr}{dt} \). Thus, the derivative of \( S \) becomes: \[ \frac{dS}{dt} = \frac{d}{dt}(4 \, \pi \, r^2) = 4 \pi \cdot 2r \frac{dr}{dt} \]

Step 4: Simplify the Expression

Simplifying the expression above gives: \[ \frac{dS}{dt} = 8 \pi r \frac{dr}{dt} \]. This equation relates the rate of change of the surface area of the sphere to the rate of change of the radius.

Key Concepts

Understanding the Chain Rule

In calculus, differentiation is a powerful tool used to understand how functions change. One important technique is the chain rule. This is essential when dealing with composite functions – functions composed of other functions. The chain rule helps us express the derivative of a composite function in terms of the derivative of its inner functions.
If you think of a function within another function, that is where the chain rule comes in handy. For example, if you have a function like \( y = f(g(x)) \), where \( g(x) \) is the inner function, the chain rule allows us to differentiate this by saying:

\( \frac{dy}{dx} = f'(g(x)) \cdot g'(x) \).
This formula states that to find the derivative of \( y \) with respect to \( x \), we simply multiply the derivative of the outer function \( f \) evaluated at \( g(x) \) by the derivative of the inner function \( g \) with respect to \( x \). It’s a method that breaks a seemingly complex problem into easier parts.
In the context of the original exercise involving a sphere, we see the chain rule illuminate how changes in one quantity (like radius) affect another (surface area). This idea is not just crucial for mastering calculus, but also for comprehending how interconnected parts of a system work together in various scientific fields.

Calculating the Surface Area of a Sphere

Delving into geometry, the surface area of a sphere can be calculated using the well-known formula \( S = 4 \pi r^2 \).
Here, \( S \) represents the surface area, and \( r \) is the sphere's radius. This formula provides a simple relationship, showing how the surface area scales with the square of the radius.
As the radius increases, the surface area grows quadratically.

When we differentiate the expression for the surface area with respect to time, it provides insights into how quickly the surface area changes as the radius changes over time. This is precisely what was done in the solution to the original problem. By applying the principles of differentiation, one can observe how the inner dynamics of a sphere's growth affect its overall size.
This knowledge is pivotal when tackling problems not only in straightforward geometric contexts but also in more complex situations where these concepts apply.

Application of Calculus in Biology and Medicine

Calculus finds extensive applications in the fields of biology and medicine where processes change over time. For instance, understanding how populations grow or decay requires the use of differential calculus.
Similarly, medical studies involving the growth of tumors or the spread of diseases often rely on calculus for accurate modeling and predictions. In these cases, differentiation helps to describe rates of change, an essential aspect in understanding dynamic biological systems.

The chain rule, as introduced earlier, is an integral part of this arsenal, particularly when dealing with complex models where one factor influences another. Such as in pharmacokinetics, where the concentration of a drug in the body as a function of time can be influenced by various factors like dosage and metabolism rate.
Calculus, especially when applied using methods such as the chain rule, enables scientists and healthcare professionals to predict changes and outcomes under different scenarios, making it a powerful tool for problem-solving in practical, real-world situations.
This underscores the importance of understanding calculus not just as an abstract mathematical concept but as a crucial lens through which we explore and model our world.