Problem 2 - University Calculus: Early Transcendentals

Question

Suppose that the radius \(r\) and surface area \(S=4 \pi r^{2}\) of a sphere are differentiable functions of \(t .\) Write an equation that relates \(d S / d t\) to \(d r / d t\).

Step-by-Step Solution

Answer

The equation is \( \frac{dS}{dt} = 8\pi r \frac{dr}{dt} \).

Step 1: Review the given formula

The surface area of a sphere is given by the formula \( S = 4 \pi r^2 \), where \( S \) is the surface area and \( r \) is the radius of the sphere.

Step 2: Differentiate the surface area formula

To relate \( \frac{dS}{dt} \) to \( \frac{dr}{dt} \), differentiate both sides of the equation \( S = 4\pi r^2 \) with respect to \( t \).

Step 3: Apply the Chain Rule

When differentiating \( S = 4 \pi r^2 \) with respect to \( t \), apply the chain rule. The derivative \( \frac{d}{dt}(S) = \frac{dS}{dt} \) and \( \frac{d}{dt}(4\pi r^2) = 4\pi \cdot 2r \cdot \frac{dr}{dt} \).

Step 4: Write the related rates equation

Set the two derivatives equal to each other: \( \frac{dS}{dt} = 8\pi r \frac{dr}{dt} \). This equation relates \( \frac{dS}{dt} \) and \( \frac{dr}{dt} \).

Key Concepts

Chain rule

The Chain Rule is a fundamental concept in calculus used to find the derivative of a composition of differentiable functions. It's essential when dealing with related rates. In our specific exercise involving the surface area of a sphere, we want to understand how the surface area changes over time based on the changes in the radius of the sphere.

The sphere's surface area equation is a function of its radius, specifically, it's given by the formula \[ S = 4 \pi r^2 \]. To find how the surface area \( S \) changes with time \( t \), we use the Chain Rule. By differentiating both sides of the equation with respect to time \( t \), we find:

• The derivative of the surface area \( \frac{dS}{dt} \)
• The derivative of \( 4\pi r^2 \), which is \( 8\pi r \frac{dr}{dt} \)

This allows us to relate the rate of change of the surface area to the rate of change of the radius, showcasing the power of the Chain Rule in connecting related rates of change.

Differentiable functions

Differentiable functions have derivatives that are continuous within their domain, which makes them very important in calculus. When we talk about the radius \( r \) and surface area \( S \) of a sphere as differentiable functions of time \( t \), we imply smooth and predictable changes with respect to \( t \).

This continuity allows us to apply differentiation rules, like the Chain Rule, effectively. In the context of our example:

• The surface area \( S \) depends on the radius \( r \)
• The radius \( r \) itself changes over time

This smooth change is essential because it guarantees that we can compute meaningful derivatives, such as \( \frac{dS}{dt} \) and \( \frac{dr}{dt} \), giving us a full picture of how these values depend on each other over time.

Implicit differentiation

Implicit differentiation is a technique used when we have equations that define functions implicitly rather than explicitly. In the case of related rates, we often use implicit differentiation to uncover the relationships between rates of change of different quantities.

For example, in our exercise, the function relating surface area and radius does not directly solve for a rate of change over time. By differentiating implicitly with respect to time, we uncover the relationship between \( \frac{dS}{dt} \) and \( \frac{dr}{dt} \).

Here’s how implicit differentiation helps:

• We differentiate both sides of \( S = 4 \pi r^2 \) with respect to \( t \)
• The derivatives reveal the hidden relationship \( \frac{dS}{dt} = 8\pi r \frac{dr}{dt} \)

Through this approach, we can effectively analyze how quantities change together without needing explicit functions directly relating these quantities to time.