Problem 79 - Calculus Volume 1

Question

For the following exercises, find the change in volume \(d V\) or in surface area \(d A\) $$ d A \text { if the sides of a cube change from } x \text { to } x+d x $$

Step-by-Step Solution

Answer

The change in surface area is \( dA = 12x \, dx \).

Step 1: Understand the Problem

We need to find the change in the surface area of a cube when the length of its side changes from \( x \) to \( x + dx \).

Step 2: Recall the Surface Area Formula

The surface area \( A \) of a cube with side length \( x \) is given by \( A = 6x^2 \).

Step 3: Differentiate the Surface Area with Respect to x

To find how the surface area changes with a change in \( x \), we differentiate \( A = 6x^2 \) with respect to \( x \). This gives:\[ \frac{dA}{dx} =\frac{d}{dx}(6x^2) = 12x \]

Step 4: Determine the Change in Surface Area

The change in surface area \( dA \) when \( x \) changes by \( dx \) can be written as:\[ dA = \frac{dA}{dx} \, dx = 12x \, dx \]

Key Concepts

Differentiation

Differentiation is a fundamental concept in calculus that allows us to find the rate at which a quantity changes. It's like a tool that tells us how steep a curve is at any point.
During the step-by-step solution, we used differentiation to determine how the surface area of a cube changes as its side length changes.

• We started by identifying the formula for the surface area of a cube, which is given by \( A = 6x^2 \) where \( x \) is the side length.
• To find how \( A \) changes with \( x \), we took the derivative of \( A \) with respect to \( x \). This involved applying the basic power rule of differentiation.
• The power rule states that if you have \( x^n \), then the derivative is \( nx^{n-1} \). For \( A = 6x^2 \), the derivative is \( 12x \), meaning that for every incremental increase in \( x \), the surface area increases by \( 12x \times dx \).

Now, we can predict how small changes in one measurement can slightly modify another, which is very useful in many real-world applications.

Surface Area

Surface area is a measure of how much exposed area the surface of an object has. For a cube, it's all about counting each square face that makes up the figure.
The formula for the surface area \( A \) of a cube is \( A = 6x^2 \), where \( x \) is the side length.

• Why 6? A cube has 6 faces, and each face is a square with area \( x^2 \). So, you multiply \( x^2 \) by 6 to get the total surface area.
• However, what happens when \( x \) changes? If there is a small increase \( dx \) in \( x \), the change in surface area \( dA \) helps us understand how the total surface area of the cube is affected.

To find \( dA \), you consider the change in the surface area equation because of the change in \( x \). We differentiate \( A = 6x^2 \) with respect to \( x \) as explained, leading to \( dA = 12x \, dx \), indicating the incremental change in surface area based on how a tiny change in side length impacts the whole figure.

Rate of Change

The rate of change is a concept that helps to describe how one quantity changes in relation to another. In the context of our problem, we're looking at how the surface area of a cube changes as its side length changes.

- When we talk about the derivative \( \frac{dA}{dx} = 12x \), we're discussing the instantaneous rate of change of the surface area with respect to the cube's side length. This tells us how sensitive the surface area is to changes in the side length.- It's like knowing the speed of a car at a specific instant. In this scenario, it’s the surface area’s speed of growth as each side increases.In practical terms, understanding the rate of change allows us to predict behavior.

• For example, if \( x \) is really small, even smaller changes in \( dx \) can result in noticeable changes in surface area compared to when \( x \) is larger.
• This knowledge helps in designing objects, optimizing structures, or even analyzing natural phenomena where such incremental changes have significant impacts.

Therefore, mastering rate of change through differentiation gives valuable insights into how different elements in systems respond to small alterations.