Problem 79 - Calculus Volume 1

Question

Find the change in volume \(d V\) or in surface area \(d A\). \(d A\) if the sides of a cube change from \(x\) to \(x+d x\).

Step-by-Step Solution

Answer

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

Step 1: Understand the Problem

We need to find the change in the surface area of a cube when its side length changes from \(x\) to \(x + dx\). The initial surface area of a cube with side length \(x\) is given by \(A = 6x^2\).

Step 2: Identify the Formula for Change in Area

To find the change in surface area \(dA\), we need to differentiate the formula for area with respect to \(x\) and then multiply by \(dx\). This gives the formula \(dA = \frac{d}{dx}(6x^2) \cdot dx\).

Step 3: Differentiate the Area Formula

Differentiate \(6x^2\) with respect to \(x\). The derivative \(\frac{d}{dx}(6x^2) = 12x\). This tells us how the surface area changes with respect to a change in side length \(x\).

Step 4: Calculate the Change in Surface Area

Now substitute the derivative into the formula for \(dA\): \(dA = 12x \cdot dx\). This gives the change in surface area in terms of \(dx\).

Key Concepts

Differentiation

Differentiation is a core concept in calculus. It is the process of finding the derivative of a function. The derivative represents how a function changes. In simpler terms, it shows the rate of change of one quantity with another.
For example, if you have a function that shows the position of an object over time, its derivative gives you the velocity of that object. In mathematical notation, the derivative of a function \( f(x) \) with respect to \( x \) is expressed as \( f'(x) \) or \( \frac{d}{dx} f(x) \).

• Instantaneous rate of change: The derivative gives us the instantaneous rate of change at any point on the function.
• Slope of a tangent: It also represents the slope of the tangent line to the function at any specific point.

In this exercise, we used differentiation to find how the surface area of a cube changes when its side length changes. By identifying the derivative of the surface area formula, we could determine the rate at which the surface area grows as the cube expands.

Surface Area of a Cube

The surface area of a cube is the total area covered by its six faces. A cube has square faces, and each face has the same side length, \( x \). To find the surface area \( A \) of a cube, we use the formula:
\[ A = 6x^2 \]Each face of the cube has an area of \( x^2 \), and there are six such faces. This is why the formula is \( 6x^2 \).

• Uniformity: Every face contributes equally to the total surface area.
• Direct relationship: The surface area of the cube increases as the square of its side length increases.

When the side of the cube changes by a small amount \( dx \), we want to know how this affects the total surface area. This leads us to use differentiation to estimate the change in the area, expressed as \( dA \).

Calculus

Calculus is the branch of mathematics focused on continuous change. It comprises two main concepts: differentiation and integration. In calculus, we analyze how quantities change and understand rates of change.

• Understanding dynamic systems: Calculus helps us understand systems where things are changing continuously. This applies to many fields like physics, engineering, and economics.
• Foundation of mathematical analysis: It provides the fundamental tools needed for mathematical modeling of systems.

In this exercise, we apply calculus techniques to determine changes in geometric properties, like surface area, when small changes occur in object dimensions. The key aspect here is using differentiation to predict how a small change in side length impacts the surface area of a cube.
Through calculus, we translate physical problems into mathematical equations, solve them, and interpret the solutions to gain insights about the real world.