Problem 46 - University Calculus: Early Transcendentals

Question

Write a differential formula that estimates the given change in volume or surface area. The change in the volume \(V=x^{3}\) of a cube when the edge lengths change from \(x_{0}\) to \(x_{0}+d x\)

Step-by-Step Solution

Answer

The change in volume \(dV\) is estimated by \(dV = 3x_0^2 \cdot dx\).

Step 1: Understand the Problem

You need to find a formula that estimates the change in volume of a cube when its edge length changes slightly from \(x_0\) to \(x_0 + dx\). The change in volume refers to the infinitesimal change as \(dx\) approaches zero.

Step 2: Recall the Volume Formula of a Cube

The volume \(V\) of a cube with edge length \(x\) is given by \(V = x^3\). Our goal is to estimate the differential change in \(V\) when there is a small change in \(x\).

Step 3: Differentiate the Volume Formula

To estimate the change in volume, you use calculus to find the derivative of \(V = x^3\) with respect to \(x\). This derivative represents the rate of change of the volume with respect to changes in \(x\).

Step 4: Compute the Derivative

Differentiate \(V = x^3\) with respect to \(x\): \[\frac{dV}{dx} = 3x^2\].

Step 5: Apply the Differential Formula

Using the derivative, the differential change in volume \(dV\) can be estimated as \(dV = \frac{dV}{dx} \cdot dx = 3x^2 \cdot dx\). This formula provides an estimate for how much the volume changes when \(x\) changes by a small amount \(dx\).

Step 6: Substitute the Initial Edge Length

Substitute \(x = x_0\) into the differential formula to find the specific change in volume: \[dV = 3x_0^2 \cdot dx\].

Key Concepts

Volume of a Cube

The volume of a cube is a fundamental concept in geometry and calculus. It refers to the amount of three-dimensional space enclosed by the cube's surfaces. For a cube with side length \(x\), the volume \(V\) can be calculated using the formula: \[ V = x^3 \]This formula arises because a cube has equal side lengths, and multiplying the length, width, and height (all of which are \(x\) for a cube) gives the volume. Understanding this calculation is crucial when analyzing how small changes in the cube's dimensions affect its overall volume. Knowing how the volume is calculated helps us understand why we use derivatives to find changes in the volume when its edge length changes. Such mathematical principles are essential in many fields, like physics and engineering, where volume calculations are regularly needed.

Derivative

The derivative is a key concept in calculus that measures how a function changes as its input changes. It can be thought of as the "instantaneous rate of change" or the "slope" of the function at any given point. When you take the derivative of a function, you're essentially finding how the output changes for very small changes in the input. In the context of the volume of a cube, we differentiate the volume formula \(V = x^3\) to determine how the volume changes with a small change in edge length. Calculating the derivative of the volume with respect to \(x\) gives us: \[ \frac{dV}{dx} = 3x^2 \]This formula tells us that for each unit change in \(x\), the volume changes by \(3x^2\). Derivatives are powerful tools in differential calculus, used to analyze and predict changes in various real-world applications.

Differential Formula

The differential formula is used to estimate small changes in a function's output. It uses the concept of derivatives to provide a linear approximation of the change. For the volume of a cube, the differential formula allows us to estimate how much the volume will change when the edge length changes slightly.Starting with the derivative \(\frac{dV}{dx} = 3x^2\), we can express the differential change in volume \(dV\) as:\[ dV = \frac{dV}{dx} \cdot dx = 3x^2 \cdot dx \]This equation gives us a way to calculate the change in volume \(dV\) for a small change \(dx\) in the cube’s edge.This approach is practical in real-world situations where exact calculations aren't feasible due to measurement limitations. By using the differential formula, tiny adjustments in variables can be accounted for accurately.

Infinitesimal Changes

Infinitesimal changes are an important concept in differential calculus, describing very tiny adjustments in a variable. In mathematical terms, they are changes that approach zero but never quite reach it. These subtle changes are important when examining how a function behaves under slight variations.In the context of a cube's volume, infinitesimal changes in the edge length \(x\) lead to infinitesimal changes in its volume \(V\). By applying the differential formula \(dV = 3x^2 \cdot dx\), we can understand how these small changes, \(dx\), result in corresponding changes, \(dV\), in volume.Understanding infinitesimals helps students appreciate the precision of calculus. Even the smallest changes can have significant, calculable effects, which is crucial in contexts like engineering, where accuracy is paramount.