Problem 43 - Calculus Early Transcendentals

Question

Consider the following functions and express the relationship between a small change in \(x\) and the corresponding change in \(y\) in the form \(d y=f^{\prime}(x) d x\). \(f(x)=1 / x^{3}\)

Step-by-Step Solution

Answer

Question: Find the relationship between the changes in x and y, given the function \(f(x) = \frac{1}{x^3}\) in the form of \(dy = f'(x) dx\). Answer: \(dy = -3x^{-4} dx\)

Step 1: Apply the power rule for differentiation

Recall that the power rule for derivative states that if \(f(x) = x^n\), then \(f'(x) = n x^{n-1}\). Now, rewrite the given function as \(f(x) = x^{-3}\). Then, apply the power rule to find the derivative: \(f'(x) = -3x^{-4}\)

Step 2: Express the relationship between \(dy\) and \(dx\)

We now have the derivative of the function, which is \(f'(x) = -3x^{-4}\). It now remains to establish the relationship between \(dy\) and \(dx\) in the form of a differential equation. The derivative \(f'(x)\) represents the change in \(y\) with respect to the change in \(x\). Therefore, we can write the relationship as \(dy = f'(x) dx\).

Step 3: Write the final equation

Substitute \(f'(x) = -3x^{-4}\) to obtain the final equation: \(dy = -3x^{-4} dx\). This equation represents the relationship between the change in \(y\) and the change in \(x\) for the given function \(f(x) = \frac{1}{x^3}\).

Key Concepts

Power Rule

The power rule is a fundamental tool in calculus for finding the derivative of a function. It's particularly useful when dealing with polynomial expressions. The rule states that if you have a function of the form \(f(x) = x^n\), the derivative \(f'(x)\) is given by \(n \cdot x^{n-1}\). This means you take the exponent \(n\), multiply it by \(x\), and reduce the exponent by one.

To apply the power rule, it’s important first to rewrite the function in an appropriate form, which often means expressing the function with positive or negative exponents. For instance, the exercise involves \(f(x) = \frac{1}{x^3}\). By rewriting this as \(f(x) = x^{-3}\), it’s easier to apply the power rule.

• Identify the exponent \(n\).
• Multiply \(x\) by \(n\).
• Reduce the exponent by one.

For this function, applying the power rule results in the derivative \(-3x^{-4}\). The simplicity of the power rule makes it a preferred choice for differentiating monomial terms.

Derivative

The derivative of a function measures how the function's output changes as its input changes, essentially capturing the rate of change or the "slope" at any given point. In the exercise, by finding the derivative \(-3x^{-4}\), we are calculating how \(y\) changes relative to \(x\) for the function \(f(x) = \frac{1}{x^3}\).

Derivatives have widespread applications:

• Determining slopes of tangents to curves.
• Finding extreme values (maximums and minimums) of functions.
• Modeling real-world phenomena where change is occurring.

To find a derivative, it is crucial to understand derivative rules like the power rule, product rule, and chain rule. For simple functions, the power rule often suffices. As the derivative \(-3x^{-4}\) shows, when \(x\) is small (close to zero), \(f(x)\) changes quickly, since the derived function's value becomes large. Understanding derivatives helps in interpreting the behavior of functions.

Differential Equation

A differential equation expresses a relationship involving cycles of derivatives. In simpler terms, it relates functions with their derivatives, often used to capture how a particular quantity changes over time or space.

The exercise results in a direct relationship between the differentials \(dy\) and \(dx\). By reaching \(dy = -3x^{-4} dx\), we establish that a small change in \(x\) leads to a change in \(y\) scaled by the factor \(-3x^{-4}\). These equations are important in many fields

• Engineering: predicting system behavior.
• Physics: modeling motion and forces.
• Biology: describing population dynamics.

Solving a differential equation often means finding a function that satisfies the relationship. In this case, the differential equation tells you how to express \(dy\) when \(dx\) is known, helping in integrations and providing solutions to problems featuring rates of change.