Problem 102 - University Calculus: Early Transcendentals

Question

A particle moves along the \(x\) -axis with velocity \(d x / d t=f(x) .\) Show that the particle's acceleration is \(f(x) f^{\prime}(x)\).

Step-by-Step Solution

Answer

The particle's acceleration is \(f(x) f'(x)\) by applying the chain rule.

Step 1: Understanding Given Information

We are given that the particle's velocity is represented by the function \(\frac{dx}{dt} = f(x)\), which means it depends on the position \(x\) of the particle on the \(x\)-axis.

Step 2: Using Chain Rule for Derivatives

Acceleration is the derivative of velocity with respect to time, \(\frac{d}{dt}\left(\frac{dx}{dt}\right)\). We can find the acceleration by applying the chain rule: \(\frac{d}{dt}\left( f(x) \right) = \frac{df}{dx} \cdot \frac{dx}{dt}\).

Step 3: Substitute Given Velocity

We substitute \(\frac{dx}{dt} = f(x)\) into \(\frac{d}{dt}( f(x) )\), changing it to \(\frac{df}{dx} \cdot f(x)\).

Step 4: Express Derivative in Terms of Functions

The derivative \(\frac{df}{dx}\) is simply the derivative of \(f(x)\) with respect to \(x\), which is \(f'(x)\).

Step 5: Combine Results

By substituting all expressions, we get the particle's acceleration as \(f(x) \cdot f'(x)\).

Key Concepts

Velocity and Acceleration

Understanding the concepts of velocity and acceleration is crucial in physics and differential calculus. Velocity indicates how fast an object moves along a path, and it is the rate at which an object changes its position. In mathematical terms, velocity is often defined as a function of time:

• In the original exercise, velocity is given as \( \frac{dx}{dt} = f(x) \), signifying how the position \( x \) of the particle changes with time.

Acceleration, on the other hand, tells us how quickly the velocity itself changes. Therefore, acceleration is the derivative of velocity with respect to time:

• The particle's acceleration is given by the derivative \( \frac{d}{dt}(\frac{dx}{dt}) \).
• This can be found using the chain rule, leading to an expression \( f(x) \cdot f'(x) \) as shown in the solution.

Chain Rule

The chain rule is a fundamental theorem in differential calculus used to differentiate composite functions. It is essential for finding the derivative of one function with respect to another. Let's break it down into simple steps:

• Suppose you have a composite function \( g(h(x)) \). The chain rule states that the derivative of this function is \( g'(h(x)) \cdot h'(x) \).

In the context of the exercise, to find the acceleration, we apply the chain rule to \( f(x) \), where:

• \( f(x) \) depends on the position \( x \).
• \( f'(x) \) is the rate of change of \( f(x) \) with respect to \( x \).

By applying the chain rule, you multiply the derivative of the outer function by the derivative of the inner function, resulting in the expression \( f(x) \cdot f'(x) \) for acceleration.

Derivatives

In calculus, derivatives provide a way to understand how a function changes as its input changes. Derivatives are foundational to differential calculus and have many practical applications, including understanding motion.

• A derivative, denoted \( \frac{d}{dt} \) or \( \frac{df}{dx} \), represents the rate of change of a function.
• In the original problem, \( \frac{dx}{dt} \) is the derivative representing velocity. It tells us how position \( x \) changes over time \( t \).
• Likewise, \( f'(x) \) is the derivative representing how \( f(x) \) changes with \( x \).

Understanding these concepts, students can effectively calculate changes, like velocity or acceleration, using derivatives. This enables them to explore deeper into changes of motion and solve various problems in calculus and physics.