Problem 69 - Calculus Single Variable

Question

In the Product Rule for \((f \cdot g)^{\prime}(t),\) there is a term \(f^{\prime}(t)\) \(g(t)\) and a term \(f(t) \cdot g^{\prime}(t)\) but no term involving \(f^{\prime}(t) \cdot g^{\prime}(t)\) Use dimensional analysis to explain why no such term should be present.

Step-by-Step Solution

Answer

The term \(f'(t) \cdot g'(t)\) has unmatched dimensions and thus should not be included.

Step 1: Understand the Function Dimensions

Consider functions \(f(t)\) and \(g(t)\) where \(t\) is a time variable with dimensions \([T]\). Assume \(f(t)\) and \(g(t)\) are physical quantities with certain units. Let \(f(t)\) have dimension \([F]\) and \(g(t)\) have dimension \([G]\). Consequently, the product \(f(t) \cdot g(t)\) has dimension \([F][G]\).

Step 2: Differentiate Using the Product Rule

The Product Rule states that the derivative of the product \((f \cdot g)'(t)\) is given by each function differentiated once and multiplied by the other, resulting in: \(f'(t) g(t) + f(t) g'(t)\). Here, \(f'(t)\) and \(g'(t)\) represent the derivatives of \(f\) and \(g\). The dimension of \(f'(t)\) is \([F]/[T]\), and similarly for \(g'(t)\) it is \([G]/[T]\).

Step 3: Analyze Dimensions of Derivative Terms

Examine the dimensions of the derivative terms: - For \(f'(t) g(t)\), the dimension is \(([F]/[T])([G]) = [F][G]/[T]\).- For \(f(t) g'(t)\), the dimension is \(([F])([G]/[T]) = [F][G]/[T]\).

Step 4: Consider the Hypothetical Third Term

If a term \(f'(t) \cdot g'(t)\) were present, its dimension would be \(([F]/[T])([G]/[T]) = [F][G]/[T^2]\). This dimension \([F][G]/[T^2]\) does not match the dimension \([F][G]/[T]\) of the other terms \(f'(t)g(t)\) or \(f(t)g'(t)\).

Step 5: Conclude Dimensional Consistency

The term \(f'(t) \cdot g'(t)\) introduces an unmatched dimension, indicating that its presence would violate dimensional consistency. Hence, such a term should not exist in the derivative according to the Product Rule.

Key Concepts

Dimensional Analysis

Dimensional analysis is a crucial tool in understanding the consistency of equations, especially in physics and engineering. It involves the study of the dimensions of physical quantities, which helps ensure equations are balanced in terms of units. This balance is vital for equations to accurately reflect real-world phenomena.
When dealing with dimensional analysis, you examine the units each component of an equation possesses.

• Dimensions are often represented by symbols such as $[F]$ for units of function, $[G]$ for another, and $[T]$ for time.
• By matching these dimensions, you can verify whether terms in an equation are consistent with each other.

In the product rule context, ensuring dimensional consistency means that each term checked should carry the same dimensions. If a proposed term had different dimensions, it would suggest an error in formulation. Therefore, dimensional analysis serves as a consistency check.

Derivative

Derivatives are a fundamental concept in calculus, representing the rate of change of a function relative to a variable. If you consider a function $f(t)$ with respect to time $t$, the derivative $f'(t)$ provides insight into how $f$ changes as $t$ changes.
Here are key aspects of derivatives you should know:

• The derivative of a function at a point gives the slope of the tangent line at that point.
• In the context of time, $f'(t)$ shows how quickly $f(t)$ is varying with time.
• Applying derivatives to function products necessitates the Product Rule, which ensures that both functions in the product are differentiated appropriately.

Understanding derivatives is essential as they are employed across physics, engineering, and many other fields to describe dynamic systems.

Function Dimensions

Grasping the dimensions of functions allows you to comprehend the nature of the quantities involved. Consider a function $f(t)$, which might represent a physical quantity like distance, velocity, or force. Its dimension is denoted by $[F]$, providing clarity on what the function actually measures.
When dealing with two functions $f(t)$ and $g(t)$:

• Their product $f(t) g(t)$ will possess combined dimensions $[F][G]$.
• This signifies the interaction between the two different quantities and their respective dimensions.

For derivatives, the dimension becomes $[F]/[T]$ for a single function's rate of change relative to time $t$. Recognizing these functional dimensions helps in analyzing the physical meaning behind mathematical calculations.

Calculus

Calculus forms the foundation of modern mathematical analysis and is pivotal in exploring how things change. It's divided mainly into two branches: Differential calculus focuses on derivatives, while integral calculus deals with integrals.
Some fundamental concepts that calculus covers include:

• Limits, which provide a way to understand behavior as inputs approach a specific value.
• Differential equations, governing systems and processes evolving over time.
• The integral, which accumulates quantities and is the inverse operation to differentiation.

Through these principles, calculus aids in modeling real-world phenomena, making it indispensable in physics, engineering, and economics. In the context of this exercise, calculus—specifically the Product Rule—enables the differentiation of products of functions by considering their individual rates of change.