Problem 48 - Applied Calculus

Question

Estimate the relative rate of change of \(f(t)=t^{2}\) at \(t=4\) Use \(\Delta t=0.01\)

Step-by-Step Solution

Answer

The relative rate of change of \( f(t) = t^2 \) at \( t = 4 \) is 0.5.

Step 1: Understand the Problem

The relative rate of change of a function at a certain point is given by the formula: \( \text{Relative Rate of Change} = \frac{f'(t)}{f(t)} \). Here, \( f(t) = t^2 \), and we are asked to find this rate at \( t = 4 \).

Step 2: Find the Derivative

First, we need to find the derivative of the function \( f(t) = t^2 \). Using power rule, the derivative \( f'(t) = 2t \).

Step 3: Calculate the Derivative at the Given Point

Substitute \( t = 4 \) into the derivative \( f'(t) = 2t \). This gives \( f'(4) = 2 \times 4 = 8 \).

Step 4: Calculate the Function Value at the Given Point

Substitute \( t = 4 \) into the original function \( f(t) = t^2 \). This gives \( f(4) = 4^2 = 16 \).

Step 5: Calculate the Relative Rate of Change

Use the previously found values in the relative rate of change formula: \( \frac{f'(4)}{f(4)} = \frac{8}{16} = 0.5 \). Therefore, the relative rate of change is 0.5.

Key Concepts

Derivative Calculation

Calculating derivatives is a fundamental concept in calculus that involves finding the rate at which a function changes at any given point. For the function \( f(t) = t^2 \), we want to determine how it changes with respect to \( t \). The derivative, denoted as \( f'(t) \), provides this valuable insight.
To find the derivative of our function \( f(t) = t^2 \), we will apply the power rule. Once determined, the derivative can be used to assess the function's behavior, such as identifying increasing or decreasing trends at specific values of \( t \). In our problem, solving for the derivative provides a critical piece in understanding the relative rate of change at \( t = 4 \).
In simple terms, think of the derivative as a tool that reveals the slope of the tangent line to the function at a specific point. This slope directly links to the concept of instantaneous rate of change, making it an essential step in evaluating the relative rate of change.

Power Rule

The power rule is an efficient and straightforward technique used to find derivatives of functions that have variables raised to a power. For a function expressed as \( f(t) = t^n \), where \( n \) is any real number, the derivative is given by \( f'(t) = nt^{n-1} \). This means you simply multiply the function by the power and decrease the power by one.
The simplicity of the power rule makes it one of the easiest differentiation methods, allowing us to quickly derive functions like \( f(t) = t^2 \). By applying the power rule, we find that the derivative of \( t^2 \) is \( f'(t) = 2t \). This form is elegant and shows how the slope of our function \( f(t) \) concerning \( t \) changes over its domain.
Understanding and effectively applying the power rule empowers students to quickly compute derivatives for polynomial functions, paving the way for more complex calculus concepts.

Function Evaluation

Function evaluation is the process of determining the output of a function for a particular input. In our exercise, evaluating the function at specific points helps us gain concrete values that aid in solving problems like finding the relative rate of change.
For \( f(t) = t^2 \), to estimate the relative rate of change at \( t = 4 \), we start by evaluating the function at \( t = 4 \). Plugging in 4, we calculate \( f(4) = 4^2 = 16 \). This provides the immediate value of the function at the point where we wish to analyze its behavior.
Evaluating the function may seem straightforward, but it becomes crucial in real-world applications where such assessments can dictate important decisions. By understanding how to evaluate functions, students not only solve mathematical problems but also understand the practical implications of the data produced.