Problem 28 - Applied Calculus

Question

Suppose \(f(x)=\frac{1}{3} x^{3} .\) Estimate \(f^{\prime}(2), f^{\prime}(3),\) and \(f^{\prime}(4)\) What do you notice? Can you guess a formula for \(f^{\prime}(x) ?\)

Step-by-Step Solution

Answer

The derivative values are squares of the input: \(f'(2)=4\), \(f'(3)=9\), \(f'(4)=16\). The formula is \(f'(x)=x^2\).

Step 1: Understand the Problem

We are given a function \(f(x) = \frac{1}{3}x^3\) and need to estimate its derivative \(f'(x)\) when \(x = 2, 3, \) and \(4\). We are also asked to predict a general formula for \(f'(x)\).

Step 2: Find the Derivative Formula

First, let's find the derivative of \(f(x) = \frac{1}{3}x^3\). Using the power rule, \(\frac{d}{dx}[x^n] = nx^{n-1}\), we find \[ f'(x) = \frac{1}{3} \cdot 3x^{3-1} = x^2. \] Therefore, the derivative function is \(f'(x) = x^2\).

Step 3: Calculate \(f'(2)\)

Substitute \(x = 2\) into the derivative formula \(f'(x) = x^2\) to estimate \(f'(2)\). \[ f'(2) = (2)^2 = 4. \]

Step 4: Calculate \(f'(3)\)

Substitute \(x = 3\) into the derivative formula \(f'(x) = x^2\) to estimate \(f'(3)\). \[ f'(3) = (3)^2 = 9. \]

Step 5: Calculate \(f'(4)\)

Substitute \(x = 4\) into the derivative formula \(f'(x) = x^2\) to estimate \(f'(4)\). \[ f'(4) = (4)^2 = 16. \]

Step 6: Identify a Pattern

The derivative values are computed as \(f'(2) = 4\), \(f'(3) = 9\), and \(f'(4) = 16\). Notice that these values are the squares of 2, 3, and 4 respectively. This matches the function \(f'(x) = x^2\).

Step 7: Conjecture a Formula

Based on the derivative calculations and the observed pattern, we conjecture that the formula for \(f'(x)\) is \(x^2\). This is consistent with the general derivative derived earlier.

Key Concepts

Power Rule

The Power Rule is a fundamental concept in calculus, particularly useful for finding derivatives. It's simple and efficient, helping to quickly derive expressions where variables are raised to a power. The general form of the Power Rule is:

• If you have a function of the form \(y = x^n\), the derivative with respect to \(x\) is \(\frac{dy}{dx} = nx^{n-1}\).

This rule is exceptionally useful for polynomial functions. In our given function \(f(x) = \frac{1}{3} x^3\), using the Power Rule simplifies the process. We multiply the current power, 3, by the coefficient \(\frac{1}{3}\) to get 1, and reduce the power by 1 to get \(x^2\). Thus, the derivative is \(f'(x) = x^2\).
This results in a much quicker and easier calculation of derivatives for each input value of \(x\), as shown in subsequent steps of evaluating \(f'(2)\), \(f'(3)\), and \(f'(4)\). Understanding this rule allows students to tackle similar problems with confidence.

Derivative Estimation

Estimating derivatives involves calculating the rate of change for a function at a specific point. With the derivative formula we got from the Power Rule, \(f'(x) = x^2\), we can substitute desired values of \(x\) to estimate specific derivative values.
In the provided exercise, we calculated:

• \(f'(2) = (2)^2 = 4\)
• \(f'(3) = (3)^2 = 9\)
• \(f'(4) = (4)^2 = 16\)

Each of these values represents the slope of the function \(f(x)\) at the corresponding \(x\) points, showing how the function is changing at that specific point. Practice like this helps in predicting future values and understanding how functions behave under different circumstances.
Recognizing this pattern is key in calculus, forming a core understanding of how to tackle real-world problems involving rates of change.

Function Analysis

Function analysis in calculus involves looking at the behavior of functions to understand their properties and implications. For the function \(f(x) = \frac{1}{3}x^3\), analyzing its derivative \(f'(x) = x^2\) helps us observe several things:

• As \(x\) increases, \(f'(x)\) also increases. This indicates a growing rate of change, meaning the function itself becomes steeper as \(x\) values rise.
• The derivative, \(f'(x) = x^2\), suggests that the graph of \(f'(x)\) is a parabolic curve opening upwards, which aligns with positive quadratic functions.
• Such analytical insights allow us to predict behavior across intervals and understand the underlying mathematics governing change.

These analyses play a significant role in engineering, physics, and various fields where prediction and understanding of variable interaction are crucial. Noticing these patterns enhances problem-solving skills and deepens one's understanding of calculus' practical applications.