Problem 67 - Calculus an Applied Approach

Question

Find the derivative of the given function \(f\). Then use a graphing utility to graph \(f\) and its derivative in the same viewing window. What does the \(x\) -intercept of the derivative indicate about the graph of \(f ?\) $$ f(x)=x^{2}-4 x $$

Step-by-Step Solution

Answer

The derivative of the function \(f(x) = x^2 - 4x\) is \(2x - 4\). Its x-intercept is at \(x = 2\). This indicates that the original function \(f(x)\) changes its rate of increase or decrease at \(x = 2\).

Step 1: Find the Derivative

The derivative of a function can be found using the power rule, which states that the derivative of \(x^n\) is \(nx^{n-1}\). Applying this rule to the function \(f(x) = x^2 - 4x\), the derivative \(f'(x)\) would be \(2x - 4\).

Step 2: Find the X-intercept

The x-intercept of the derivative is the value of x for which \(f'(x) = 0\). From the equation \(2x - 4 = 0\), solving for x leads to \(x = 2\).

Step 3: Interpret the Meaning of the X-intercept

The x-intercept of the derivative indicates where the original function changes from increasing to decreasing, or vice versa. This is because the derivative of a function measures its rate of change: a positive derivative indicates the function is increasing, while a negative derivative indicates it's decreasing. Since \(x = 2\) is the x-intercept of the derivative, at this point the original function \(f(x)\) changes its rate of increase or decrease.

Key Concepts

Power Rule

The Power Rule is a fundamental technique used to find the derivative of functions, particularly polynomials. It's really simple and extremely useful. Essentially, if you have a function like \( x^n \), where \( n \) is any real number, the derivative is found by multiplying \( n \) by \( x \) raised to the power of \( n-1 \).
For example, if \( f(x) = x^2 \), applying the Power Rule gives \( f'(x) = 2x^{2-1} = 2x \).
In the original problem, the function \( f(x) = x^2 - 4x \) has been differentiated using the Power Rule. The derivative \( f'(x) = 2x - 4 \) reflects both terms in the original function:

• The term \( x^2 \) becomes \( 2x \).
• The term \(-4x \) becomes \(-4 \) as the derivative of a linear function \( ax \) is simply \( a \).

Makes sense, right? This process helps us understand how the function behaves as \( x \) changes.

Graphing Derivatives

Graphing the derivative of a function alongside the original function helps illustrate how the rate of change behaves. This can be seen visually in a graphing utility. When you plot both the function and its derivative:

• The original function is represented by the curve \( f(x) \).
• The derivative, \( f'(x) \), is often a linear or simpler curve that shows where and how fast \( f(x) \) changes.

Seeing both graphs can help you understand the relationship between a function and its derivative, such as where the original function is increasing or decreasing.
The derivative graph shows where the changes occur, helping pinpoint critical points, such as maxima and minima.
For \( f(x) = x^2 - 4x \), the graph reveals the point where the behavior of \( f(x) \) transforms, visually tying together the calculated derivative and the original function.

X-Intercept

The X-intercept of the derivative's graph is a crucial point of analysis. This is where the derivative, \( f'(x) \), equals zero, indicating no change at that point.
In the exercise, the X-intercept of the derivative \( 2x - 4 = 0 \) is found where \( x = 2 \).
Here's why it's important:

• At the X-intercept, the slope of the tangent to the graph of \( f(x) \) is zero. This often corresponds to a peak or valley on the graph of \( f(x) \).
• This point is where the function changes from increasing to decreasing or vice versa. In other words, it's a critical point.

By finding the X-intercept of the derivative, you can identify where \( f(x) \) experiences a change in its rate of change, making it an essential tool for analyzing functions.

Rate of Change

The rate of change of a function tells us how fast or slow the function's value is changing. The derivative provides this rate of change. A positive derivative means an increase in the function, while a negative derivative signifies a decrease.
In mathematical terms:

• A positive value of the derivative \( f'(x) \) means the function \( f(x) \) is increasing.
• A negative value of \( f'(x) \) suggests that \( f(x) \) is decreasing.

From the exercise, the derivative \( 2x - 4 \) shows that between specific intervals, the function \( f(x) = x^2 - 4x \) will rise or fall.
When interpreted graphically or numerically, the rate of change gives valuable insight into motion, growth, or any variable we're examining over time or space, aiding in comprehensive understanding and prediction.