91Ó°ÊÓ

In mathematics, the rate of change of a function measures how a quantity changes with respect to another. It is a crucial concept that appears in various forms across different fields. In calculus, the rate of change is often represented by the derivative. For the function given, \( f(x) = 4x^2 \), the derivative represents how the function's output changes concerning changes in \( x \).

The rate of change tells us the slope of the tangent line at any point \( x \) on the curve of the function. This slope gives information about how steep the curve is at that point and in which direction it is moving. Understanding this concept allows us to predict how small adjustments to \( x \) will influence \( f(x) \).

• A positive rate of change indicates the function is increasing.
• A negative rate of change shows the function is decreasing.
• A rate of change of zero suggests a horizontal tangent, indicating a local maximum, minimum, or inflection point.

The algebraic method involves using the limit definition of the derivative to find the rate of change function. This process is analytical and requires establishing a difference quotient, which captures how \( f(x) \) changes over a small interval \( h \).

Using the limit definition, \( f'(x) = \lim_{{h \to 0}} \frac{f(x+h) - f(x)}{h} \), allows us to derive an expression that represents the rate of change for \( f(x) = 4x^2 \). By substituting the given function into this formula, we form the difference quotient:

• Expand \((x+h)^2\) to simplify \( f(x+h) = 4(x+h)^2 = 4x^2 + 8xh + 4h^2\).
• Subtract \( f(x) = 4x^2 \) from \( f(x+h) \) to form \( \frac{8xh + 4h^2}{h} \).
• Factor out \( h \) and simplify to \( 8x + 4h \).

When \( h \to 0 \), we can cancel \( h \) in the expression, enabling us to solve for \( f'(x) = 8x \), which is the general formula for the rate of change of the function.

Derivative evaluation is the process of calculating the value of the derivative for a specific point on the function's curve. This step is vital for practical applications, such as determining the rate of change at particular points.

Once the derivative function \( f'(x) = 8x \) is established, evaluating it means substituting a specific \( x \) value into this expression. For this exercise, we calculate the derivative at \( x = 2 \).

• Substitute \( x = 2 \) into \( f'(x) = 8x \).
• Solve for \( f'(2) = 8(2) = 16 \).

This calculation shows the function \( f(x) = 4x^2 \) is increasing at a rate of 16 units of change per unit \( x \) at the point where \( x = 2 \). Understanding the result aids in predicting how rapid changes occur at this specific value. It effectively demonstrates how derivatives inform us about the dynamics of functions in real-world contexts.