91Ó°ÊÓ

To understand graphing functions, the concept of the first derivative plays a key role. The first derivative of a function reveals how the function is changing. For the function \( f(x) = x^{1/2} \), the first derivative is calculated using the power rule. The resulting derivative is \( f'(x) = \frac{1}{2\sqrt{x}} \). This formula allows us to determine how fast or slow the function is rising or falling.

The first derivative indicates the slope of the tangent line at any given point \( x \). A positive derivative suggests the function is increasing, while a negative derivative suggests it is decreasing. In our case, the derivative \( f'(x) = \frac{1}{2\sqrt{x}} \) is positive for all \( x > 0 \), indicating that the function is increasing across its entire domain.

Creating a sign diagram is a helpful tool to analyze the behavior of the first derivative across the function's domain. A sign diagram shows where the derivative is positive, negative, or zero.

For the function \( f(x) = x^{1/2} \), the domain is \( x > 0 \), since the derivative \( f'(x) = \frac{1}{2\sqrt{x}} \) is undefined at \( x = 0 \) and the function itself is not defined for \( x < 0 \).

In our sign diagram for \( f'(x) \), the entire domain for \( x > 0 \) shows a positive sign. This uniform positivity indicates that \( f(x) \) is consistently increasing as the value of \( x \) gets larger.

Critical points are locations on a graph where the derivative is either zero or undefined. These are key points because they can signify potential maxima, minima, or change in the behavior of the graph.

In the function \( f(x) = x^{1/2} \), we analyze the critical points by setting the first derivative to zero or checking where it is undefined. The derivative \( f'(x) = \frac{1}{2\sqrt{x}} \) does not equal zero for any \( x > 0 \) but is undefined at \( x = 0 \). Since the function cannot go below zero, \( x = 0 \) acts as a boundary, contributing to it being a critical point even if there's no relative maximum or minimum at this location.

An increasing function is one where the output grows larger with an increase in input. Mathematically, when the first derivative \( f'(x) > 0 \) over an interval, the function is increasing across that interval.

For \( f(x) = x^{1/2} \), since \( f'(x) = \frac{1}{2\sqrt{x}} \) is positive for the entire domain \( x > 0 \), the function is classified as increasing. This means any increase in \( x \) will result in a higher \( f(x) \) value, and graphically, the curve will rise from left to right. This characteristic shapes the entire behavior and appearance of the graph, consistently directing upwards without any dips or turns.