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Graphing functions is an essential tool in calculus, allowing us to visualize the behavior and key features of an equation.
For the function \( y = \frac{1}{2}x - \ln x \), the process involves combining the elements of a linear and a logarithmic function.
The linear portion, \( \frac{1}{2}x \), suggests a straight line with a gentle upward slope. However, the logarithmic portion, \(-\ln x \), adds a twist to this story.
Since \( \ln x \) increases slowly and remains positive for \( x > 0 \), it starts to take a dominant role when \( x \) is small.

• When graphing, observe that for very small \( x \), the logarithmic part tends to decrease more quickly than the linear term can increase.
• For larger values of \( x \), the linear component becomes more significant, causing the graph to rise.

Understanding how to visualize the influence of each component provides a solid grasp of graphing functions, paving the way for deeper analysis.

A tangent line to a curve at a given point is a straight line that touches the curve at that exact point and has the same slope as the curve does at that point.
This concept is crucial when examining a function like \( y = \frac{1}{2}x - \ln x \) because it gives insight into the function's rate of change.
At any point \( x = a \), the slope of the tangent line to the curve is determined by the derivative of the function at that point.

• This means it literally "matches" the behavior of the curve at just that single touch point.
• In terms of geometry, a tangent line could rise, fall, or remain flat depending on whether the curve is turning upwards or downwards at that spot.

In essence, analyzing tangent lines helps to consume more profound ideas about the "steepness" or direction in which a function is headed at any given point.

Derivatives are a cornerstone of calculus, representing the rate of change of a function.
For the function \( y = \frac{1}{2}x - \ln x \), the derivative, \( \frac{dy}{dx} = \frac{1}{2} - \frac{1}{x} \), provides a powerful tool.
This derivative informs us how the function changes concerning \( x \).

• At any point, the derivative tells us whether the function is increasing (upwards slope), decreasing (downwards slope), or staying the same (slope of zero).
• Calculating the derivative enables us to understand when these slopes change, and what these changes mean in terms of the original function.

By analyzing derivatives, we gain a deeper understanding of the behaviors of functions and can identify important features like turning points where the tangent may be horizontal.

A horizontal tangent line occurs where the derivative of a function equals zero. This indicates that there is no steepness at that point—essentially, the function is neither increasing nor decreasing at that exact moment.
For \( y = \frac{1}{2}x - \ln x \), setting \( \frac{dy}{dx} = \frac{1}{2} - \frac{1}{x} = 0 \) identifies these points. Solving this equation shows that \( x = 2 \) is where the slope becomes zero.
Why is this important?

• Horizontal tangents often signify local maxima or minima on a graph: points where a curve turns back on itself.
• For this function, identifying \( x=2 \) as a point of horizontal tangency reveals a critical transition point in the graph.

Through such analysis, horizontal tangent lines serve not only as markers of change but as vital checkpoints in understanding the broader picture of a function's journey.