91Ó°ÊÓ

A derivative is a fundamental concept in calculus, capturing how a function changes at any given point. When we find the derivative of a function, we are looking for the rate of change or the slope of the tangent line to the curve at a specific point. In the given exercise, you started with a function described as

• \(y = x - \frac{1}{4} x^2\)

By taking the derivative of this expression, we derive its rate of change with respect to \(x\). The derivative \(f'(x)\) for your function was computed by applying basic rules of differentiation, resulting in

• \(f'(x) = 1 - \frac{1}{2}x\)

This tells us how the curve's slope alters at different values of \(x\). Notice how the derivative expression changes as \(x\) changes, revealing insights regarding the nature of the curve.

Differentiation is the process of finding a derivative, which mathematically represents the slope of a function at any point. It plays a crucial role in understanding the behavior of graphs and their geometry. For instance, to determine how 'curvy' our function \(y = x - \frac{1}{4}x^2\) is at any spot, we start by differentiating the function.
Differentiating gently peels back layers of complexity, moving from the original function to its derivative \(f'(x) = 1 - \frac{1}{2}x\), and further to the second derivative \(f''(x) = -\frac{1}{2}\).

• The first derivative \(f'(x)\) reveals the slope of the tangent.
• The second derivative \(f''(x)\) sheds light on how this slope evolves, indicating the concavity or convexity of the curve.

Here, the constant value for \(f''(x)\) simplifies our journey in finding curvature, as it tells us the rate of change of the slope is consistent across \(x\).

Curvature is a measure of how sharply a curve bends at a particular point. The formula for curvature \(\kappa\) is expressed as

• \(\kappa = \frac{|f''(x)|}{(1 + (f'(x))^2)^{3/2}}\)

This equation elegantly combines both the first and second derivatives of a function to quantify its bending. In our given math problem, we need to utilize this formula at \(x = 2\).
First, evaluate the derivatives at this point:

• \(f'(2) = 0\)
• \(f''(2) = -\frac{1}{2}\)

Then, plug these values into the curvature formula to find

• \(\kappa = \frac{| -\frac{1}{2} |}{(1 + 0^2)^{3/2}} = \frac{1/2}{1} = \frac{1}{2}\)

Therefore, the curvature of the curve at \(x=2\) is \(\frac{1}{2}\), providing the exact measure of its sharpness and confirming how the curve bends precisely at that point.