91Ó°ÊÓ

The Distance Formula is a vital concept in geometry and algebra, helping us measure the straight-line distance between two points in a plane. Imagine you have two points, say \((x_1, y_1)\) and \((x_2, y_2)\). The formula for calculating the distance between these points is given by: \[D = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2}\]

• "D" represents the distance between the two points.
• The expression inside the square root is the sum of the squares of the differences in the x-coordinates and y-coordinates.

By squaring these differences, we ensure that everything remains positive, allowing us to find the true distance without worrying about the direction.

In our exercise, the Distance Formula helps us calculate how far \(\left(\frac{3}{2}, 0\right)\) is from any point \(x, \sqrt{x}\)\ on the curve \(y = \sqrt{x}\). This foundational step allows us to proceed towards finding the shortest possible distance by using optimization techniques.

Optimization is all about finding the best solution to a problem. In the context of this exercise, we are searching for the shortest distance between a specific point and a curve. When you want to optimize a function, it's often useful to consider minimizing or maximizing it. With distance problems, minimizing the squared distance instead of the actual distance is usually more convenient mathematically.

• We work with \(f(x) = (x - \frac{3}{2})^2 + x\), which is the function for the squared distance.
• We find a point where the slope of \(f(x)\) is zero, meaning we've found a local minimum or maximum.

In this case, using calculus we find that the derivative \(f'(x)\) is zero at \(x = 1\), indicating a critical point. This signifies that at \(x = 1\), we've identified the closest point on the curve to the given point, achieving our optimization goal.

Curve Analysis involves understanding how a curve behaves and is applied here to find the least distance from a point to a curve. In this problem, analyzing \(y = \sqrt{x}\) is crucial.The curve \(y = \sqrt{x}\) is part of the family of functions known as root functions. This particular curve starts at \(x = 0\) and gently rises as \(x\) increases, with a characteristic curve.

• The point \((1, 1)\) is found during evaluation at the critical point \(x = 1\). It lies on our curve \(y = \sqrt{x}\).
• This corresponds to the shortest path from \(\left(\frac{3}{2}, 0\right)\) to the curve.

By applying the distance formula to this point on the curve, we can confirm that it provides the minimum possible distance, showing how useful curve analysis is when solving real-world optimization problems.