Closest point on a curve What point on the parabola y = 1 - x² is closest to the point (1, 1)?

Verified step by step guidance

1. First, understand that the problem is asking for the point on the parabola y = 1 - x² that is closest to the point (1, 1). This involves finding the minimum distance between a point on the curve and the given point.
2. The distance between any point (x, y) on the parabola and the point (1, 1) can be expressed using the distance formula: d = √((x - 1)² + (y - 1)²).
3. Substitute y = 1 - x² into the distance formula to express the distance in terms of x only: d = √((x - 1)² + ((1 - x²) - 1)²).
4. Simplify the expression inside the square root: d = √((x - 1)² + (-x²)²) = √((x - 1)² + x⁴).
5. To find the minimum distance, take the derivative of the distance function with respect to x, set it equal to zero, and solve for x. This will give the x-coordinate of the point on the parabola closest to (1, 1).

Key Concepts

Distance Formula

The distance formula calculates the distance between two points in a Cartesian plane. For points (x₁, y₁) and (x₂, y₂), the distance d is given by d = √((x₂ - x₁)² + (y₂ - y₁)²). This formula is essential for determining how far a point on the parabola is from the point (1, 1), which is necessary for finding the closest point.

Optimization

Optimization in calculus involves finding the maximum or minimum values of a function. In this context, we need to minimize the distance from the point (1, 1) to points on the parabola y = 1 - x². This typically involves taking the derivative of the distance function, setting it to zero, and solving for critical points.

Parabola Properties

A parabola is a symmetric curve defined by a quadratic function, such as y = 1 - x². Understanding its shape, vertex, and direction is crucial for identifying points on the curve. The vertex of this parabola is at (0, 1), and it opens downward, which helps in visualizing and determining the closest point to (1, 1).