Problem 14 - Calculus of a Single Variable

Question

Minimum Distance In Exercises \(13-16,\) find the point on the graph of the function that is closest to the given point. $$ f(x)=(x-1)^{2},(-5,3) $$

Step-by-Step Solution

Answer

The point on the graph of the function \(f(x) = (x-1)^2\) that is closest to the point (-5,3) is (-1,4).

Step 1: Set up the distance formula

First, set up the distance formula with the given point (-5,3) and a general point on the function \(f(x) = (x-1)^2\). This will be denoted as \(D = \sqrt{(x - (-5))^2 + ((x-1)^2 - 3)^2}\). We will aim to minimize the function \(D^2\) instead of \(D\) for simplicity as \(\sqrt{x}\) and \(x\) attain minimum at the same point.

Step 2: Take the derivative of the distance function

Next, simplify the equation to \(D^2 = (x + 5)^2 + (x-1)^2 - 6\). Then, take the derivative of \(D^2\) to find \(\frac{dD^2}{dx} = 2(x + 5) + 2(x - 1)\).

Step 3: Set the derivative equal to zero

To find the x-coordinate of the point on the function closest to (-5,3), set the derivative equal to zero and solve for x. So, \(2(x + 5) + 2(x - 1) = 0\) and solve for \(x\). After solving the equation we find that \(x = -1\).

Step 4: Identify the corresponding y-coordinate

Now that we know the x-coordinate of the point on the function that is closest to (-5,3), we find the corresponding y-coordinate by plugging \(x = -1\) into our function. So, \(f(-1) = ((-1)-1)^2 = 4\). Therefore, the closest point on the function to (-5,3) is (-1,4).

Key Concepts

Distance Formula

The distance formula is a fundamental expression used to determine the straight-line distance between two points in a coordinate system. In the context of the minimum distance problem in calculus, we use the distance formula to find the shortest path between a fixed point and a point on a curve.

The formula is based on the Pythagorean theorem and in a two-dimensional space it is given by:
\[D = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2}\]
where \((x_1, y_1)\) and \((x_2, y_2)\) represent the coordinates of two points. In the exercise, the distance formula was modified to include the function's definition \(f(x) = (x - 1)^2\) to describe the y-coordinate of a point on the curve, which was compared to the fixed point \((-5, 3)\).

Derivative Application

Derivatives represent the rate of change of a function and are a pivotal tool in finding maxima and minima—crucial tasks in optimization. In the minimum distance problem, the derivative application involves finding where the function has a minimum value, which corresponds to the shortest distance from the fixed point to the curve.

To illustrate, the derivative of the distance squared function, \(D^2\), with respect to \(x\) was found in the solution step. The derivative, \(\frac{dD^2}{dx}\), reveals how rapidly the square of the distance changes as the x-coordinate varies. By setting the derivative to zero, we identify potential points where the function could have a local minimum, indicating a possible shortest distance.

Optimization in Calculus

Optimization in calculus involves finding the maximum or minimum value of a function within a given interval or set of constraints. In solving the minimum distance problem, we optimize by searching for the point on the function that results in the smallest possible value of the distance squared, \(D^2\).

We prefer to work with \(D^2\) in optimizations involving distance because it eliminates the square root, simplifying the derivative calculation without affecting the location of the minimum, since the square root function is monotonic.

Function Analysis

Function analysis is the study of the properties of functions, such as their behavior, continuity, and differentiability. It helps us understand the function's structure and provides tools to evaluate its critical points, which are essential in identifying optimization solutions.

In this minimum distance problem, after setting the derivative to zero, we solve for \(x\) and analyze the result within the context of the original function. Computing the corresponding \(y\)-value with this \(x\) gives us the specific point on the curve that is nearest to the given point. This final synthesis is part of a thorough function analysis.