Question

Let \(P=(x, y)\) be a point on the graph of \(y=\sqrt{x}\) (a) Express the distance \(d\) from \(P\) to the point (1,0) as a function of \(x\). (b) Use a graphing utility to graph \(d=d(x)\). (c) For what values of \(x\) is \(d\) smallest? (d) What is the smallest distance?

Short answer

The distance function is \( d(x) = \sqrt{(x - 1)^2 + x} \). From the graph, \( x = 0.5 \) gives the smallest distance, which is \sqrt{0.5} \.

Step-by-step solution

Define the Distance Formula

To find the distance from point \(P=(x, y)\) to the point \( (1,0) \), use the distance formula: \ d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2} \.

Substitute Coordinates into Distance Formula

The coordinates of point \(P\) are \( (x, \sqrt{x}) \) and the coordinates of point \( (1,0) \) are given. Substituting these into the distance formula gives: \ d = \sqrt{(x - 1)^2 + (\sqrt{x} - 0)^2} = \sqrt{(x - 1)^2 + x} \.

Express Distance as a Function of x

The expression \ d = \sqrt{(x - 1)^2 + x} \ is already the distance as a function of \( x \). Therefore, \( d(x) = \sqrt {(x - 1)^2 + x} \).

Graph the Function d(x)

Use a graphing utility (such as a graphing calculator or software) to graph the function \( d(x) = \sqrt{(x - 1)^2 + x} \).

Determine the Minimum Distance

Using the graph, identify the value of \( x \) at which the function \( d(x) \) reaches its minimum. This corresponds to the smallest distance.

Calculate the Smallest Distance

Substitute the value of \( x \) found in Step 5 into the function \( d(x) \) to find the smallest distance.

Key concepts

Distance from a Point to a Curve

When determining the distance from a point to a curve, we often use the distance formula. This formula calculates the distance between any two points in a plane. Given two points \( (x_1, y_1) \) and \( (x_2, y_2) \), the distance \( d \) between them is given by:
\[ d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2} \]
In our exercise, point \( P \) on the curve of \( y = \sqrt{x} \) has coordinates \( (x, \sqrt{x}) \). Another point is given as \( (1, 0) \). Substituting these coordinates into the distance formula, we get:
\[ d = \sqrt{(x - 1)^2 + (\sqrt{x} - 0)^2} = \sqrt{(x - 1)^2 + x} \]
This distance formula helps us express the distance \( d \) as a function of \( x \), which becomes critical in analyzing other concepts within the exercise.

Minimizing Distance Functions

To solve for when the distance function is minimized, we generally look for the smallest value of the function. In other words, we must determine the minimum of the function \( d(x) \). The function provided is:
\[ d(x) = \sqrt{(x - 1)^2 + x} \]
Minimizing this function means finding the value of \( x \) that results in the smallest \( d \). One efficient way to determine this value by hand is to use calculus principles such as finding the derivative of the distance function and solving for \( x \) where this derivative equals zero (critical points). However, in this particular exercise, we use a graphing utility to visually identify the minimum value of \( d(x) \).
Once the minimum distance is identified on the graph, you can substitute the corresponding \( x \) value back into the distance function to find the smallest distance.

Graphing Utility in Algebra

Graphing utilities are invaluable tools in algebra for visualizing functions and analyzing their behavior. They help us understand how the function changes over different intervals and make it easier to identify key points such as maximums, minimums, and intercepts. In our exercise, a graphing utility is used to plot the function \( d(x) = \sqrt{(x - 1)^2 + x} \).
Here's how you can use a graphing utility effectively:

• Input the function into the graphing calculator or software.
• Adjust the viewing window to ensure the relevant portions of the graph are visible.
• Observe the plotted graph and look for the point at which the function reaches its lowest value. This is the minimum distance.

Using these steps, you can confirm the minimum distance visually, and gain a better intuition of the function's behavior.
For our exercise, after plotting \( d(x) \), we identify the minimum distance and corresponding \( x \) value directly from the graph.