Problem 12 - Thomas Calculus

Question

A point \(P\) in the first quadrant lies on the graph of the function \(f(x)=\sqrt{x} .\) Express the coordinates of \(P\) as functions of the slope of the line joining \(P\) to the origin.

Step-by-Step Solution

Answer

The coordinates of \( P \) are \( \left( \frac{1}{m^2}, \frac{1}{m} \right) \)."

Step 1: Write the coordinates of the point

Any point \( P \) on the graph of the function \( f(x) = \sqrt{x} \) has coordinates \( (x, \sqrt{x}) \). Since point \( P \) lies in the first quadrant, both \( x \) and \( \sqrt{x} \) are positive.

Step 2: Determine the slope of line OP

The slope \( m \) of the line joining the origin \( O(0,0) \) to the point \( P(x, \sqrt{x}) \) is given by the formula \( m = \frac{y_2 - y_1}{x_2 - x_1} \). Substituting the coordinates of the origin and point \( P \), we get \( m = \frac{\sqrt{x} - 0}{x - 0} = \frac{\sqrt{x}}{x} = \frac{1}{\sqrt{x}} \).

Step 3: Solve for x in terms of m

From the equation \( m = \frac{1}{\sqrt{x}} \), solve for \( x \). Rearranging, we find \( \sqrt{x} = \frac{1}{m} \), and then squaring both sides gives us \( x = \frac{1}{m^2} \).

Step 4: Express the y-coordinate in terms of m

We know the y-coordinate is \( \sqrt{x} \). Using \( x = \frac{1}{m^2} \), we find \( \sqrt{x} = \sqrt{\frac{1}{m^2}} = \frac{1}{m} \). So, the y-coordinate is \( \frac{1}{m} \).

Step 5: Express coordinates of P in terms of m

The coordinates \( P(x, \sqrt{x}) \) can now be expressed as functions of \( m \). With \( x = \frac{1}{m^2} \) and \( \sqrt{x} = \frac{1}{m} \), the coordinates of \( P \) are \( \left( \frac{1}{m^2}, \frac{1}{m} \right) \).

Key Concepts

Slope of a line

The slope of a line is a measure of its steepness, often represented by the letter "m." In mathematics, particularly in calculus and coordinate geometry, knowing the slope is essential for understanding the behavior of lines. For any line connecting two points \( (x_1, y_1) \) and \( (x_2, y_2) \), the slope is calculated as:

• \( m = \frac{y_2 - y_1}{x_2 - x_1} \).

If the calculated value is positive, the line goes upwards as we move from left to right. Conversely, if the value is negative, the line slopes downwards. In the context of a point on a function curve, such as \( f(x) = \sqrt{x} \), the slope is derived by comparing the vertical and horizontal distances from the origin \( (0,0) \) to the point \( P(x, \sqrt{x}) \).
Using the formula \( m = \frac{\sqrt{x}}{x} \), we see how the slope \( m = \frac{1}{\sqrt{x}} \) changes, illustrating how quickly or slowly the curve approaches a vertical position as x increases. This measurement informs us about the rate of change in the curve's height relative to horizontal distance.

First Quadrant

In coordinate geometry, the first quadrant refers to the section of the Cartesian coordinate plane where both x and y values are positive. The Cartesian plane is divided into four quadrants, each corresponding to a different combination of signs for x and y. In mathematical analyses, the first quadrant is often considered because it involves all non-negative values, making it simpler to visualize and calculate.
When we are dealing with points in this quadrant, like \( P(x, \sqrt{x}) \) from the exercise, it's guaranteed that these values (x, \sqrt{x}) remain positive as long as x is positive. This positivity is vital in real-world applications where negative values might not be feasible, such as in measuring distance or time. Understanding the first quadrant helps students focus on positive solutions and is often a starting point for graphing and analyzing functions.

Coordinate Geometry

Coordinate geometry, also known as analytical geometry, involves studying geometric figures using a coordinate system. This field combines algebra and geometry to solve problems involving lines and curves in a structured way. By using a coordinate system, every geometric shape can be represented algebraically, allowing for precise calculation and analysis.
The given exercise revolves around plotting and analyzing a point \( P \) on the function \( f(x) = \sqrt{x} \), emphasizing the importance of coordinates in defining positions on a graph. When finding the slope or the position of points like \( O(0,0) \) to \( P(x, \sqrt{x}) \), coordinate geometry provides tools to not only describe but also manipulate these positions mathematically. Understanding how to switch between algebraic and geometric interpretations aids in comprehending complex calculus concepts.

• The conversion of the slope into terms of coordinates, \( (\frac{1}{m^2}, \frac{1}{m}) \), is an example of applying coordinate geometry principles.

This field also lays the groundwork for more advanced studies in calculus and vector analysis, bridging a fundamental gap between algebraic equations and their geometric counterparts.