Problem 14 - 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 are \( (m^{-2}, m^{-1}) \).

Step 1: Understanding the Problem

We are given a point \( P \) on the curve \( f(x) = \sqrt{x} \). We need to express its coordinates as a function of the slope of the line joining \( P \) and the origin (0,0).

Step 2: Expressing Coordinates of P

Let the coordinates of point \( P \) be \( (x, \sqrt{x}) \). Since \( P \) lies on the graph of \( y = \sqrt{x} \), if we know \( x \), then \( y = \sqrt{x} \).

Step 3: Finding the Slope of the Line from Origin to P

The slope \( m \) of the line joining the origin \( (0,0) \) to the point \( P(x, \sqrt{x}) \) is given by \( m = \frac{\sqrt{x} - 0}{x - 0} = \frac{\sqrt{x}}{x} \).

Step 4: Expressing x in terms of the Slope

We have the equation \( m = \frac{\sqrt{x}}{x} = x^{-1/2} \). Solving for \( x \), we get \( x = m^{-2} \).

Step 5: Expressing y in terms of Slope

Using \( x = m^{-2} \) in \( y = \sqrt{x} \), we find \( y = \sqrt{m^{-2}} = m^{-1} \).

Step 6: Final Answer

The coordinates of \( P \) can be expressed as functions of the slope \( m \) as \( (m^{-2}, m^{-1}) \).

Key Concepts

Slope of a Line

The slope of a line is a measure of how steep the line is. To find the slope, you simply divide the change in the vertical direction, known as the "rise," by the change in the horizontal direction, known as the "run." Given two points on a line,

• The slope formula is: \( m = \frac{y_2 - y_1}{x_2 - x_1} \).
• This formula gives you a number that represents the steepness and direction of the line.
• If \( m > 0 \), the line rises as it moves from left to right.
• If \( m < 0 \), the line falls as you move from left to right.
• If \( m = 0 \), the line is horizontal.
• For a vertical line, the slope is undefined as the denominator will be zero.

Understanding slope is crucial for graphing lines and analyzing the relationship between variables in different applications. It allows you to predict how a change in one variable affects another.

Functions

A function is a relationship between two sets of numbers or variables, where each input (typically called \( x \)) is related to one output (typically called \( y \)). In mathematical language, you can write this as \( y = f(x) \). Here are some key points about functions:

• A function assigns a unique output for every input.
• Common examples of functions include linear functions and quadratic functions.
• In the function given in the exercise, \( f(x) = \sqrt{x} \), each \( x \) in the domain is paired with the square root of \( x \).

Functions are fundamental in mathematics because they help model real-world situations by describing how outputs react when inputs change. For students tackling functions, the idea of expressing one variable as a function of another, like finding coordinates as functions of a slope, is a key skill to develop.

Graph of a Function

The graph of a function is a visual representation of all the possible pairs of a function’s inputs and outputs, which means plotting \( f(x) \) on the y-axis against \( x \) on the x-axis. When graphing a function:

• The x-axis is the independent variable, and the y-axis is the dependent variable.
• Each point on the graph corresponds to a solution of the equation \( y = f(x) \).
• The shape of the graph gives insights into the behavior of the function. For example, an increasing graph indicates that as \( x \) increases, \( f(x) \) also increases.
• Key features like intercepts, slopes, and curvature are often analyzed.

Understanding the graph of a function is critical as it provides a whole set of solutions to the function’s equation at a glance. In our exercise, the graph of the square root function \( f(x) = \sqrt{x} \) helps pinpoint the location of point \( P \) and relate it to the slope of the line joining it to the origin, illustrated by the solutions \( (m^{-2}, m^{-1}) \). This reflects how changes in the slope affect the position along the graph.