Problem 69 - Calculus and its applications

Question

For each function, find the points on the graph at which the tangent line is horizontal. If none exist, state that fact. $$ y=-2 x+5 $$

Step-by-Step Solution

Answer

There are no points where the tangent line is horizontal for the function.

Step 1: Understand What a Horizontal Tangent Means

A horizontal tangent line to a curve occurs where the derivative of the function is equal to zero. Therefore, for the function \( y = -2x + 5 \), we need to find the derivative and set it equal to zero to find the points where the tangent line is horizontal.

Step 2: Find the Derivative of the Function

The given function is \( y = -2x + 5 \). Computing the derivative with respect to \( x \), we have \( \frac{dy}{dx} = -2 \).

Step 3: Set the Derivative Equal to Zero

To find where the tangent line is horizontal, set the derivative equal to zero: \( -2 = 0 \). This equation is not true for any value of \( x \), meaning that the derivative cannot be zero.

Step 4: Conclusion on Horizontal Tangents

Since \( -2 = 0 \) is not valid, there are no points on the graph where the derivative is equal to zero. Thus, the graph has no points where the tangent line is horizontal.

Key Concepts

Finding Derivatives

Understanding derivatives is essential to analyzing functions and their behaviors, such as identifying points with horizontal tangents. A derivative represents the rate of change of a function with respect to its variable. In simple terms, it describes how a function's output value changes as the input changes.
For linear functions, finding the derivative is straightforward. You only need to worry about the slope. The standard form of a linear function is expressed as \( y = mx + b \), with \( m \) being the slope. The derivative of this equation, \( \frac{dy}{dx} \), is simply equal to \( m \). This means, for any linear function, the derivative is constant. In our example, where \( y = -2x + 5 \), the derivative is \( -2 \).

• Slope \( m \): The coefficient of \( x \) in a linear function.
• Constant Derivative: Linear functions have constant derivatives equal to their slopes.

When the derivative of a function is zero, it implies there's no change in the function's value at that point, indicating a potential horizontal tangent. Always start by finding the derivative if you need to analyze movement or change in function graphs.

Tangent Lines

Tangent lines give valuable insight into a function's behavior at a specific point. Essentially, a tangent line touches a curve at precisely one point and follows the direction of the curve at that spot.
In mathematics, the concept of a "horizontal tangent line" specifically refers to the specific scenario where this tangent line runs parallel to the x-axis. It means the curve neither increases nor decreases at that point—also implied by a zero derivative.

• Tangent Line: A line touching a curve at only one point, reflecting the curve's immediate direction.
• Horizontal Tangent: Occurs when the derivative of a function at a point equals zero.

In our example, because the derivative \( -2 \) never equals zero, the function does not possess any horizontal tangents. This example highlights how essential tangents are in visualizing and determining function behaviors and trends.

Graphing Linear Functions

Graphing linear functions involves plotting points for lines defined by equations like \( y = mx + b \). Linear functions result in straight lines in the Cartesian coordinate plane, where \( m \) indicates the line's slope, and \( b \) represents where the line crosses the y-axis (the y-intercept).
For the function \( y = -2x + 5 \), where \( m = -2 \) and \( b = 5 \):

• Slope \( m = -2 \): Shows the line's steepness and direction. In this case, the line slopes downwards as it goes from left to right.
• Y-intercept \( b = 5 \): States where the line intersects the y-axis.

Graphing this function helps understand it visually. Notice how the negative slope ensures the function never levels out to have a horizontal tangent. The graph's linearity and constant descent demonstrate how consistent first derivatives are with linear equations.