Question

Horizontal Tangent Line In Exercises \(73-76\) , determine the point(s) at which the graph of the function has a horizontal tangent line. $$ f(x)=\frac{2 x-1}{x^{2}} $$

Short answer

The point where the graph of the function \(f(x)=\frac{2 x-1}{x^{2}}\) has a horizontal tangent line is \((1/2, 0)\).

Step-by-step solution

Compute the derivative of the function

The function is given by \(f(x)=\frac{2 x-1}{x^{2}}\), which is the quotient of two functions \(u(x)=2x-1\) and \(v(x)=x^{2}\). Therefore, its derivative is calculated using the quotient rule for derivatives, which states that \((\frac{u}{v})'=\frac{u'v-uv'}{v^{2}}\), where \(u'\) and \(v'\) are the derivatives of \(u\) and \(v\) respectively. Thus, the derivative of \(f(x)\) \((f'(x))\) is calculated as follows: \(f'(x)=\frac{(2\cdot x^{2})-((2x-1) \cdot 2x)}{(x^{2})^{2}} =\frac{2}{x^{3}} - \frac{4}{x}}\).

Equate the derivative to zero and solve for x

For the function to have a horizontal tangent line, the derivative at that point must be zero. Therefore, equate \(f'(x)=0\) to find the \(x\) value, \(0 =\frac{2}{x^{3}} - \frac{4}{x}\). To simplify the equation, it could be rewritten as \(0 = 2x^{-2} - 4x^{-1}\), in standard form \(0 = 4x^{-1}-2x^{-2}\), from which \( x = 1/2\) is obtained.

Input the x-value into the original function

After obtaining the \(x\) value, substitute it into the original function \(f(x)\) to obtain the \(y\) coordinate of the tangency point. Therefore, \(f(1/2) =\frac{2 \cdot (1/2) - 1}{(1/2)^{2}} = 0\). Thus, the tangency point is at \((1/2, 0)\).

Key concepts

Quotient Rule

In calculus, functions are often represented as quotients of two other functions. The process of finding the derivative of such functions involves the use of the quotient rule.
If you have a function in the form of a quotient \(\frac{u(x)}{v(x)}\), the quotient rule states that its derivative \((\frac{d}{dx} \frac{u}{v})\) is given by:

• \[\left(\frac{u}{v}\right)' = \frac{u'v - uv'}{v^2}\]

Here,

• \(u'\) is the derivative of the numerator function \(u(x)\).
• \(v'\) is the derivative of the denominator function \(v(x)\).
• The denominator in the result, \(v^2\), ensures that the entire expression is divided properly.

This rule ensures you accurately calculate the rate of change of a function that is a ratio of two other functions. Understanding this rule is crucial for solving problems where functions are expressed as quotients.

Derivative Calculation

Calculating the derivative of a function helps us understand how the function behaves or changes. When our function is a fraction of two simpler functions, as in \(f(x) = \frac{2x-1}{x^2}\), we apply the quotient rule to find its derivative.
First, identify your functions:

• \(u(x) = 2x - 1\)
• \(v(x) = x^2\)

Calculate their derivatives:

• \(u'(x) = 2\)
• \(v'(x) = 2x\)

Use these to apply the quotient rule:

• \[f'(x) = \frac{2\cdot x^2 - (2x-1)\cdot 2x}{(x^2)^2} = \frac{2}{x^3} - \frac{4}{x}\]

The derivative provides insight into the slope, helping determine where the function rises, falls, or stabilizes, which aids in identifying the points where the tangent is horizontal.

Tangent Line at a Point

A tangent line is a straight line that just "touches" the curve at a given point without crossing it. If the tangent line to the graph at a specific point is horizontal, it means the slope of the tangent (the derivative) is zero at that point.
For our function, to find this horizontal tangent line, we set the derivative equal to zero:

• \[f'(x) = 0 = \frac{2}{x^3} - \frac{4}{x}\]

Simplifying, we solve for \(x\):

• \[0 = 2x^{-2} - 4x^{-1}\] leading to \[x = \frac{1}{2}\]

Substitute back into the original function to find \(y\):

• \[f(\frac{1}{2}) = \frac{2\cdot \frac{1}{2} - 1}{(\frac{1}{2})^2} = 0\]

Thus, the horizontal tangent point is \((\frac{1}{2}, 0)\), where the tangent line is flat, indicating no change in the value of \(y\).

Function Graph Analysis

Analyzing the function graph involves understanding its behavior by examining the resulting curve from plotting the function. This graph can reveal important features like intercepts, slopes, and the nature of its tangent lines.
For \(f(x) = \frac{2x-1}{x^2}\), the curve changes direction at points where its slope, the derivative, equals zero. Previous calculations showed this occurs at:\( x = \frac{1}{2} \).

• Horizontally tangent at \((\frac{1}{2}, 0)\): Here the function neither rises nor falls.
• This point also informs us about local extremum behavior, crucial for sketching or understanding broader function trends.

Graph analysis allows us to visualize and predict future behavior of the function, providing more context than just algebraic manipulation. It grounds usage of derivatives, making concepts like horizontal tangents and slope more intuitive.