Question

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

Short answer

To find the point where the function has a horizontal tangent line, first compute the derivative of the function. Then equate the derivative function to zero and solve the equation for x. The solution for x will be the x-coordinates where the function has a horizontal tangent. Note that x = 1 should be excluded because the function has a vertical asymptote at x = 1.

Step-by-step solution

Computation of the Derivative

First, we need to find the derivative f'(x) of the function f(x) = \( \frac{x^{2}}{x-1} \). As it's a fractional function, we can use the quotient rule which states that the derivative of \( \frac{u}{v} \) is \( \frac{vu'-uv'}{v^{2}} \). Here, u(x) = x² and v(x) = x-1. Compute the derivative u'(x) and v'(x) and then apply the quotient rule to evaluate f'(x).

Equating the Derivative to Zero

Once we've obtained the derivative, the next step is to set it equal to 0 and solve for x. This is because the slope of the tangent line at any x-coordinate of the function f(x) is equal to the value of the derivative at that point, and we're looking for an x-value where the slope is zero.

Solve for x

Simplify and solve the equation f'(x) = 0 to get x. Be aware that x - 1 in the denominator of f(x) indicates a vertical asymptote. Thus, x = 1 is the value at which f(x) isn't defined, and it's excluded from the domain of the function.

Key concepts

Calculus

Calculus is a branch of mathematics that allows us to explore changes and motion. It provides a powerful toolset to understand functions and their behaviors. The concept of calculus is based on two main operations: differentiation and integration.
Differentiation involves finding the rate of change of a function, while integration is concerned with finding areas under curves or accumulating quantities. By using calculus, we can make predictions and understand patterns not immediately obvious by just looking at equations or graphs.
Calculus plays an essential role in solving problems related to rates of change, which is fundamental in physics, engineering, and economics.

Derivative

The derivative of a function measures how the function changes as its input changes. In simpler terms, the derivative tells us the slope of the function at any given point.
Mathematically, if you have a function denoted as \( f(x) \), the derivative is often represented as \( f'(x) \) or \( \frac{df}{dx} \). For a function describing a curve, the derivative at a particular point is the slope of the tangent line to that curve at that point.
Understanding derivatives is key to determining where functions increase, decrease, or have a rate of change that is constant. They help us not only in learning about function behavior but also in analyzing graphs and optimizing functions in real-world applications.

Quotient Rule

When dealing with fractional functions, the quotient rule is a valuable method for differentiating such functions. This rule is specific to functions that can be expressed as one function divided by another, like \( f(x) = \frac{u(x)}{v(x)} \).
The quotient rule states that the derivative of \( \frac{u}{v} \) is given by \[ \frac{vu' - uv'}{v^2} \], where \( u' \) and \( v' \) are the derivatives of \( u \) and \( v \) respectively.
Applying this rule helps us find the slope of functions that are not straightforward to differentiate by conventional methods. It is particularly useful in engineering and physics, where complex ratio equations often need to be managed.

Fractional Function

A fractional function is a type of function in which the output is expressed as the ratio of two polynomials. They are identifiable by their typical form \( \frac{p(x)}{q(x)} \).
Such functions might present unique challenges, like undefined points or asymptotes, especially when the denominator becomes zero. In the study of calculus, knowing how to handle fractional functions is crucial.
For horizontal tangent lines, it's important to set the derivative of such a fractional function to zero to find points where the slope of the tangent is zero. This helps in analyzing the behavior of the function and determining key characteristics of its graph.