Problem 50 - Calculus an Applied Approach

Question

Find the point(s), if any, at which the graph of has a horizontal tangent. $$f(x)=\frac{x^{4}+3}{x^{2}+1}$$

Step-by-Step Solution

Answer

The points where the graph has a horizontal tangent are: (0,3) and \((\sqrt{3}, 4), (-\sqrt{3}, 4)

Step 1: Calculate the Derivative

First, calculate the derivative of \(f(x) = \frac{x^{4} + 3}{x^{2} + 1}\) using the quotient rule for derivatives. The quotient rule states that \((\frac{u}{v})' = \frac{vu' - uv'}{v^2}\), where \(u\) and \(v\) are differentiable functions of x. Here, \(u = x^{4} + 3\), \(v = x^{2} + 1\), \(u' = 4x^{3}\), and \(v' = 2x\). Plugging into the quotient rule gives: \(f'(x) = \frac{(x^2 + 1) * 4x^3 - (2x * (x^4 + 3))}{(x^2 + 1)^2}\)

Step 2: Set the Derivative Equal to Zero

Set the derivative equal to zero and solve for x. Simplifying gives: \(0 = \frac{4x^{3} - 2x^3 - 6x}{(x^2 + 1)^2}\). This simplifies further to: \(0 = \frac{2x^{3} - 6x}{(x^2 + 1)^2}\). Solving this equation for x gives the x-value(s) of the points where the graph of the function has a horizontal tangent.

Step 3: Solve for X

Extract common factors and factor the equation to: \(0 = \frac{2x(x^2 - 3)}{(x^{2} + 1)^2}\). Setting this equation to zero, we get \(2x = 0\) and \(x^2 - 3 = 0\), which yield solutions \(x = 0, \sqrt{3}, -\sqrt{3}\).

Step 4: Find the Corresponding Y-values

Substitute each x value into the function in order to find the corresponding y values. These (x, y) coordinates are the points where the graph has a horizontal tangent. Therefore, the points are: when \(x = 0, f(0) = \frac{0^4 + 3}{0^2 + 1} = 3\). When \(x = \sqrt{3}, f(\sqrt{3}) = \frac{(\sqrt{3})^4 + 3}{(\sqrt{3})^2 + 1} = 4\). When \(x = -\sqrt{3}, f(-\sqrt{3}) = \frac{(-\sqrt{3})^4 + 3}{(-\sqrt{3})^2 + 1} = 4\).

Key Concepts

Horizontal Tangent

When we discuss a horizontal tangent, we're looking at points on a graph where the slope of the tangent line is zero.
This means that derivative of the function at that point equals zero. Identifying horizontal tangents is essential because they indicate

• where the function can have local maxima or minima,
• aid in understanding the function's overall behavior,
• and help point out where abrupt changes in growth or decline occur.

In the given exercise, finding horizontal tangents means solving for the x-values where the derivative equals zero, since these are where the function flattens out.

Quotient Rule

The quotient rule is a handy tool used in calculus to find the derivative of a quotient of two functions.
When dealing with a function expressed as one expression divided by another, say \[f(x) = \frac{u(x)}{v(x)}\], the quotient rule is applied to differentiate it: \[ \left( \frac{u}{v} \right)' = \frac{vu' - uv'}{v^2} \]Here's how each part contributes to the rule:

• \(u\) is the function in the numerator.
• \(v\) is the function in the denominator.
• \(u'\) is the derivative of \(u\).
• \(v'\) is the derivative of \(v\).

In our exercise, applying the quotient rule correctly allows us to determine where the derivative is zero, leading to identifying horizontal tangents.

Derivative

A derivative is a fundamental concept in calculus. It represents how a function changes as its input changes. In simpler terms, it's like measuring the speed of change at any point on a function's graph. In math terms, for a function \(f(x)\), its derivative \(f'(x)\) helps us:

• Find the slope of the tangent line to the graph of \(f\) at any point,
• Know how the function increases or decreases,
• Identify critical points where the function might change direction.

In the exercise, finding the derivative of the given function using the quotient rule helps us find where the graph has a horizontal tangent. Once we have the derivative, setting it to zero provides the x-values we're interested in.

Function Analysis

Function analysis in calculus includes examining a function to understand its behavior and characteristics. This involves looking at the function’s graph for notable features like slopes, curves, maxima, minima, and points of inflection. Steps in function analysis can include:

• Finding derivatives to assess slope changes,
• Determining intercepts and critical points,
• Evaluating second derivatives for concavity and points of inflection.

For the function \(f(x) = \frac{x^4 + 3}{x^2 + 1}\) in the exercise, our analysis identifies horizontal tangents. This involves setting its derivative to zero, solving for x, and checking resulting coordinates. This full understanding aids in sketching the graph accurately and predicting how its shape will look in different regions.