Problem 48 - Calculus an Applied Approach

Question

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

Step-by-Step Solution

Answer

The graph of the function \(f(x)=\frac{x^{2}}{x^{2}+1}\) has a horizontal tangent at the point (0,0).

Step 1: Finding the Derivative

Use the quotient rule to find the derivative of \(f(x)\). Quotient rule states that if you have a function of the form \(g(x) = \frac{u(x)}{v(x)}\), then \(g'(x)=\frac{u'(x)v(x)-u(x)v'(x)}{[v(x)]^{2}}\). Given \(f(x)=\frac{x^{2}}{x^{2}+1}\), \(u(x) = x^{2}\) and \(v(x)=x^{2}+1\). So, \(u'(x) = 2x\) and \(v'(x) = 2x\). Plugging these values back into the quotient rule formula, we get \(f'(x)=\frac{2x(x^{2}+1)-x^{2}*2x}{(x^{2}+1)^{2}}\). Simplifying this yields \(f'(x)=\frac{2x}{(x^{2}+1)^{2}}\).

Step 2: Finding x-coordinates for Horizontal Tangents

For a horizontal tangent, the derivative \(f'(x)\) must be equal to zero. Therefore, set \(\frac{2x}{(x^{2}+1)^{2}} = 0\). The only solution for this equation is \(x=0\). We are basically looking for values of x that would make the numerator of our derivative to be zero.

Step 3: Finding the Corresponding y-coordinate

Plug \(x=0\) into \(f(x)=\frac{x^{2}}{x^{2}+1}\) to get the corresponding y-coordinate. On substitution, we find that when \(x=0\), \(y=0\). Therefore, the point at which the graph has a horizontal tangent is (0,0).

Key Concepts

Quotient Rule

To tackle the problem of finding the derivative of the given function, we use the quotient rule, a key tool in calculus for differentiating functions expressed as a quotient. This rule applies when you have a function that can be divided into two parts, the numerator ( u(x)) and the denominator ( v(x)) . The formula for the quotient rule is:

• \( g'(x) = \frac{u'(x)v(x) - u(x)v'(x)}{[v(x)]^2} \)

In our exercise, the function is \( f(x) = \frac{x^2}{x^2 + 1} \). We identify:

• \( u(x) = x^2 \) with its derivative \( u'(x) = 2x \)
• \( v(x) = x^2 + 1 \) with its derivative \( v'(x) = 2x \)

Substituting these values into the quotient rule formula, we compute:\[ f'(x) = \frac{2x(x^2 + 1) - x^2 \cdot 2x}{(x^2 + 1)^2} \]By simplifying this expression, we find the derivative to be:\[ f'(x) = \frac{2x}{(x^2 + 1)^2} \]This process allows us to express the rate of change of the function, a key step in analyzing the behavior of the graph.

Horizontal Tangents

A horizontal tangent line to a function graph is a line that is completely flat, indicating a point where the rate of change or the slope is zero. This concept is crucial to understanding moments of transition or peaks in the behavior of a function. In our exercise, the task is to determine where such tangents occur for \( f(x) = \frac{x^2}{x^2 + 1} \).

To find horizontal tangents, we set the derivative from the previous section to zero and solve for x. Given:\[ f'(x) = \frac{2x}{(x^2 + 1)^2} = 0 \]The fraction can only be zero if the numerator equals zero. This gives:

• \( 2x = 0 \)

With this, we find that the only value of x that satisfies the equation is \( x = 0 \). This means the slope of the tangent at \( x = 0 \) is zero, indicating a horizontal tangent.
Finding such points is helpful because they often represent maxima, minima, or points of inflection on the graph.

Derivative of a Function

The derivative of a function is a fundamental concept in calculus that measures how a function's value changes as its input changes. It is essentially the rate of change or the slope of the function at any given point. Derivatives help us understand and predict the behavior of functions by providing insights into their slopes and curvatures.

In the context of our problem, we started with the function \( f(x) = \frac{x^2}{x^2 + 1} \) and calculated its derivative to grasp how the function behaves across its domain.
The derivative was found to be:\[ f'(x) = \frac{2x}{(x^2 + 1)^2} \] This expression shows how the function's slope changes at each point. When the derivative equals zero at a point, it implies no change in the height of the curve—a significant insight pointing to a horizontal tangent.

By using derivatives, we can determine critical points and understand the overall shape and direction of the graph. They are essential for studying the curve and making connections between analytical results and graphical representations.