Problem 47 - Calculus an Applied Approach

Question

Use a symbolic differentiation utility to find the derivative of the function. Graph the function and its derivative in the same viewing window. Describe the behavior of the function when the derivative is zero. $$ f(x)=\frac{\sqrt{x}+1}{x^{2}+1} $$

Step-by-Step Solution

Answer

The derivative of the function \(f(x) = \frac{\sqrt{x}+1}{x^{2}+1}\) is \(f'(x)=\frac{(x^2 +1)(1/2)x^{-1/2} - (\sqrt{x} +1)(2x)}{(x^2+1)^2}\). The function at points where its derivative equals zero reaches local extremes (maximum or minimum) or inflection points.

Step 1: Differentiate the function

Use the Quotient Rule for differentiation, which states that the derivative of a fraction \(u/v\) is \((v \cdot u’ - u \cdot v’)/v^{2}\). Here, let \(u=\sqrt{x}+1\) and \(v= x^2 +1\). Differentiating \(u\) with respect to \(x\) gives \(u'=(1/2)x^{-1/2}\), and differentiating \(v\) with respect to \(x\) yields \(v'=2x\). Now using the quotient rule, \n\(f'(x)=\frac{(x^2 +1)(1/2)x^{-1/2} - (\sqrt{x} +1)(2x)}{(x^2+1)^2}\).

Step 2: Graph the function and its derivative

Now that we have \(f(x)\) and it's derivative \(f'(x)\), we can graph them using a graphing tool. Plot \(f(x)\) and \(f'(x)\) on the same graph. Notice the points where \(f'(x)\) crosses or touches the x-axis. Those are the points where the derivative is zero.

Step 3: Analyze the behavior of the function when the derivative is zero

The derivative of a function equals zero at points where the function reaches a local extreme (minimum or maximum point), or an inflection point. Depending on the nature of the function, the derivative can either intersect or just touch the x-axis. In the points where the derivative \(f'(x)\) equals zero, look at the graph of \(f(x)\). If the function seems to reach an utmost or lowest point, then it's a local maximum or minimum. If the function changes its concavity (i.e., it goes from curving upwards to curving downwards, or vice versa), then it's an inflection point.

Key Concepts

Quotient Rule

The quotient rule is an essential tool for differentiating functions that are expressed as a ratio of two other functions. If you have a function represented as a fraction \( \frac{u}{v} \), the quotient rule allows you to find its derivative using the formula:

• \( f'(x) = \frac{v \cdot u' - u \cdot v'}{v^2} \)

This formula might seem a little complex, but it can be tackled step-by-step. Begin by identifying which part of your function is \( u \) (the numerator) and which is \( v \) (the denominator). In our exercise, \( u = \sqrt{x} + 1 \) and \( v = x^2 + 1 \). Calculating the derivative of \( u \) gives \( u' = \frac{1}{2}x^{-1/2} \), and for \( v \), we find \( v' = 2x \).
Plug these values into the quotient rule to obtain the derivative of the original function. This systematic approach helps keep operations organized and makes solving for the derivative more manageable.

Derivative of a Function

Finding the derivative of a function involves determining how the function changes at any given point. Derivatives are the building blocks of calculus, representing the slope or rate of change of a function.
In the context of our problem, once we have identified \( f(x) = \frac{\sqrt{x}+1}{x^{2}+1} \), we apply differentiation rules, like the quotient rule, to find \( f'(x) \).

• We've calculated that \( f'(x) \) is: \( \frac{(x^2 +1)\cdot \frac{1}{2}x^{-1/2} - (\sqrt{x} +1)\cdot 2x}{(x^2+1)^2} \)

This expression provides valuable information about the function's behavior. It helps us understand where the function is increasing or decreasing. When \( f'(x) = 0 \), the function could potentially have a local maximum, minimum, or be transitioning through an inflection point. Calculating and understanding derivatives are crucial for analyzing any function effectively.

Graphing Functions

Graphing both the function and its derivative offers a comprehensive view of how the function behaves across its domain. By plotting \( f(x) = \frac{\sqrt{x}+1}{x^{2}+1} \) and its derivative \( f'(x) \), you gain a visual understanding of the function.
When both graphs are visualized in the same window, it's easier to see the areas where \( f'(x) \) equals zero. These are pivotal points - the spots where the behavior of \( f(x) \) shifts.

• If \( f'(x) \) touches or crosses the x-axis, those points are either local maxima or minima, indicating peaks or valleys in \( f(x) \).
• Observing where \( f'(x) \) does not change sign helps find inflection points, where the concavity changes direction.

Graphing is a crucial tool, providing insights that might not be obvious from algebraic manipulation alone. It lets you see how theoretical calculus translates into visual interpretation, affirming analytical results.