Problem 48 - Calculus Early Transcendentals

Question

Behavior at the origin Using calculus and accurate sketches, explain how the graphs of \(f(x)=x^{p} \ln x\) differ as \(x \rightarrow 0\) for \(p=\frac{1}{2}, 1,\) and 2

Step-by-Step Solution

Answer

For \(p = \frac{1}{2}\), the function approaches 0 as \(x \rightarrow 0^+\), and its graph is increasing and steep near the origin. For \(p = 1\), the function approaches 0 as \(x \rightarrow 0^+\), and its graph is increasing and less steep near the origin compared to the case for \(p = \frac{1}{2}\). For \(p = 2\), the function approaches 0 as \(x \rightarrow 0^+\), and its graph is increasing and less steep near the origin compared to the case for \(p = 1\).

Step 1: Analyze the function for \(p = \frac{1}{2}\)

To analyze the function \(f(x) = x^{1/2} \ln{x}\) as \(x \rightarrow 0\), we need to find the limit as x approaches 0. Since \(\ln{0}\) is undefined, we will use L'Hôpital's rule to evaluate the limit: $$\lim_{x\to 0^+} \frac{\ln{x}}{x^{-1/2}}$$ Applying L'Hôpital's rule, we need to find the derivatives of the numerator and denominator with respect to \(x\): $$\lim_{x\to 0^+} \frac{\frac{1}{x}}{-\frac{1}{2}x^{-3/2}} = \lim_{x\to 0^+} \frac{-2x^{1/2}}{1}$$ Since the limit as \(x \rightarrow 0^+\) is 0, the function approaches 0 as x approaches 0. To get an idea of the shape of the graph, we can find its derivative: $$f'(x) = \frac{d}{dx}(x^{1/2} \ln{x}) = x^{1/2} \frac{1}{x} + \frac{1}{2}x^{-1/2} \ln{x} = \frac{1}{\sqrt{x}} + \frac{\ln{x}}{2\sqrt{x}}$$ As \(x \rightarrow 0^+\), the first term approaches infinity while the second term approaches negative infinity. Therefore the graph will be increasing and steep near the origin.

Step 2: Analyze the function for \(p = 1\)

Now let's analyze the function \(f(x) = x \ln{x}\) as \(x \rightarrow 0\). The limit to find here is: $$\lim_{x\to 0^+} x \ln{x}$$ This limit evaluates to 0 directly, so the function also approaches 0 as \(x\) approaches 0. For the shape of the graph, we will find its derivative: $$f'(x) = \frac{d}{dx}(x \ln{x}) = \ln{x} + 1$$ As \(x \rightarrow 0^+\), the derivative approaches negative infinity. Therefore the graph will be increasing and less steep near the origin compared to the case for \(p = \frac{1}{2}\).

Step 3: Analyze the function for \(p = 2\)

Finally, let's analyze the function \(f(x) = x^2 \ln{x}\) as \(x \rightarrow 0\). The limit to find is: $$\lim_{x\to 0^+} x^2 \ln{x}$$ This limit evaluates to 0 directly, so the function also approaches 0 as \(x\) approaches 0. For the shape of the graph, we will find its derivative: $$f'(x) = \frac{d}{dx}(x^2 \ln{x}) = 2x \ln{x} + x^2 \frac{1}{x} = 2x \ln{x} + x$$ As \(x \rightarrow 0^+\), the derivative approaches 0. This means that the graph will be increasing and less steep near the origin than the case for \(p=1\).

Step 4: Conclusion

Now we know the behavior of the function \(f(x) = x^p \ln{x}\) as \(x \rightarrow 0\) for each value of \(p\): - For \(p = \frac{1}{2}\), the function approaches 0 as \(x \rightarrow 0^+\), and its graph is increasing and steep near the origin. - For \(p = 1\), the function approaches 0 as \(x \rightarrow 0^+\), and its graph is increasing and less steep near the origin compared to the case for \(p = \frac{1}{2}\). - For \(p = 2\), the function approaches 0 as \(x \rightarrow 0^+\), and its graph is increasing and less steep near the origin compared to the case for \(p = 1\). Using this information, one can accurately sketch the graphs to visualize their differences as \(x \rightarrow 0\).

Key Concepts

L'Hôpital's rule

L'Hôpital's Rule is a powerful tool in calculus used to calculate limits that are difficult to address directly. This rule helps us find limits for functions that approach an indeterminate form like \( \frac{0}{0} \) or \( \frac{\infty}{\infty} \) as \( x \to a \). It's a way to transform complicated limits into something easier to manage.

To apply L'Hôpital's Rule, follow these steps:

• Verify that your limit is of an indeterminate form \( \frac{0}{0} \) or \( \frac{\infty}{\infty} \).
• Differentiate the numerator and the denominator separately.
• Re-evaluate the limit.

This is done until the limit can be resolved through direct evaluation. For example, in the function \( f(x) = x^{1/2} \ln{x} \) as \( x \rightarrow 0^+ \), we faced an indeterminate form. After differentiating, the problem simplified, helping us understand that this function approaches 0 at the origin while its derivative offered insights into its steepness. L'Hôpital's Rule eases the process of analyzing difficult limits, especially near boundaries or critical points of a function.

Behavior at the origin

Understanding the behavior of functions as \( x \to 0 \) is crucial in many calculus problems. It helps us sketch curves and predict how functions will behave near the origin. When analyzing a function's behavior at the origin, consider:

• Limits: Ensure that the function is approaching a defined value as \( x \to 0 \). In our exercise, regardless of \( p \), the limit evaluated to 0.
• Steepness: Analyze how quickly or slowly the function increases or decreases. This is often done by finding the function's derivative.

In the given examples of \( f(x) = x^p \ln{x} \) for different values of \( p \), each variation shared a common trend: approaching zero as \( x \to 0 \).

The steepness of the graph at the origin differed based on \( p \):

• \( p = \frac{1}{2} \): The graph is very steep, shooting upwards quickly.
• \( p = 1 \): The steepness is less pronounced compared to \( p = \frac{1}{2} \).
• \( p = 2 \): The graph rises even more gently near the origin.

This behavior helps in sketching curves that accurately capture how functions look at and near \( x = 0 \). Utilizing these concepts allows us to predict functional behavior and communicate these ideas visually through graphing.

Function analysis

Function analysis involves breaking down a function into understandable parts to understand its characteristics and predict its behavior. This includes:

• Identifying its limit behavior at crucial points like \( x \to 0 \).
• Evaluating its first derivative to understand the rate of change and determine steepness.
• Using derivative information to sketch graphs and infer concavity or points of inflection.

In our particular problem, the function \( f(x) = x^p \ln{x} \) was analyzed for various values of \( p \). We computed limits to know where the function heads as \( x \to 0 \), revealing they all approached 0.

Next, by evaluating the derivatives, information about the function's steepness was extracted, vital for sketching graphs and understanding how the behavior changes with \( p \). This analysis allows us to:

• Spot critical points.
• Understand increasing or decreasing behavior.
• Analyze potential asymptotes or boundaries.

These insights allow for more accurate representations in sketching. Function analysis, therefore, empowers predictions about behavior across a range of values, simplifying the process of understanding complex functions.