Problem 89 - University Calculus: Early Transcendentals

Question

The continuous extension of \((\sin x)^{x}\) to \([0, \pi]\) a. Graph \(f(x)=(\sin x)^{x}\) on the interval \(0 \leq x \leq \pi .\) What value would you assign to \(f\) to make it continuous at \(x=0 ?\) b. Verify your conclusion in part (a) by finding \(\lim _{x \rightarrow 0^{+}} f(x)\) with I'Hôpital's Rule. c. Returning to the graph, estimate the maximum value of \(f\) on \([0, \pi] .\) About where is max \(f\) taken on? -d. Sharpen your estimate in part (c) by graphing \(f^{\prime}\) in the same window to see where its graph crosses the \(x\) -axis. To simplify your work, you might want to delete the exponential factor from the expression for \(f^{\prime}\) and graph just the factor that has a

Step-by-Step Solution

Answer

At \(x=0\), set \(f(0)=1\) for continuity. Maximum \(f(x)\) is about 1 near \(x=\frac{\pi}{2}\).

Step 1: Understanding the Function

Consider the function \( f(x) = (\sin x)^x \) defined over \([0, \pi]\). At \( x = 0 \), the function needs careful analysis due to the \( 0^0 \) form.

Step 2: Graph Examination

Graph \( f(x) = (\sin x)^x \) over the interval \( 0 \leq x \leq \pi \). Observe that as \( x \rightarrow 0^+ \), the value of \( (\sin x)^x \) approaches a particular limit from the graph.

Step 3: Determining Continuity at x=0

To make the function continuous at \( x = 0 \), assign \( f(0) = 1 \) since \( \lim_{x \to 0^+} (\sin x)^x = 1 \), observed via graphical trend.

Step 4: Verification Using L'Hôpital's Rule

Rewrite \( (\sin x)^x \) in logarithmic form to utilize L'Hôpital's Rule:\[ \ln f(x) = x \ln(\sin x) \].Compute \( \lim_{x \to 0^+} x \ln(\sin x) \) as follows:\[ \lim_{x \to 0^+} x \ln(\sin x) = \lim_{x \to 0^+} \frac{\ln(\sin x)}{1/x} \].Apply L'Hôpital's Rule as this is an indeterminate form \( \frac{\infty}{\infty} \). Differentiate numerator and denominator:\[ \lim_{x \to 0^+} \frac{\cos x/\sin x}{-1/x^2} = \lim_{x \to 0^+} -x \cot x = 0 \].Since \( e^0 = 1 \), we conclude \( \lim_{x \to 0^+} (\sin x)^x = 1 \).

Step 5: Examine Maximum Graphically

By examining the graph of \( f(x) = (\sin x)^x \), estimate the maximum value \( f(x) \) takes is close to 1, along with the interval it appears in, likely at \( \frac{\pi}{4} \) or nearby.

Step 6: Sharpen Maximum Estimate with First Derivative

Determine \( f'(x) \) by differentiating \( f(x) = (\sin x)^x \) using the chain rule and product rule:\[ f'(x) = (\sin x)^x \left[ \ln(\sin x) + x \cot x \right] \].Graph \( \ln(\sin x) + x \cot x \) to find where it crosses the x-axis, indicating the maximum is around \( x = 1.57 \) or \( \frac{\pi}{2} \).

Step 7: Conclusion of Maximum Value

Use the graph and calculations to reconfirm the maximum value of \( f(x) \) is approximately 1, occurring just before or at \( \frac{\pi}{2} \).

Key Concepts

L'Hôpital's Rule

L'Hôpital's Rule is a vital tool used in calculus to evaluate limits that result in indeterminate forms, specifically when limits yield results such as \(0/0\) or \(\infty/\infty\). When encountering these forms, you can apply L'Hôpital's Rule by differentiating the numerator and the denominator separately and then taking the limit of the resulting fraction.

In the problem concerning the function \( f(x) = (\sin x)^x \) as \( x \to 0^+ \), we transform the expression into its logarithmic form: \( \ln f(x) = x \ln(\sin x) \). By doing so, we aim to evaluate \( \lim_{x \to 0^+} x \ln(\sin x) \) using L'Hôpital's Rule since it results in an indeterminate form \( \frac{-\infty}{-\infty} \).

• First, rewrite the limit as \( \lim_{x \to 0^+} \frac{\ln(\sin x)}{1/x} \).
• Next, apply L'Hôpital's Rule by differentiating the numerator and the denominator:
• Numerator differentiation: \( \cos x / \sin x \)
• Denominator differentiation: \(-1/x^2 \)
• The limit simplifies to \( \lim_{x \to 0^+} -x \cot x = 0 \).

This result demonstrates the utility of this rule in simplifying complex limits, leading us to conclude that \( e^0 = 1 \), indicating that \( \lim_{x \to 0^+} (\sin x)^x = 1 \). Here, L'Hôpital's Rule provides an elegant solution to a potentially complex problem by taming indeterminate forms.

Indeterminate Forms

Indeterminate forms often arise in calculus when trying to find the limit of a function as a variable approaches a certain value, leading to results like \(0/0\), \(\infty/\infty\), \(\infty - \infty\), among others. Such expressions do not provide immediate information about the behavior of the functions involved, necessitating further analysis.

In our specific exercise with \( f(x) = (\sin x)^x \), we encounter the indeterminate form \(0^0\) as \( x \) approaches zero. To handle this, we need to convert the expression into a solvable form. By taking the natural logarithm of the function, \( \ln f(x) = x \ln(\sin x) \), we transform it into a familiar limit form \(0 \cdot (-\infty)\).

Then, by switching to the fraction \( \frac{\ln(\sin x)}{1/x} \), we can apply L'Hôpital's Rule to resolve this indeterminate form. This thoughtful conversion allows for breaking down an otherwise troublesome calculation into manageable steps.

Understanding and recognizing these indeterminate forms are crucial, as they frequently occur in applications involving limits. Utilizing techniques such as L'Hôpital's Rule and logarithmic transformations helps in managing their complexity effectively.

Maximum Value Estimation

Estimating the maximum value of a function involves understanding its behavior over the given interval. This is where calculus shines, offering analytical and graphical methods to locate and evaluate such points.

For the function \( f(x) = (\sin x)^x \) on \([0, \pi]\), estimating its maximum value involves:

• Graphically observing the function, noting where \( f(x) \) reaches its peak value.
• Using derivatives to precisely pinpoint where this maximum occurs.

Initially, the graph suggests that the maximum approaches a value close to 1. By differentiating the function, applying the chain rule and product rule, we obtain \( f'(x) = (\sin x)^x \left[ \ln(\sin x) + x \cot x \right] \). This derivative offers insights into the function's increasing or decreasing nature.

• To accurately locate the maximum, graph the expression \( \ln(\sin x) + x \cot x \).
• Find where the graph of this expression crosses the \(x\)-axis, indicating potential maximum points.

The analysis reveals that the peak probably happens around \( x \approx \frac{\pi}{2} \) since \( f'(x) \) changes sign there. Hence, this method allows us to sharpen our estimates significantly, ensuring accurate conclusions about the maximum's location and value.