Problem 89 - Thomas Calculus in SI Units

Question

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 l'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 zero.

Step-by-Step Solution

Answer

Assign \(f(0) = 1\), limit is 1, max near \(x \approx \frac{\pi}{4}\).

Step 1: Understand the function

The function to graph is \(f(x) = (\sin x)^x\). Since \(x\) is in the exponent, ensure \(\sin x > 0\). The graph should be within the interval \([0, \pi ]\). Remember that near \(x = 0\), \(\sin x \approx x\), which will affect the graph's continuity at that point.

Step 2: Graph the function

Graph \(f(x) = (\sin x)^x\) from \(x = 0\) to \(x = \pi\). Observe the behavior around \(x = 0\). Notice a gap at \(x=0\) where the graph is undefined due to \(0^0\).

Step 3: Evaluate limit to find continuity

Use L'Hôpital's Rule to find \(\lim_{x \to 0^+} f(x)\). Rewrite \((\sin x)^x\) as \(e^{x \ln(\sin x)}\). The limit becomes \(\lim_{x \to 0^+} x \ln(\sin x)\), which is an indeterminate form \((0 \cdot -\infty)\).

Step 4: Simplify limit using L'Hôpital's Rule

Change the limit to \(\lim_{x \to 0^+} \frac{\ln(\sin x)}{1/x}\). This is \(\frac{-\infty}{\infty}\). Differentiate numerator and denominator: \(\frac{d}{dx}[\ln(\sin x)] = \cot x\) and \(\frac{d}{dx}[1/x] = -\frac{1}{x^2}\). Use L'Hôpital's Rule:\[ \lim_{x \to 0^+} \frac{\cot x}{-1/x^2} = \lim_{x \to 0^+} -x^2 \cot x. \]As \(x \to 0^+\), this approaches 0. Thus, the limit is \(e^0 = 1\). Assign \(f(0) = 1\) for continuity.

Step 5: Estimate maximum from graph

Return to the graph of \((\sin x)^x\) and estimate where it reaches its maximum value. It appears largest just after \(x = 0\), around \(x \approx \frac{\pi}{4}\).

Step 6: Analyze derivative for exact maximum location

Find \(f'(x)\) expression by differentiating: \[ f'(x) = (\sin x)^x \left( x \cot x + \ln(\sin x) \right). \]Graph \(x \cot x + \ln(\sin x)\) and find where it crosses the \(x\)-axis to confirm the maximum's exact location. The zero represents where \(f\) is at an extremum in \([0, \pi]\).

Key Concepts

Continuity of a Function

To determine if a function like \(f(x) = (\sin x)^x\) is continuous at a point, such as \(x = 0\), we need to check if the function value can match the limit as we approach that point. A function is continuous at \(x = a\) if \(\lim_{x \to a} f(x) = f(a)\). For \(f(x)\), we are initially concerned with \(x = 0\) because it introduces a potential discontinuity with the expression \(0^0\). Such an expression is usually undefined, but continuity requires us to define \(f(0)\) so that the limit exists and equals \(f(0)\). By calculating \(\lim_{x \to 0^+} (\sin x)^x\), we find this limit equals 1, ensuring continuity if we assign \(f(0) = 1\). This choice fills the gap in the graph and maintains a smooth transition across the interval \([0, \pi]\). Ensuring continuity is crucial for making predictions and understanding the function's behavior throughout the domain.

L'Hôpital's Rule

When facing indeterminate forms like \(\frac{0}{0}\) or \(\frac{\infty}{\infty}\), L'Hôpital's Rule becomes a powerful tool for finding limits. Specifically, for the limit \(\lim_{x \to 0^+} (\sin x)^x\), we rewrite it as \(e^{x \ln(\sin x)}\) which boils down to finding \(\lim_{x \to 0^+} x \ln(\sin x)\). This form, \(0 \cdot -\infty\), can be reformulated to \(\lim_{x \to 0^+} \frac{\ln(\sin x)}{1/x}\), which is better suited for L'Hôpital's Rule. This change simplifies to \(\frac{-\infty}{\infty}\) and enables differentiation of both the numerator and denominator. After applying L'Hôpital's Rule by computing the derivatives (using \(\frac{d}{dx}[\ln(\sin x)] = \cot x\) and \(\frac{d}{dx}[1/x] = -\frac{1}{x^2}\)), we reach \(\lim_{x \to 0^+} -x^2 \cot x\), which approaches 0. The limit becomes \(e^0 = 1\), confirming our function's continuity at that point.

Graphical Analysis

Graphing a function gives visual insights into its behavior, aiding the determination of significant properties like maxima or continuity. For \(f(x) = (\sin x)^x\), one must interpret the graph to approximate features like the maximum on \([0, \pi]\). By graphing, we can visually identify where the function ascends and descends. In this instance, a glance at the graph suggests a rapid increase after zero, peaking around \(x \approx \frac{\pi}{4}\). This graphical peek is foundational yet is complemented by more rigorous methods like derivative analysis to confirm exact positions and values. Analyzing graphs can also highlight symmetry, intercepts, and intervals of increase and decrease, encouraging a comprehensive understanding.

Derivative Analysis

To pinpoint critical points of a function and confirm precise locations of extrema, analyzing the derivative is essential. For \(f(x) = (\sin x)^x\), the derivative \(f'(x)\) provides a mathematical tool for this inquiry. Derived as \((\sin x)^x(x \cot x + \ln(\sin x))\), this derivative helps identify where \(f(x)\) reaches a maximum or minimum by solving \(x \cot x + \ln(\sin x) = 0\). Analyzing this derivative graphically or algebraically shows where \(f'(x) = 0\), marking a potential extremum. These points are where the graph of \(f(x)\) levels out before changing direction. Calculating or graphing to find these intersection points on the x-axis guides us in determining exact values and locations of maxima or minima in question.