Problem 75 - Calculus Single Variable

Question

The sine integral Si is defined to be the antiderivative of \(\sin (x) / x\) such that \(\mathrm{Si}(0)=0 .\) Analyze the graph of \(\operatorname{Si}(x)\) over \(-4 \pi \leq x \leq 4 \pi\) for intervals of increase and decrease and for upward and downward concavity. Explain your analysis. Then use a computer algebra system to graph Si \((x)\) over this interval.

Step-by-Step Solution

Answer

\( \mathrm{Si}(x) \) increases where \( \sin(x) > 0 \) and decreases where \( \sin(x) < 0 \); it is concave up where \( \mathrm{Si}''(x) > 0 \) and concave down where \( \mathrm{Si}''(x) < 0 \) across \([-4\pi, 4\pi]\). Use a graphing tool for visualization.

Step 1: Understanding the Sine Integral Function

The sine integral, denoted as \( \mathrm{Si}(x) \), is defined as the antiderivative of the function \( \frac{\sin(x)}{x} \). It is given that \( \mathrm{Si}(0)=0 \). This function helps us analyze certain aspects of its graph, such as intervals of increase and decrease, as well as concavity.

Step 2: Determining Intervals of Increase and Decrease

To find where \( \mathrm{Si}(x) \) is increasing or decreasing, consider the derivative of \( \mathrm{Si}(x) \). Since \( \mathrm{Si}(x) \) is the antiderivative, the derivative is \( \frac{\sin(x)}{x} \). The function is increasing where \( \frac{\sin(x)}{x} > 0 \) and decreasing where \( \frac{\sin(x)}{x} < 0 \). Analyze the behavior of \( \sin(x) \) based on the intervals \([-4\pi, 4\pi]\) to find these intervals.

Step 3: Analyzing Concavity Using the Second Derivative

The concavity can be determined using the second derivative of \( \mathrm{Si}(x) \). The second derivative \( \mathrm{Si}''(x) \), is \( \left(\frac{\sin(x)}{x} \right)' \). Using quotient rule, find \( \mathrm{Si}''(x) = \frac{x \cos(x) - \sin(x)}{x^2} \). \( \mathrm{Si}(x) \) is concave up where \( \mathrm{Si}''(x) > 0 \) and concave down where \( \mathrm{Si}''(x) < 0 \). Solve \( \mathrm{Si}''(x) \) for these intervals within \([-4\pi, 4\pi]\).

Step 4: Visualizing Si(x) Using a Computer Algebra System

Use a computer algebra system (such as Desmos, GeoGebra, or a graphing calculator) to graph \( \mathrm{Si}(x) \) over the interval \([-4\pi, 4\pi]\). Observe the intervals of increase, decrease, and concavity directly on this graph for verification.

Key Concepts

Antiderivative

The concept of an antiderivative is an integral component of calculus, which helps us understand the accumulation of quantities. The sine integral, denoted as \( \mathrm{Si}(x) \), is a specific type of antiderivative. An antiderivative of a function \( f(x) \) is any function whose derivative is \( f(x) \). For \( \mathrm{Si}(x) \), the function that we are finding the antiderivative of is \( \frac{\sin(x)}{x} \).
This means that when we differentiate \( \mathrm{Si}(x) \), we get \( \frac{\sin(x)}{x} \). This definition helps define the sine integral's starting point, where \( \mathrm{Si}(0) = 0 \). For other values, the sine integral will build upon this, accumulatively summing the effects of \( \frac{\sin(x)}{x} \).
Understanding the antiderivative provides the foundation for analyzing the function's behavior, such as determining intervals of increasing and decreasing.

Derivative Test

The derivative test is a key tool in calculus used to determine whether a function is increasing or decreasing at different intervals. In the case of the sine integral \( \mathrm{Si}(x) \), its derivative is \( \frac{\sin(x)}{x} \). By analyzing this derivative, one can identify where \( \mathrm{Si}(x) \) increases or decreases.
To check for intervals of increase, look for where \( \frac{\sin(x)}{x} > 0 \). This means the numerator, \( \sin(x) \), must align with the value of \( x \) to retain the positive effect. Conversely, \( \mathrm{Si}(x) \) decreases when \( \frac{\sin(x)}{x} < 0 \).
Frequently, you can find these intervals by observing the behavior of \( \sin(x) \) over one period and scaling it appropriately along \( x \). Consider interval patterns such as \([-4\pi, -3\pi], [-2\pi, -\pi], [\pi, 2\pi], [3\pi, 4\pi]\), where you can check for these sign changes.

Second Derivative Test

The second derivative test extends the understanding of a function's curvature or concavity. After establishing the behavior of \( \mathrm{Si}(x) \) with the first derivative, the second derivative \( \left(\frac{\sin(x)}{x}\right)' \) offers insight into the function's shape. Using the quotient rule, this derivative is \( \frac{x\cos(x) - \sin(x)}{x^2} \).
To determine where the graph of \( \mathrm{Si}(x) \) is concave up or down, check where \( \mathrm{Si}''(x)>0 \) or \( \mathrm{Si}''(x)<0 \). When \( \mathrm{Si}''(x) > 0 \), the graph is concave up, resembling the shape of a bowl. Conversely, \( \mathrm{Si}''(x) < 0 \) indicates a concave down graph, similar to an upside-down bowl.
Exploring this within the interval \([-4\pi, 4\pi]\), calculate points where \( \mathrm{Si}''(x) = 0 \)—these could be potential inflection points where the concavity changes.

Graphing with Technology

Graphing with technology provides a visual confirmation of the analytical results derived from derivative tests. Software tools like Desmos or GeoGebra allow you to plot \( \mathrm{Si}(x) \) over the desired interval \([-4\pi, 4\pi]\). This visual representation highlights the intervals of increase and decrease, as well as changes in concavity, making it easier to understand complex functions.
To use these tools, input the sine integral's equation and define the specific interval limits. You can observe how the graph behaves, identifying visually where it turns upwards or downwards, and where it's rising or falling. Graphing calculators and computer algebra systems provide features like sliders or trace functions to track precise values and their corresponding slopes or curvatures.
This graphical analysis complements your earlier findings and ensures accuracy, providing a simple way to convey complex mathematical behavior through visual aids.