Problem 24 - A Graphical Approach to Precalculus with Limits

Question

Use a graphing calculator to graph $$\begin{aligned}&y_{1}=e^{-x} \sin x\\\&y_{2}=e^{-x}\\\&\text { and } \quad y_{3}=-e^{-x}\end{aligned}$$ in the viewing window \([0, \pi]\) by \([-0.5,0.5]\) (a) Find the \(x\) -intercepts of the graph of \(y_{1}\). Explain the relationship of these \(x\) -intercepts to those of the graph of \(y=\sin x\) (b) Find the \(x\) -coordinate of any points of intersection of \(y_{1}\) and \(y_{2}\) or \(y_{1}\) and \(y_{3}\)

Step-by-Step Solution

Answer

(a) x-intercepts of y₁ are 0 and π, relating to sin x's zeros. (b) Intersection of y₁ and y₂ occurs at x = π/2; no intersection for y₁ and y₃ within [0, π].

Step 1: Graph Setup

Set up your graphing calculator by entering the functions. Use the following inputs: \( y_1 = e^{-x} \sin x \), \( y_2 = e^{-x} \), and \( y_3 = -e^{-x} \). Ensure your calculator is set to the viewing window \([0, \pi]\) for the x-axis and \([-0.5, 0.5]\) for the y-axis.

Step 2: Observe x-intercepts of y1

Graph \( y_1 = e^{-x} \sin x \). The x-intercepts are the points where the graph crosses the x-axis. These happen when \( \sin x = 0 \), which occurs at \( x = 0, \pi \) within the given range.

Step 3: Verify Relationship to y = sin x

The x-intercepts of \( y_1 \) are related to the zeros of \( y = \sin x \), since \( y_1 = e^{-x} \sin x \) is zero when \( \sin x = 0 \). Hence, the x-intercepts match the x-intercepts of the graph of \( y = \sin x \) in the interval \([0, \pi]\).

Step 4: Identify Intersections of y1 and y2

On the graph, look for where \( y_1 = e^{-x} \sin x \) intersects \( y_2 = e^{-x} \). Set the equations equal: \( e^{-x} \sin x = e^{-x} \). Simplifying gives \( \sin x = 1 \), occurring at \( x = \frac{\pi}{2} \) within the interval \([0, \pi]\).

Step 5: Identify Intersections of y1 and y3

Look for intersections between \( y_1 \) and \( y_3 \). Set \( e^{-x} \sin x = -e^{-x} \), which simplifies to \( \sin x = -1 \). This does not occur within \([0, \pi]\) since the sine function does not reach \(-1\) in this interval.

Key Concepts

Exponential Functions

Exponential functions are mathematical expressions where a constant base is raised to a variable power. In the expression \(e^{-x}\), \(e\) represents Euler's number, approximately 2.718, and \(-x\) is the exponent. This expression defines a decaying exponential function, where the value decreases as \(x\) increases.
Exponential functions have distinctive characteristics:

• They always produce positive outputs because they involve raising a positive constant to any real power.
• As the exponent becomes more negative, the function value approaches zero. This is apparent in functions like \(e^{-x}\), where larger \(x\) values result in smaller outputs.
• They have a horizontal asymptote, often the x-axis, indicating the line the function approaches but never reaches as \(x\) increases or decreases.

When combined with other functions, such as in \(e^{-x} \sin x\), the behavior of the exponential function influences how the product behaves, contributing to the function’s overall shape, ranges, and intercepts.

Trigonometric Functions

Trigonometric functions are foundational in mathematics, especially in the circle and wave-related problems. The sine function, denoted \(\sin x\), is defined based on the coordinates of a point on a unit circle. It oscillates between -1 and 1 as the angle \(x\) varies.
Key properties of the \(\sin x\) function include:

• Periodic nature: The function repeats its values in regular intervals. The period of \(\sin x\) is \(2\pi\), meaning it completes one cycle every \(2\pi\) radians, or approximately 6.28 units of \(x\).
• Zero crossings: \(\sin x\) is zero at integer multiples of \(\pi\), such as \(x = 0, \pi, 2\pi, \ldots\)
• Amplitudes: The maximum and minimum values are 1 and -1, respectively.

When \(\sin x\) is multiplied by \(e^{-x}\), resulting in \(e^{-x} \sin x\), the oscillations of the sine are modulated by the exponential decay, leading to diminishing oscillations as \(x\) increases.

Function Intercepts

Function intercepts are the points where a graph crosses an axis, and they provide crucial information about the behavior of functions.
For an \(x\)-intercept, we typically solve for \(x\) when \(f(x) = 0\). This is where the function's graph intersects the x-axis.

• In the function \(y_1 = e^{-x} \sin x\), the \(x\)-intercepts occur when \(\sin x = 0\). Since \(e^{-x}\) is always positive, the zeros are determined by the sine factor, which is zero at \(x = 0, \pi\).
• These \(x\)-intercepts match the zeros of \(y = \sin x\), preserving the property that multiplication by an ever-positive exponential function doesn't introduce new intercepts.

Understanding intersections of functions can also be seen with their points of intersection:

• For example, \(y_1 = e^{-x} \sin x\) intersects with \(y_2 = e^{-x}\) when \(\sin x = 1\), which occurs at \(x = \frac{\pi}{2}\).
• No intersection takes place between \(y_1\) and \(y_3\) within \([0, \pi]\) since \(\sin x = -1\) does not occur in this range.