Problem 52 - Calculus Early Transcendentals: Pearson New International Edition

Question

Trigonometric functions that have high frequencies pose special problems for graphing. We now explore how to plot such functions. Graph the function \(f(x)=\sin 50 x\) using the window given by a \(y\) range of \(-1.5 \leq y \leq 1.5\) and the \(x\) range given by (a) \([-15,15]\) (b) \([-10,10]\) (c) \([-8,8]\) (d) \([-1,1]\) (e) \([-0.25,0.25]\) Indicate briefly which \(x\)-window shows the true behavior of the function, and discuss reasons why the other \(x\)-windows give results that look different.

Step-by-Step Solution

Answer

Use x-window [-0.25, 0.25] to see true behavior; larger windows appear too dense.

Step 1: Understand the Function

The given function is \(f(x) = \sin(50x)\). This function is a sine wave with a very high frequency because of the multiplier 50 in the argument. High frequency means the wave completes many cycles in a small interval along the x-axis.

Step 2: Identify the Windows

The window refers to the range of x-values used to plot the function. We will be plotting \(f(x)\) for the given x-ranges, checking how the function appears in each case. The x-ranges provided are: (a) \([-15, 15]\)(b) \([-10, 10]\)(c) \([-8, 8]\)(d) \([-1, 1]\)(e) \([-0.25, 0.25]\) with a y-range of \([-1.5, 1.5]\) across all plots.

Step 3: Graph the Function Locally for Each X-window

For each interval, graph \(f(x) = \sin(50x)\). By using a plotting tool, adjust the x-axis to the given intervals while keeping the y-axis from \(-1.5\) to \(1.5\). Observe how the wave appears denser or less dense.

Step 4: Analyze the Results

(a) In \([-15, 15]\), the function appears very dense due to the high frequency completing many cycles. (b) In \([-10, 10]\), the function is also very dense and consistent with (a).(c) In \([-8, 8]\), the function is easier to discern but still somewhat dense.(d) In \([-1, 1]\), the wave appears less dense and begins to show individual cycles more distinctly.(e) In \([-0.25, 0.25]\), distinct cycles of the wave can be seen, showing the true behavior.

Step 5: Determine the True Behavior Window

The window \([-0.25, 0.25]\) shows the true sinusoidal wave behavior of the function most clearly because it captures a small enough interval to make the high-frequency oscillations visible and distinct. Larger intervals make the function's oscillations appear as a blur due to high density.

Key Concepts

High Frequency

High-frequency functions like \(f(x) = \sin(50x)\) are intriguing in how they complete several oscillations over a small span of the x-axis.
These oscillations arise because the sine function cycles between -1 and 1, and the multiplier 50 in this function increases the number of these cycles significantly.
Understanding high frequency is important when graphing because:

• In larger x-windows, the oscillations are so frequent they appear as a blur.
• High frequency helps to capture a detailed snapshot of how wave patterns behave within small intervals.

To fully capture the essence of a high-frequency function, choosing the proper viewing window is crucial.

Sine Wave

A sine wave is characterized by its smooth, periodic oscillations, repeatedly moving above and below the x-axis.
This is the core shape behind functions like \(f(x) = \sin(x)\), and by understanding this function's behavior, we can better grasp its variations with different multipliers.
Sine waves have intrinsic traits:

• The amplitude dictates the wave's height, from peak to valley.
• The frequency indicates how many cycles are completed within a given x-interval.

For \(f(x) = \sin(50x)\), the frequency is heightened, hence the waves are tightly packed, making the cycles less discernable within larger intervals. It's vital to visualize them in a correctly sized x-window to truly appreciate the sine wave.

Plotting Function

Plotting trigonometric functions involves unveiling them within proper visual constraints.
For this, tools can assist in capturing an accurate representation, especially when handling high-frequency functions.
To effectively plot \(f(x) = \sin(50x)\):

• Ensure the x-axis range allows visibility of repeating patterns.
• Choose a stable y-axis to keep the peak and valley visible, like [-1.5, 1.5].

Experimentation with various x-windows shows how repeating cycles become a blur in larger ranges. This highlights the importance of choosing the right plotting parameters to discern the function’s true behavior.

X-Window Range

The x-window range is crucial in function graphing as it defines the span of x-values over which the function is plotted.
For trigonometric functions, selecting the correct x-window range determines how well the behavior of the function is captured and understood.
In the case of \(f(x) = \sin(50x)\), different ranges reveal different aspects:

• Wider x-windows like [-15, 15] condense many cycles, making distinctions difficult.
• Narrower x-windows like [-0.25, 0.25] allow clearer vision of individual oscillations.

The choice of window greatly impacts the clarity and interpretation of the function, making it a fundamental part of analyzing high-frequency trigonometric functions.