Problem 61 - Calculus: Early Transcendentals

Question

(a) Graph the function \( f(x) = \sin x - \frac{1}{1000} \sin (1000x) \) in the viewing rectangle \( [-2 \pi, 2 \pi] \) by \( [-4, 4] \). What slope does the graph appear to have at the origin? (b) Zoom in to the viewing window \( [-0.4, 0.4] \) by \( [-0.25, 0.25] \) and estimate the value of \( f'(0) \). Does this agree with your answer from part (a)? (c) Now zoom in to the viewing window \( [-0.008, 0.008] \) by \( [-0.005, 0.005] \). Do you wish to revise your estimate for \( f'(0) \)?

Step-by-Step Solution

Answer

The slope of the graph at the origin is approximately 1; thus, \( f'(0) \approx 1 \). Further zooming confirms this estimate.

Step 1: Graph the Function

Graph the function \( f(x) = \sin x - \frac{1}{1000} \sin (1000x) \) in the viewing rectangle \( [-2\pi, 2\pi] \) by \( [-4, 4] \). Observe the graph to determine the apparent slope at the origin. The main component \( \sin x \) oscillates between \(-1\) and \(1\). The \(-\frac{1}{1000} \sin(1000x)\) term is a high-frequency oscillation with a much smaller amplitude, primarily impacting the graph only slightly on a large scale. Therefore, the graph predominantly resembles \( \sin x \). At the origin, it appears that the slope is approximately \(1\) since \( \sin'(0) = \cos(0) = 1 \).

Step 2: Zoom In to Estimate Derivative

Zoom into the viewing window \( [-0.4, 0.4] \) by \( [-0.25, 0.25] \). Analyzing this more detailed region will help us estimate \( f'(0) \). In this zoomed-in view, \( \sin x \) still dominates as \( \frac{1}{1000} \sin(1000x) \) oscillates rapidly with a small amplitude. Close to \( x = 0 \), the function will closely resemble its tangent line due to the damping effect of the coefficient \( \frac{1}{1000} \), suggesting that \( f'(0) = 1 \).

Step 3: Fine-Tune Zoom for Clarity

Further zoom into the viewing window \( [-0.008, 0.008] \) by \( [-0.005, 0.005] \) for an even closer inspection at the origin. This refined view barely shows the oscillation of the high-frequency term given its tiny amplitude. Hence, the graph again suggests that the derivative at the origin is approximately \(1\), as \( \sin x \) remains the predominant factor governing the slope at \( x = 0 \). Thus, further scrutiny reinforces that \( f'(0) = 1 \).

Key Concepts

Derivative estimation

When dealing with functions like \[ f(x) = \sin x - \frac{1}{1000} \sin(1000x) \]understanding the derivative is important as it tells us the function's rate of change at a given point. Here, the derivative estimation involves analyzing how the function behaves at specific points, most importantly at the origin, point \( x = 0 \).By initially viewing the function over a larger interval \([-2\pi, 2\pi]\), the graph suggests that the derivative at the origin is similar to the derivative of \( \sin x \),which is \( \cos(0) = 1 \).

• Consider the dominant component: In \( f(x) \), \( \sin x \) dominates due to the small coefficient of the second term.
• Simplify to estimate: At \( x=0 \), estimating \( f'(0) \)simplifies to understanding the behavior of \( \sin x \)

Even when zooming in more closely on the graph, this estimation remains consistent.

High-frequency oscillation

High-frequency oscillation refers to the term \(-\frac{1}{1000} \sin(1000x) \)in our function \( f(x) = \sin x - \frac{1}{1000} \sin(1000x) \).

• Rapid Changes: \( \sin(1000x) \) oscillates very rapidly over a small region due to the high frequency \( 1000 \).
• Low Amplitude Impact: Since it's multiplied by \(-\frac{1}{1000} \),the effect on the graph is softened and does not drastically change the shape of the graph.

High-frequency oscillations might seem complex but often have only slight visual impact, as they are buried in the dominant, lower frequency signals when observed over wide ranges. By zooming in, finer details like these oscillations become visible but still consistently show they are negligible in determining the slope at specific points.

Slope at a point

The slope at a point on a graph is the measure of how steep the graph is at that point. Calculating this for our function \( f(x) = \sin x - \frac{1}{1000} \sin(1000x) \)at \( x=0 \) is key for understanding its behavior.

• Understanding at \( x=0 \): The main term \( \sin x \) has a derivative of \(\cos x\), so at \( x=0 \), \( f'(0) = \cos(0) = 1 \).
• Graphical Appearance: Even when the high-frequency term is present, the slope appears as if determined by \( \sin x \)

Despite potential complexities from added terms, focusing on major components like \( \sin x \) can provide clear insights into slope behavior.

Zooming in on graphs

Zooming into specific parts of a graph allows us to see details not visible from the initial, broader view. This is especially useful for functions like \( f(x) = \sin x - \frac{1}{1000} \sin(1000x) \) where multiple components interact.

• Initial View: The graph seems dominated by \( \sin x \), indicating a slope at the origin around 1.
• Narrower Focus: Zooming into a window like \([-0.4, 0.4]\) provides an idea of behavior closer to the origin, filtering out smaller oscillations.
• Fine Detail: With a view of \([-0.008, 0.008]\), it becomes clear through graph inspection that the high-frequency term's oscillations have minimal effect near \( x = 0 \).

Thus, zooming informs the understanding of the function's behavior and reinforces derivative estimations, showing that additional details may not alter initial observations when evaluating slopes at particular points.