Problem 88 - Precalculus: Functions and Graphs

Question

Graph \(f\) in the viewing rectangle \([0,3]\) by \([-1.5,1.5]\). (a) Approximate to within four decimal places the largest solution of \(f(x)=0\) on \([0,3]\) (b) Discuss what happens to the graph of \(f\) as \(x\) becomes large. (c) Examine graphs of the function \(f\) on the interval \([0, c]\) where \(c=0.1,0.01,0.001 .\) How many zeros does \(f\) appear to have on the interval \([0, c],\) where \(c>0 ?\) $$f(x)=\sin \frac{1}{x^{2}}$$

Step-by-Step Solution

Answer

(a) The largest solution is approximately 0.5642. (b) Flatten to zero. (c) Appears to have infinite zeros on \([0, c]\).

Step 1: Understanding the Function

The function given is \( f(x) = \sin \left(\frac{1}{x^2}\right) \). This means that the function is defined except at \(x = 0\) and becomes oscillatory as \(x\) approaches 0 from the right.

Step 2: Graphing the Function on the Provided Interval

Using a graphing utility, plot \( f(x) = \sin \left(\frac{1}{x^2}\right) \) with the viewing rectangle of \([0,3]\) along the x-axis and \([-1.5,1.5]\) along the y-axis. Notice the oscillations increase as \(x\) approaches zero.

Step 3: Finding the Largest Solution on \([0,3]\) for \(f(x) = 0\)

Calculate where \(\sin \left(\frac{1}{x^2}\right) = 0\). This implies \(\frac{1}{x^2} = n\pi\) for integer \(n\). For the largest \(x\) in \([0,3]\), evaluate possible roots using trial or computational methods. Check values around \(x \approx 0.5642\) to identify the zero with desired precision.

Step 4: Discussing the Behavior as \(x\) Becomes Large

As \(x\) increases, \(\frac{1}{x^2}\) approaches zero, making \(\sin \left(\frac{1}{x^2}\right)\) approach \(\sin(0)\), which equals zero. Hence, \(f(x)\) flattens to 0 as \(x\) increases.

Step 5: Examining the Number of Zeros in Small Intervals

For intervals \([0, 0.1]\), \([0, 0.01]\), and \([0, 0.001]\), note that as \(c\) decreases, the number of oscillations, and consequently zeros, increases rapidly. Analyze or compute to realize infinite zeros would trend as \(c\) approaches zero broader but steeper.

Key Concepts

Oscillatory Functions

When we say a function is oscillatory, we mean that it swings back and forth, or up and down, repeatedly within a certain interval. The function under discussion, \( f(x) = \sin \left( \frac{1}{x^2} \right) \), is a classic example of an oscillatory function.

• As \( x \) approaches zero from the right, \( \frac{1}{x^2} \) becomes extremely large.
• This results in the sine function cycling rapidly through its values from \(-1\) to \(1\), thereby showing repeated oscillations.

The closer \( x \) gets to zero, the faster the oscillations occur within a smaller range of \( x \). This implies that the oscillatory nature of \( f(x) \) intensifies as it nears zero.
Visually, on a graph, you would see these oscillations become more frequent and tight as \( x \) becomes smaller near zero. This characteristic makes understanding and graphing oscillatory functions an intriguing challenge, as they exhibit a high degree of dynamic behavior in localized regions of the input space.

Zeros of Functions

Zeros of a function are the values of \( x \) for which the function \( f(x) \) equals zero. For \( f(x) = \sin \left( \frac{1}{x^2} \right) \), zeros occur when \( \sin \left( \frac{1}{x^2} \right) = 0 \).

• This implies \( \frac{1}{x^2} = n\pi \) for integer \( n \).
• Solving for \( x \), we find \( x = \pm \sqrt{1/(n\pi)} \).

Given the constraint of the viewing rectangle \([0,3]\), we focus on finding the largest solution. We approximate this value by examining where the oscillations cross the x-axis at specific, calculable points.
A likely candidate for the largest zero is calculated around \( x \approx 0.5642 \), found by inspecting where the function crosses the axis and refining with computational tools or deeper analysis.
This process allows us to pinpoint zeros with high precision, essential for understanding the detailed behavior of functions that frequently oscillate yet are confined to a restricted domain.

Behavior at Infinity

The behavior of \( f(x) = \sin \left( \frac{1}{x^2} \right) \) as \( x \) becomes large is distinct from its behavior near zero. As \( x \) increases:

• The term \( \frac{1}{x^2} \) approaches zero.
• Consequently, \( \sin \left( \frac{1}{x^2} \right) \) approaches \( \sin(0) = 0 \).

Thus, the graph of \( f(x) \) tends to flatten out towards zero simplifying the oscillations seen around the origin. This convergent behavior suggests that for very large \( x \), the function produces values closer and closer to zero, eventually appearing almost flat.
This characteristic is important for graphing and real-world interpretations where understanding a function's end behavior can aid in predicting long-term trends or behaviors.
So, although \( f(x) \) exhibits dramatic oscillations for small \( x \), it behavies more predictably and stably as \( x \) reaches higher values, moving from the turbulence of its oscillatory phase outside the zero vicinity towards a calm horizon.