Problem 87 - 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 \(c>0 ?\) $$f(x)=\cos \frac{1}{x}$$

Step-by-Step Solution

Answer

(a) Largest zero is near x=0.2811. (b) \(f(x)\to 1\) as \(x\to \infty\). (c) Infinite zeros for \([0,c]\) as \(c \to 0\).

Step 1: Understand the Function

The function given is \( f(x) = \cos(\frac{1}{x}) \). This is defined for \( x > 0 \) because the denominator cannot be zero. At \( x = 0 \), the function is undefined. Therefore, we need to consider plotting the graph starting slightly above 0.

Step 2: Setup the Graph Viewing Rectangle

The problem specifies the viewing rectangle to be \([0,3]\) by \([-1.5,1.5]\). You will need graphing software or a graphing calculator capable of plotting trigonometric functions for precise observation. Ensure the x-axis scale covers from just above \( x=0 \) to \( x=3 \), and the y-axis scale covers from \( y=-1.5 \) to \( y=1.5 \).

Step 3: Graph the Function

Using the graphing tool, plot \( f(x) = \cos(\frac{1}{x}) \). Notice how rapidly oscillating behaviors appear as \( x \) gets closer to 0. This behavior smooths out as \( x \) increases towards 3. The graph will have continuous oscillations, but the amplitude remains bounded between \(-1\) and \(1\).

Step 4: Find the Largest Zero in [0,3]

To approximate the largest zero of \( f(x)=0 \) in \([0,3]\), look for points where the graph crosses the x-axis. Use numerical or graphical tools to find this more accurately. It may be easiest by setting a high zoom level on a calculator or software and inspecting it numerically or visually.

Step 5: Investigate Graph Behavior as \(x\) Increases

As \( x \) becomes large, the term \( \frac{1}{x} \) approaches zero, making \( \cos(\frac{1}{x}) \) approach \( \cos(0) = 1 \). Therefore, the graph tends towards 1 as \( x \rightarrow \infty \).

Step 6: Explore the Graph for Small Intervals [0,c]

Investigate the graph on very small intervals \([0, c]\) for \( c = 0.1, 0.01, 0.001 \). You will observe an increase in oscillations. Due to the behavior near zero, the function appears to have infinitely many zeros, since \( \cos(\frac{1}{x}) = 0 \) is achieved whenever \( \frac{1}{x} = \frac{\pi}{2} + n\pi \) for integers \( n \).

Key Concepts

Zeros of a Function

In the world of mathematics, a zero of a function is any point where the function value is zero. These points are often where the graph intersects the x-axis. For the function \( f(x) = \cos(\frac{1}{x}) \), determining zeros involves understanding when the cosine function equals zero.

• A function is zero when \( \cos(\theta) = 0 \), typically where the angle \( \theta \) equals \( \frac{\pi}{2} + n\pi \), where \( n \) is an integer.
• For \( f(x) = \cos(\frac{1}{x}) \), this means \( \frac{1}{x} = \frac{\pi}{2} + n\pi \).
• By rearranging, we find zeros when \( x = \frac{1}{\frac{\pi}{2} + n\pi} \).

Finding these zeros accurately requires a graphical or numerical approach, especially on intervals such as \([0, 3]\). It is important to understand that the oscillations occur increasingly faster as \( x \) approaches zero, giving a large number of zeros near zero. This concept helps to deepen understanding when creating or reading the function's graph.

Graphing Techniques

Graphing is an essential skill for visualizing how functions behave. Effective graphing techniques are particularly important in exploring trigonometric functions such as \( f(x) = \cos(\frac{1}{x}) \).

• Ensure your graphing tool covers the specified viewing rectangle, which is \([0,3]\) by \([-1.5,1.5]\) in this problem.
• Since the function is undefined at \( x=0 \), start your graph slightly above zero to show behavior accurately.
• Notice the oscillatory pattern of \( \cos(\frac{1}{x}) \) becomes extremely frequent as \( x \) approaches zero.
• To capture detailed zeros and behavior, zoom into smaller segments of the graph. Using software that can handle zoom levels is vital for precise plotting.

Graphical representations let you see the dynamic feature of trigonometric functions, clarifying how the oscillations spread out as \( x \) increases and assuring you of the bounded amplitude between \([-1,1]\).

Trigonometric Behavior

Understanding trigonometric behavior is fundamental to graphing and interpreting functions such as \( f(x) = \cos(\frac{1}{x}) \). The characteristics of trigonometric functions contribute to their unique graphical form.

• Oscillation: Trigonometric functions inherently oscillate. With increasing \( x \), \( \cos(\frac{1}{x}) \) moves from rapid oscillations close to zero to less frequent oscillations.
• Approaching Limits: As \( x \rightarrow \infty \), \( \frac{1}{x} \rightarrow 0 \), thus \( \cos(\frac{1}{x}) \rightarrow \cos(0) = 1 \), meaning the graph will level near 1.
• Bounded Amplitude: Whether near-zero or as \( x \) increases, the cosine function maintains an output bounded between -1 and 1.

This predictable behavior is key to comprehending the nature of the function as \( x \) varies. With these insights, observing graphs helps solidify visualization and understanding, making the function's characteristics clear through its plotted form.