Problem 51 - Precalculus Mathematics for Calculus

Question

Determine an appropriate viewing rectangle for each function, and use it to draw the graph. $$f(x)=\cos 100 x$$

Step-by-Step Solution

Answer

View the function from \(x = 0\) to \(x = \frac{3\pi}{25}\) horizontally and from \(y = -1\) to \(y = 1\) vertically.

Step 1: Identify the Function's Features

The function given is \[ f(x) = \cos 100x \]This is a cosine function with a modified frequency. The standard frequency of \(\cos x\) is 1, but in this case, it's multiplied by 100, indicating that the frequency is 100.

Step 2: Determine the Period

To find the period of the function \( f(x) = \cos 100x \), use the formula for the period of a cosine function, \( T = \frac{2\pi}{b} \), where \( b \) is the frequency. Thus,\[ T = \frac{2\pi}{100} = \frac{\pi}{50} \]This means the graph of the function completes a full cycle every \( \frac{\pi}{50} \) units.

Step 3: Decide on an Appropriate Viewing Interval

Select a range that captures multiple periods to show repeated behavior. A reasonable choice would be an interval that includes several periods, for instance, from \[ x = 0 \text{ to } x = \frac{3\pi}{25} \]This interval includes 5 complete periods and showcases the behavior clearly, as \[ \frac{3\pi}{25} \approx 5 \times \frac{\pi}{50} \]

Step 4: Determine the Appropriate Vertical Range

The amplitude of \( \cos 100x \) is the same as \( \cos x \), which oscillates between -1 and 1. Thus, the vertical viewing range should be \[ y = -1 \text{ to } y = 1 \]to encompass the full amplitude of the cosine function.

Step 5: Setting the Viewing Rectangle

Combine the intervals from the previous steps to determine the viewing rectangle: - Horizontal range: \( [0, \frac{3\pi}{25}] \)- Vertical range: \([-1, 1]\)This rectangle will allow the function's many oscillations to be visualized effectively.

Step 6: Sketch the Graph

Using the determined viewing rectangle, sketch the graph of \( f(x) = \cos 100x \) inside the rectangle.The graph should show high frequency oscillation between -1 and 1, repeating every \( \frac{\pi}{50} \) units on the x-axis.

Key Concepts

Cosine Function

The cosine function is one of the fundamental trigonometric functions, often represented as \( \cos \theta \). It originates from the unit circle, being the x-coordinate of a point as an angle sweeps around the circle. For a regular cosine function, \( y = \cos x \), its graph is a smooth, continuous wave that oscillates between -1 and 1.Key characteristics include:

• Amplitude: The peak height of the wave, which for \( \cos x \) is always 1.
• Frequency: The number of cycles the function completes per unit interval, which for \( \cos x \) is 1.
• Period: The distance it takes for the function to complete one full cycle, typically \( 2\pi \) for \( \cos x \).

These properties help in identifying and graphing the cosine function. Adjustments to these properties, such as changing the frequency, will significantly alter the graph's appearance.

Frequency of a Function

When discussing trigonometric functions like cosine, frequency refers to how often the full wave cycle repeats within a given interval. A higher frequency means more cycles within the same interval. Mathematically, the frequency is affected by the coefficient of \( x \) in the function's argument. For example, in \( f(x) = \cos (bx) \), the frequency is \( b \).Consider this function: \( f(x) = \cos 100x \). Here, the frequency is drastically increased by the factor of 100. Therefore:

• The graph of \( \cos 100x \) will have 100 times more waves than \( \cos x \) within the same horizontal span.
• This results in a graph with much tighter oscillations.

Understanding frequency changes is crucial in predicting how the trigonometric graph will look.

Period of a Function

The period of a trigonometric function is the length of the interval that the function needs to complete one cycle. It is an integral property that helps determine the repetitive nature of the function.For the basic cosine function \( y = \cos x \), the period is \( 2\pi \). However, for functions like \( f(x) = \cos (bx) \), the period changes.Using the formula for the period of a cosine function, \( T = \frac{2\pi}{b} \), we can calculate the period for any modified function:

• For \( f(x) = \cos 100x \), the period becomes \( \frac{2\pi}{100} = \frac{\pi}{50} \).
• This signifies a much shorter distance for a full cycle compared to \( \cos x \).

Having a shorter period translates to more cycles fitting into the same space, creating a denser graph. Recognizing the period helps in setting appropriate viewing windows to capture the essence of the graph.