Problem 20 - Calculus: An Integrated Approach to Functions and Their Rates of Change

Question

Graph \(f(x)=\frac{1}{\cos x}\) on \([-\pi, 2 \pi]\).

Step-by-Step Solution

Answer

The graph of the given function is undefined at points \(\pm \frac{\pi}{2}, \pm \frac{3\pi}{2}\), etc., where there will be vertical asymptotes. Between the asymptotes, the graph will resemble either \(f(x)=\frac{1}{x}\) for \(x>0\) or \(f(x)=-\frac{1}{x}\) for \(x<0\), depending on the sign of \(\cos x\) in that interval. Because the \(\cos x\) pulls towards 0, the given function tends to \(\pm\infty\).

Step 1 Identify the cosine function

First, remember the typical graph of cosine function, \(\cos x\). It is a wave oscillating between -1 and 1. The key points are \(\cos 0 = 1\), \(\cos \frac{\pi}{2} = 0\), \(\cos \pi = -1\), \(\cos \frac{3\pi}{2} = 0\), and \(\cos 2\pi = 1\).

Step 2 Identify where \(1/ \cos x\) is undefined

This function will be undefined where the denominator, \(\cos x\), is zero. These points are \(\pm \frac{\pi}{2}, \pm \frac{3\pi}{2}\), etc. Hence, there will be vertical asymptotes at these points.

Step 3 Sketch the main graph

Sketch the graph of \(f(x)=\frac{1}{\cos x}\) knowing what a graph of \(\cos x\) looks like. Between the asymptotes, the plot will resemble either \(f(x)=\frac{1}{x}\) for \(x>0\) or \(f(x)=-\frac{1}{x}\) for \(x<0\), depending on the sign of \(\cos x\) in that interval. Pay special attention around the asymptotes, cosine tends to 0, and \(f(x)=\frac{1}{\cos x}\) tends to \(\pm\infty\).

Key Concepts

Cosine Function

The cosine function, denoted as \(\cos x\), is a fundamental part of trigonometry and wave-like behaviors in mathematics. In its graph form, \(\cos x\) exhibits a periodic wave that oscillates between -1 and 1. Key points on this wave include \(\cos 0 = 1\), \(\cos \frac{\pi}{2} = 0\), and \(\cos \pi = -1\). This periodicity repeats every \(2\pi\), meaning the waveform continues its pattern over every interval of \(2\pi\).

• Periodicity: The function completes one full cycle over the interval \(0\) to \(2\pi\).
• Symmetry: \(\cos x\) is an even function, meaning \(\cos(-x) = \cos x\).
• Amplitude: The maximum absolute value the function reaches is 1.

Understanding \(\cos x\) sets the foundation for graphing its reciprocal.

Vertical Asymptotes

Vertical asymptotes occur in the reciprocal graph \(f(x)=\frac{1}{\cos x}\) when the denominator \(\cos x\) is zero. At these points, the function is undefined, causing the graph to shoot upwards or downwards to infinity. For \(\cos x\), this occurs at specific angles:

• Zero Points: \(\cos x = 0\) at \(\pm \frac{\pi}{2}, \pm \frac{3\pi}{2}\), etc.
• Asymptotic Behavior: As the graph approaches these x-values, \(\frac{1}{\cos x}\) trends towards \(+\infty\) or \(-\infty\).

This vertical asymptote behavior is central to interpreting the curve behavior of reciprocal trigonometric functions.

Trigonometric Graphing

Graphing reciprocal trigonometric functions like \(f(x) = \frac{1}{\cos x}\) requires knowledge of both the original function's behavior and how reciprocal operations work. Here are a few steps to help:

• Analyze Key Points: Begin with a sketch of \(\cos x\) to identify where the original function is zero, positive, or negative.
• Locate Asymptotes: Add vertical lines where \(\cos x = 0\), indicating undefined points.
• Sketch Behavior: Between the asymptotes, observe that when \(\cos x > 0\), \(f(x)\) mimics \(\frac{1}{x}\), and when \(\cos x < 0\), \(f(x)\) mirrors \(-\frac{1}{x}\).

By combining understanding from \(\cos x\) and concepts of asymptotic behavior, the graph of \(\frac{1}{\cos x}\) can be drawn accurately over the specified interval.